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0000000000000000000000000000000000000000..cbfff87bc31dccb4829789072c8804c1c2858af4 --- /dev/null +++ b/0NE0T4oBgHgl3EQfdADd/content/tmp_files/2301.02372v1.pdf.txt @@ -0,0 +1,1052 @@ +A Multi-Objective Planning and Scheduling +Framework for Community Energy Storage Systems +in Low Voltage Distribution Networks +K.B.J. Anuradha +The Australian National University +Canberra, Australia +Jayaminda.KariyawasamBovithanthri@anu.edu.au +Chathurika P. Mediwaththe +The Australian National University & CSIRO +Canberra, Australia +chathurika.mediwaththe@csiro.au +Masoume Mahmoodi +The Australian National University +Canberra, Australia +masoume.mahmoodi@anu.edu.au +Abstract—This paper presents a methodology for optimizing +the planning and scheduling aspects of a community energy +storage (CES) system in the presence of solar photovoltaic (SPV) +power in low voltage (LV) distribution networks. To this end, we +develop a multi-objective optimization framework that minimizes +the real power loss, the energy trading cost of LV customers and +the CES provider with the grid, and the investment cost for the +CES. Distribution network limits including the voltage constraint +are also taken into account by combining the optimization +problem with a linearized power flow model. Simulations for the +proposed optimization framework with real power consumption +and SPV generation data of the customers, highlight both real +power loss and energy trading cost with the grid are reduced +compared with the case without a CES by nearly 29% and 16%, +respectively. Moreover, a case study justifies our methodology +is competent in attaining the three objectives better than the +optimization models which optimize only the CES scheduling. +Keywords—Community energy storage, distribution networks, +multi-objective optimization, planning and scheduling, power +flow +NOMENCLATURE +Sets and Indices +V, i, j +Set of nodes, node indices +E +Set of lines in the network +Wj +Set of downstream nodes of node j includ- +ing itself +Cj, c +Set of customers at node j, customer index +T , t +Set of time intervals, time index +X, x +Feasible set, decision variable vector +Model Parameters +rij, xij +Resistance and reactance of line (i, j) - (Ω) +Umin, Umax +Minimum and maximum squared voltage +magnitude limits - (V 2) +ηch, ηdis +Charging and discharging efficiencies of the +CES +λmin, λmax +Percentage coefficients of the CES capacity +aj +Binary variable to find the optimal CES +location +pRate +j +Optimal CES rated power- (kW) +Ecap +j +Optimal CES capacity - (kWh) +pRate +min , pRate +max +Minimum and maximum allowable CES +rated power- (kW) +ECES +min , ECES +max +Minimum and maximum allowable CES +capacity - (kWh) +λp(t) +Grid energy price at time t - (AUD/kWh) +γCES +Fixed part of the CES investment cost - +(AUD) +δCES +Cost of the CES for a unit capacity - +(AUD/kWh) +∆t +Time difference between two adjacent time +instances - (h) +wi +Weight coefficient of the ith objective func- +tion +Power Flows and Injections +pL +cj(t), qL +cj(t) +Real and reactive power consumption of +the customer c at node j at time t - +(kW, kV AR) +pP V +cj (t) +SPV generation of the customer c at node +j at time t - (kW) +Pij(t), Qij(t) +Real and reactive power flow from i to j +node at time t - (kW, kV AR) +pj(t), qj(t) +Real and reactive power absorption at node +j at time t - (kW, kV AR) +pG +cj(t)) +Real power exchange with the grid by the +customer c at node j at time t - (kW) +pCES +cj +(t)) +Real power exchange with the CES by the +arXiv:2301.02372v1 [eess.SY] 6 Jan 2023 + +customer c at node j at time t - (kW) +pG +CES(t)) +Real power exchange with the grid by the +CES at time t- (kW) +pCES,ch +j +(t) +Charging power of the CES at node j at +time t - (kW) +pCES,dis +j +(t) +Discharging power of the CES at node j at +time t - (kW) +Other Notations +ECES +j +(t) +Energy level of the CES at node j at time +t - (kWh) +Vj(t) +Voltage magnitude of node j at time t - (V ) +Uj(t) +Squared voltage magnitude of node j at +time t - (V 2) +Iij(t) +Current flow from node i to j at time t - +(A) +I. INTRODUCTION +In the recent past, there has been a notable interest among +the power systems research community and the industry for +the uptake of community energy storage (CES) in low voltage +(LV) power systems. This trend is driven by the benefits +gained from a CES such as providing the opportunity to +increase the hosting capacity of the network, enhancing the +solar energy self consumption of the customers, and increasing +the community access to renewable energy [1]. Additionally, +CES devices can be deployed to gain technical merits such as +real power loss minimization and economic benefits including +the curtailment of energy purchase cost of the customers [2]. +As discussed in literature, a CES may be used in energy +management problems to earn technical and monetary benefits +together [3], [4]. Those merits can be fully exploited if the +CES planning aspects including its location, the rated power +and the capacity are optimized simultaneously with the CES +scheduling aspects namely, its charging and discharging. +The existing literature on CES utilization in LV distribution +networks can be divided into two categories as; (i) optimiza- +tion of CES scheduling only, (ii) optimization of both CES +planning and scheduling. In the first category, the authors +have presented optimization frameworks for CES scheduling +without accounting for its planning aspects. For instance, a +method built up on game theory concepts to maximize the +revenue for the CES provider and minimize energy costs for +the customers is discussed in [3]. A multi-objective framework +to minimize the real energy loss and energy costs of the +customers and the CES provider for trading energy with the +grid is discussed in [4]. A method based on model predictive +control to optimize the CES scheduling is presented in [5]. +In addition to the papers which have presented methods for +optimizing only the CES scheduling, there are research work +which have proposed methods for optimizing both planning +and scheduling of CES simultaneously. For instance, a method +for maximizing the hosting capacity in a distribution network +in the presence of a CES is proposed in [6]. Analytical meth- +ods to minimize the real energy loss of a network by finding +the optimal CES location and its capacity are discussed in [7], +[8]. A common feature of these methods is that the optimal +CES planning aspects are determined based on analytical +(such as graphical or numerical methods) and sensitivity based +approaches (methods which decide the optimal values based +on a calculated sensitivity parameter). These approaches can +be computationally exhaustive as the optimal CES location and +the capacity are found upon computing a sensitivity parameter +for a large number of location-capacity combinations. Also, +even after an exhaustive search, it is not always guaranteed to +reach an optimal solution [7]. Thus, a robust formulation to +optimize the capacity, the rated power and the location of a +CES while generating the techno-economic benefits associated +with such storage devices would be an effective alternative to +overcome the challenges in the literature. +In this paper, we study the extent to which the location, +the capacity and the power rating of a CES in addition to its +scheduling, affect network and economic benefits achievable +from it. For this, we develop an optimization framework +that optimizes both planning and scheduling of a CES. The +optimized planning and scheduling aspects are then leveraged +to minimize the network power loss, cost incurred by the +customers and the CES provider for trading energy with the +grid and the investment cost of the CES simultaneously. To the +best of our knowledge, this problem has not been addressed in +the literature. The contributions of this paper are as follows. +• A linearized power flow model is exploited with the +CES operational constraints to develop a multi-objective +optimization framework. It is then solved as a mixed +integer quadratic program according to the optimization +algorithms in [9]. The analytic hierarchy process (AHP) +is used for fairly weighting the objective functions [10]. +• The performance of the proposed optimization framework +is evaluated on a real LV distribution network. Here, +we do a comparison between our proposed optimization +framework, and the models that arbitrarily choose the +CES planning aspects such as its location, to assess the +impact of it on the objectives. Finally, a comprehensive +analysis of the results is also presented. +The rest of the paper is structured as follows. Section II +presents the mathematical models used in our problem. The +proposed CES planning and scheduling optimization frame- +work is illustrated in Section III. Section IV is about the +numerical and graphical results along with their discussion. +Eventually, the conclusion of the work and possible future +developments are given in Section V. +II. SYSTEM MATHEMATICAL MODELLING +In this paper, the positive power absorption convention is +considered for all nodes. Also, it is considered that there are +multiple customers at each node. All the real and reactive +power quantities are measured in kW and kVAR, respectively. +It is assumed that power consumption (both real and reactive) +and SPV generation of each customer are known ahead from +their forecasts. A summary of the notations used in this paper, +together with their definitions are given in the Nomenclature. + +Fig. 1: Possible power exchanges between a customer, the CES +and the grid +A. Power Flow Model +The typical mutual power exchanges that can ensue between +different entities (i.e. customers, CES and grid) in the presence +of a CES is shown in Fig. 1. pCES +cj +(t) > 0 suggests a power +import by the customer c at node j at time t from the CES, +and pCES +cj +(t) < 0 occurs when that customer exports power +to the CES. The same sign convention used for pCES +cj +(t) is +valid for pG +cj(t) and pG +CES(t). The mathematical relationships +between the power flows shown in Fig. 1 are described later. +The relationship between the line power flows and node +absorptions which follows the LinDistflow model are given +by (1), (2) [11]. +Pij(t) = pj(t) + +� +k:j→k +Pjk(t) +(1) +Qij(t) = qj(t) + +� +k:j→k +Qjk(t) +(2) +The nodal real and reactive power absorptions are illustrated +by the equations (3) and (4). The equation (3a) governs the +real power absorption for the CES connected node and for the +rest of the nodes (except the slack node), it is the equation +(3b). Additionally, we assume all the SPV units and the CES +operate at unity power factor. +pj(t) = +� +c∈Cj +pL +cj(t) − +� +c∈Cj +pP V +cj (t) + pCES,ch +j +(t) +−pCES,dis +j +(t) +∀j ∈ V\ {0} , t∈ T +(3a) +pj(t) = +� +c∈Cj +pL +cj(t) − +� +c∈Cj +pP V +cj (t) +∀j ∈ V\ {0} , t∈ T +(3b) +qj(t) = +� +c∈Cj +qL +cj(t) +∀j ∈ V\ {0} , t∈ T +(4) +The equations (5) and (6) demonstrate how a customer +exchanges power with the CES and the grid, when that +customer encounters a mismatch of its real power consumption +and SPV generation. When a customer experiences a deficit of +its SPV generation to supply its real power consumption, that +deficit can be fulfilled in share by the CES and the grid. On +the other hand, if a customer has a surplus SPV generation, +that customer exports the excess to both CES and the grid. +The mathematical relationship between pG +CES(t), pCES +cj +(t), +pCES,ch +j +(t) and pCES,dis +j +(t) can be written as (7). +If pL +cj(t) ≥ pP V +cj (t): +0 ≤ pG +cj(t) + pCES +cj +(t) = pL +cj(t) − pP V +cj (t) +(5a) +0 ≤ pG +cj(t) ≤ pL +cj(t) − pP V +cj (t) +∀j ∈ V\ {0} , c ∈ Cj, t∈ T +(5b) +Otherwise: +pG +cj(t) + pCES +cj +(t) = pL +cj(t) − pP V +cj (t) ≤ 0 +(6a) +pL +cj(t) − pP V +cj (t) ≤ pG +cj(t) ≤ 0 +∀j ∈ V\ {0} , c ∈ Cj, t∈ T +(6b) +pG +CES(t) = +N +� +j=1 +� +� +� +� +c∈Cj +pCES +cj +(t) + pCES,ch +j +(t) − pCES,dis +j +(t) +� +� +� +(7) +The Lindistflow equations given in (1)-(4) can be written in +matrix format as ((8) [11] +U = U01 − 2˜Rp − 2˜Xq +∀t∈ T +(8) +where U = |V(t)|2 is the vector of squared voltage magni- +tudes of nodes, 1 is a vector of all ones and U0 = |V0|2 is +the squared voltage magnitude of the slack node. Also, p and +q are the vectors of nodal real and reactive power absorption. +The matrices ˜R and ˜X ∈ RN×N have the elements Rij = +� (a.b) ∈ Li ∩ Ljrab and Xij = � (a.b) ∈ Li ∩ Ljxab, re- +spectively where Li is the set of lines on the path connecting +node 0 and “i” [3], [11]. +The squared voltage magnitudes at each node needs to be +maintained within its allowable voltage magnitude limits. This +is guaranteed by the inequality given in (9). Here, Umin = +Umin1 and Umax = Umax1. +Umin ≤ U ≤ Umax +∀t∈ T +(9) +B. Community Energy Storage Model +In this section we present the mathematical modelling of +the CES. We consider the CES is owned by a third party, and +the owner is designated as the CES provider. +The set of constraints listed from (10) to (17) model the +CES. The equations (10) and (11) imply that the CES charging +and discharging power should not exceed the rated power +pRate +j +of the CES. The temporal variation of the energy level +of the CES is expressed by (12). Also, the CES energy level + +External Grid +Grid +pCEs(t) < 0 +pCes(t) > 0 +0 > ()号d +pgj(t) > 0 +田田 +pCES(t) < 0 +cth Customer +CES Device +at node jat any time should exist within its upper and lower state of +charge (SoC) limits. This is handled by (13). The continuity +of the CES operation over the next day is guaranteed by the +inequality given in (14) which is bounded by a small positive +number ε [3], [4]. Note that td in (14) represents the day num- +ber of the year. Here td ∈ TD, where TD = {1, 2, ...., NT /24} +and NT is the cardinality of set T . +0 ≤ pCES,ch +j +(t) ≤ pRate +j +∀j ∈ V\ {0} , t∈ T +(10) +0 ≤ pCES,dis +j +(t) ≤ pRate +j +∀j ∈ V\ {0} , t∈ T +(11) +ECES +j +(t) = ECES +j +(t − 1) + (ηchpCES,ch +j +(t) +− +1 +ηdis pCES,dis +j +(t))∆t +∀j ∈ V\ {0} , t∈ T +(12) +λminEcap +j +≤ ECES +j +(t) ≤ λmaxEcap +j +∀j ∈ V\ {0} , t∈ T +(13) +��ECES +j +(24td) − ECES +j +(0) +�� ≤ ε +∀j ∈ V\ {0} , td∈ T D +(14) +The equation (15) is used to find the optimal CES location. +Also, (15) ensures that only one CES is installed in the +network. If aj = 0, it implies that there is no CES at node j. +If aj = 1, then the CES is connected to node j. To determine +the optimal CES capacity Ecap +j +, the inequality given in (16) +is utilized. For a case aj = 0, (16) makes Ecap +j +also to be +zero. When Ecap +j += 0, the values ECES +j +(t), pCES,ch +j +(t) and +pCES,dis +j +(t) in (12) and (13) also turn out to be zero. The +inequality in (17) guarantees the rated power of the CES is +bounded by its minimum and maximum allowable values. +N +� +j=1 +aj = 1 +∀j ∈ V\ {0} , aj ∈ {0, 1} +(15) +ajEcap +min ≤ Ecap +j +≤ ajEcap +max +∀j ∈ V\ {0} , aj ∈ {0, 1} +(16) +ajpRate +min ≤ pRate +j +≤ ajpRate +max +∀j ∈ V\ {0} , aj ∈ {0, 1} +(17) +III. OPTIMIZATION FRAMEWORK & PROBLEM +FORMULATION +In our paper, it is expected to minimize the real power +loss of the network, energy trading costs of the customers +and the CES provider with the grid, and to minimize the +CES investment cost. Therefore, a multi-objective function +is obtained by combining those objectives functions, and its +formulation is given as follows. +A. Objective Functions +1) Minimizing the Real Power Loss of the Network: The +real power loss in a network can be written as (19), in terms +of (18), and by taking Ui(t) ≈ U0(t) ∀i ∈ V\ {0} [4], [11]. +|Iij(t)|2 = Pij(t)2 + Q2 +ij(t) +Ui(t) +∀(i, j)∈ E, t∈ T +(18) +fP loss = +� +t∈T +� +(i,j)∈E +rij |Iij(t)|2 +(19) +2) Minimizing the Energy Trading Cost of the Customers +and the CES Provider with the Grid: The first term of the +objective function given in (20) relates to the energy trading +cost with the grid by customers, and latter for the CES +provider. +fEn,cost = +� +t∈T +λp(t) +� +� +� +N +� +j=1 +� +c∈Cj +pG +cj(t) + pG +CES(t) +� +� +� ∆t (20) +Here, it is considered a one-for-one non-dispatchable energy +buyback scheme such that the same energy price for both +imports and exports of energy from the grid by the customers +and the CES is used [12]. This kind of an energy pricing +scheme can effectively value the SPV power as being same as +the power imported from the grid, which is usually generated +by a conventional generation method. +3) Minimizing the Investment Cost of the CES: The third +objective is to minimize the investment cost of the CES device +which is given by (21) [2]. +fInv,cost = γCES + δCESEcap +j +(21) +B. Problem Formulation +The three objective functions are normalized and weighted +to form the multi-objective function in (22), according to the +techniques described in [13]. The normalization guarantees +the objective functions are converted into a form which can +be added together (since fP loss is measured in kW, and +fEn,cost, fInv,cost are measured in AUD ). +x∗ = argmin +x∈X +w1 +� fP loss−f utopia +P loss +f Nadir +P loss −f utopia +P loss +� ++ w2 +� +fEn,cost−f utopia +En,cost +f Nadir +En,cost−f utopia +En,cost +� ++w3 +� +fInv,cost−f utopia +Inv,cost +f Nadir +Inv,cost−f utopia +Inv,cost +� +(22) +where X is the feasible set which is constrained by (1)-(17). +The utopia values, individual minimum point values and nadir +values of the multi-objective function are found by (23), (24) +and (25), respectively. Besides, the decision variable vector +can be explicitly expressed as (26). +f utopia +P loss = fP loss(x∗ +Ploss) +(23a) + +Fig. 2: 7-Node LV radial distribution network +f utopia +En,cost = fEn,cost(x∗ +En,cost) +(23b) +f utopia +Inv,cost = fInv,cost(x∗ +Inv,cost) +(23c) +x∗ +Ploss = argmin +x∈X +fP loss +(24a) +x∗ +En,cost = argmin +x∈X +fEn,cost +(24b) +x∗ +Inv,cost = argmin +x∈X +fInv,cost +(24c) +f Nadir +P loss = Max +� +fP loss(x∗ +Ploss), fP loss(x∗ +En,cost), +fP loss(x∗ +Inv,cost) +� +(25a) +f Nadir +En,cost = Max +� +fEn,cost(x∗ +Ploss), fEn,cost(x∗ +En,cost), +fEn,cost(x∗ +Inv,cost) +� +(25b) +f Nadir +Inv,cost = Max +� +fInv,cost(x∗ +Ploss), fInv,cost(x∗ +En,cost), +fInv,cost(x∗ +Inv,cost) +� +(25c) +x = (aj, pRate +j +, Ecap +j +, pCES,ch +j +, pCES,dis +j +, pG +CES, pG +cj) (26) +In summary, the optimization framework can be written as +(22), subject to a set of constraints (1)-(17). Also, as (22) being +a quadratically-constrained convex multi-objective function, it +is solved as a mixed-integer quadratic program. +IV. NUMERICAL AND SIMULATION RESULTS +In the simulations, a 7-node LV radial distribution network +given in Fig. 2 is used and its line data can be found +in [14]. Also, real power consumption and SPV generation +data of 30 customers in an Australian residential community +were used for simulations [15]. To be more practical, we +randomly allocated multiple customers for each node. Hence, +�N +j=1 |Cj| = 30, and the number of customers at each node +are marked in Fig. 2. Here, all the customers generate SPV +power in addition to their real power consumption. However, +reactive power consumption of the customers is not considered +due to the lack of sufficient real data. As the optimization +Fig. 3: Variation of grid energy price for 24 hours +involves not only a scheduling problem but also a planning +problem, the optimization is performed over a long time +period. Thus, we consider one year time period split in one +hour time intervals (i.e.|T | = 8760 ) for the simulations. +The voltage and power base are taken as 400V and +100 kVA, respectively. In addition to that, V0 += 1p.u., +Umin = 0.9025p.u., Umax = 1.1025p.u., λmin = 0.05, +λmax = 1, ηch = 0.98, ηdis = 1.02, Ecap +min = 200kWh, +Ecap +max = 2000kWh, pRate +min += 20kW, pRate +max += 200kW, +ε = 0.0001kWh and ∆t = 1h are used as the model +parameters. The values of γCES and δCES are taken as 24000 +AUD and 300 AUD/kWh as specified in [2]. Additionally, the +weighting factors w1, w2 and w3 were calculated according +to the principles of AHP specified in [10]. We considered +a moderate plus importance for both fEn,cost and fInv,cost +compared to fP loss, and an equal importance for fEn,cost and +fInv,cost. Hence, based on the AHP method, the values of +w1, w2 and w3 were calculated as 1/9, 4/9 and 4/9, respec- +tively. Fig. 3 depicts how the grid energy price varies with the +time of the day following a time of use (ToU) tariff scheme. +As seen in Fig. 3, the grid energy price is 0.24871 AUD/kWh +during T1(from 12am-7am) & T5(from 10pm-12am), 0.31207 +AUD/kWh during T2(from 7am-3pm) & T4 (from 9pm-10pm) +and 0.52602 AUD/kWh during T3(from 3pm-9pm) [16]. +A. Case Study - Proposed Optimization Framework Vs Opti- +mization Models With Arbitrary CES Locations +We did a case study to compare the results of our model with +four different cases by arbitrarily changing the CES location. +For this, we considered our optimization framework as Case I, +while the rest as Case II-V. The same optimization framework +(except the constraint that finds the optimal CES location), +and the model parameters as for Case I were used for Case II- +V. A synopsis of the results for the five cases are tabulated in +Table I. The Case I lists the planning results and the minimized +objective function values for our proposed model. The Cases +II and III suggest the same optimal CES capacity and the rated +power. Nevertheless, due to their difference in CES location, +Case II provides less real energy loss and energy trading cost +compared with the Case III. When the CES is at node 6, the +optimization suggests the same optimal capacity as in Case I. +However, as node 6 is not the optimal location for CES, the +real energy loss and energy trading cost for Case IV are higher + +2 +6 +External +Transformer +IC2l = 4 +IC6l = 4 +Grid +22/0.4 kV +0 +1 +3 +4 +IC1l = 3 +IC3l = 5 +C4 = 6 +ICzl = 5 +5 +ICsl = 30.55 +I (AUD/kWh) +0.50 +0.45 + Signal ( +0.40 +0.35 +Price +Energy I +0.30 +0.25 +T4 T5 +0.20 +0 +5 +10 +15 +20 +25 +Time Duration (24 HoursTABLE I: SUMMARY OF THE RESULTS FOR CASE STUDIES +CES +Location +(Node) +Optimal CES +Capacity (kWh) +Optimal CES +Power Rating (kW) +Real Energy +Loss1 (kWh) +Energy Trading +Cost With +Grid1 (AUD) +CES Investment +Cost (AUD) +Base Case (Without CES) +Not applicable +Not applicable +Not applicable +110116.68 +45585 +Not applicable +Case I (Proposed Model) +4 (optimal) +482.15 +200 +78200.88 (71.02%) +38520 (84.50%) +168645 +Case II +3 (chosen) +601.32 +200 +80250.48 (72.88%) +43362 (95.12%) +204396 +Case III +5 (chosen) +601.32 +200 +81961.28 (74.43%) +43840 (96.17%) +204396 +Case IV +6 (chosen) +482.15 +200 +80761.16 (73.34%) +44154 (96.86%) +168645 +Case V +7 (chosen) +547.69 +200 +86082.52 (78.17%) +43625 (95.70%) +188307 +1 Percentage values are calculated with respect to their corresponding values without a CES +Fig. 4: Total power exchange with the grid by the customers +Fig. 5: Total power exchange with the CES by the customers +than in Case I. Also, our model has produced the highest cost +reduction percentages for real energy loss (28.98%) and the +energy trading cost with the grid (15.5%), compared to all the +other cases. Hence, it is clear that Case I yields the minimum +values for all the three objective functions, and this justifies +the effectiveness of our optimization framework compared to +the models that optimize only the CES scheduling. +B. Analysis of the Results-Mutual Power Exchanges Between +the customers, the CES and the grid +In order to understand the CES scheduling and power +exchanges between different entities, we select a single day (24 +hours) for our discussion. Fig. 4 shows the variation of total +Fig. 6: Power exchange with the grid by the CES +Fig. 7: CES charging and discharging power pattern +power exchange that occurs with the grid by the customers. +Since �N +j=1 +� +c∈Cj pG +cj(t) being a positive value approxi- +mately during T1, T3, T4, T5 time intervals, it implies that the +customers tend to import certain amount of power from the +grid for satisfying their real power consumption during those +time periods. On the other hand, during T2 (time period of the +day usually the SPV generation is high), the customers have a +tendency to export a portion of their surplus SPV generation +to the grid. This is evident as �N +j=1 +� +c∈Cj pG +cj(t) < 0 during +T2. This behavior guarantees a cost benefit for the customers +for their exported power according to equation (20). +The Fig. 5 depicts how the customers exchange power + +) (kw) +50 +()53 +0 +-50 +-100 +0 +5 +10 +15 +20 +2550 +(kw) +0 +-50 +-100 +-150 +-200 +0 +5 +10 +15 +20 +25100 +(kW) +50 +0 +CG +-50 +-100 +0 +5 +10 +15 +20 +25 +Time +e Duration(24 Hours100 +(kW) +50 +0 +and +-50 +-100 +p +0 +5 +10 +15 +20 +25Fig. 8: Temporal variation of the CES energy level +with the CES. During T2, the customers export a part of +their surplus SPV generation to the CES. On the contrary, +during rest of the time periods, the customers import a certain +amount of power from the CES for satisfying their real power +consumption. This action results in reducing the cost for the +customers as the amount of power imported from the grid is +minimized. +The Fig. 6 illustrates how the CES exchanges power with +the grid. As the grid energy price during T1 being the lowest, +the CES tends to import power from the grid (i.e. pCES +G +(t) > +0) during T1. This guarantees that the CES is charged with +low priced energy from the grid. However, during T2, T3 and +T4, it is seen that the CES exports its power back to the grid +(i.e. pCES +G +(t) < 0). This happens as the CES provider can +maximize its revenue by exporting power back to grid. +In Fig. 7 and 8, it is observed that during T1, the CES +charges (from the low priced grid energy) and partially dis- +charges by the end of T1. During T2, the CES continues +to charge and by the end of this time period, it reaches its +maximum energy level. The stored energy in the CES is fully +utilized during T3 and T4 for partially supplying the real +power consumption of the customers. This facilitates monetary +benefits for both the customers as the amount of expensive +power imported from the grid is lowered. Additionally, when +observing the temporal variation of the CES energy level, it +is visualized that it is the peak value of the CES energy level +which was obtained as the optimal CES capacity (i.e. 482.15 +kWh). +V. CONCLUSION & FUTURE WORK +In this work, we have explored how the optimization of +the planning and scheduling aspects of a community energy +storage (CES) can benefit both the network and the customers. +To this end, we developed a multi-objective mixed-integer +quadratic optimization framework to minimize three objec- +tives: (i) network real power loss, (ii) energy trading cost of +the customers and the CES provider with the grid, and (iii) +the CES investment cost. The simulation results highlighted +our optimization framework is competent in acquiring the +expected merits compared with the case without a CES, and +optimization models that optimize only the scheduling of CES. +As future work, we expect to develop the work considering +a stochastic model taking into account the uncertainties of +real power consumption and SPV generation of the customers. +Moreover, we look forward to extend the work by considering +the unbalanced nature of LV distribution networks, and reac- +tive power control capabilities of solar photovoltaic (SPV) and +CES inverters. +REFERENCES +[1] M. Shaw, B. Sturmberg, C.P. Mediwaththe, H. Ransan-Cooper, D. Taylor +and L. Blackhall “Community batteries: a cost/benefit analysis,” Tech- +nical Report, Australian National University, 2020. +[2] Y. +Zheng, Y. Song, A. Huang, and D.J. Hill, “Hierarchical Optimal +Allocation of Battery Energy Storage Systems for Multiple Services in +Distribution Systems,” IEEE Trans. Sust. Energy, vol. 11, no. 3, pp. +1911–1921, 2020. +[3] C.P. Mediwaththe, and L. Blackhall, “Network-Aware Demand-Side +Management Framework With A Community Energy Storage System +Considering Voltage Constraints,” IEEE Trans.Power Syst., vol. 36, +no. 2, pp. 1229–1238, 2021. +[4] C.P. Mediwaththe, and L. Blackhall, “Community Energy Storage-based +Energy Trading Management for Cost Benefits and Network Support,” +in Proc. Int. Conf. Smart Grids and Energy Syst., 2020, pp. 516–521. +[5] R. Zafar, J. Ravishankar, J.E. Fletcher and H.R. Pota, “Multi-Timescale +Model Predictive Control of Battery Energy Storage System Using +Conic Relaxation in Smart Distribution Grids,” IEEE Trans.Power Syst., +vol. 33, no. 6, pp. 7152–7161, 2018. +[6] P. Hasanpor Divshali , and L. S¨oder, “Improving Hosting Capacity of +Rooftop PVs by Quadratic Control of an LV-Central BSS,” IEEE Trans. +Smart Grid, vol. 10, no. 1, pp. 919–927, 2019. +[7] D.Q. Hung , and N. 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Grodzevich , and O. Romanko, “Normalization and other topics in +multi-objective optimization,” in Fields MITACS Indust. Prob. Workshop, +2006. +[14] M. Zeraati, M.E. Hamedani Golshan, and J.M. Guerrero, “Distributed +control of battery energy storage systems for voltage regulation in +distribution networks with high pv penetration,” IEEE Trans. Smart Grid, +vol. 9, no. 4, pp. 3582–3593, 2018. +[15] “Solar Home Electricity Data,” [Online]. Available: https://www.ausgrid. +com.au/Industry/Our-Research/Data-to-share/Solar-home-electricity- +data/.” +[16] “Origin, “VIC residential energy price fact sheet,” 2018.” [Online]. +Available: shorturl.at/gkmV5” + +500 +400 +(kWh) +300 +200 +100 +0 +5 +10 +15 +20 +25 +Time Duration (24 Hours) \ No newline at end of file diff --git a/0NE0T4oBgHgl3EQfdADd/content/tmp_files/load_file.txt b/0NE0T4oBgHgl3EQfdADd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dea1da65c20dd5142481e7951f075a1c5cfcc460 --- /dev/null +++ b/0NE0T4oBgHgl3EQfdADd/content/tmp_files/load_file.txt @@ -0,0 +1,455 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf,len=454 +page_content='A Multi-Objective Planning and Scheduling Framework for Community Energy Storage Systems in Low Voltage Distribution Networks K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Anuradha The Australian National University Canberra, Australia Jayaminda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='KariyawasamBovithanthri@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='au Chathurika P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Mediwaththe The Australian National University & CSIRO Canberra, Australia chathurika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='mediwaththe@csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='au Masoume Mahmoodi The Australian National University Canberra, Australia masoume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='mahmoodi@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='au Abstract—This paper presents a methodology for optimizing the planning and scheduling aspects of a community energy storage (CES) system in the presence of solar photovoltaic (SPV) power in low voltage (LV) distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' To this end, we develop a multi-objective optimization framework that minimizes the real power loss, the energy trading cost of LV customers and the CES provider with the grid, and the investment cost for the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Distribution network limits including the voltage constraint are also taken into account by combining the optimization problem with a linearized power flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Simulations for the proposed optimization framework with real power consumption and SPV generation data of the customers, highlight both real power loss and energy trading cost with the grid are reduced compared with the case without a CES by nearly 29% and 16%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Moreover, a case study justifies our methodology is competent in attaining the three objectives better than the optimization models which optimize only the CES scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Keywords—Community energy storage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' distribution networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' multi-objective optimization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' planning and scheduling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' power flow NOMENCLATURE Sets and Indices V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' j Set of nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' node indices E Set of lines in the network Wj Set of downstream nodes of node j includ- ing itself Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' c Set of customers at node j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' customer index T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' t Set of time intervals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' time index X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' x Feasible set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' decision variable vector Model Parameters rij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' xij Resistance and reactance of line (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' j) - (Ω) Umin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Umax Minimum and maximum squared voltage magnitude limits - (V 2) ηch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' ηdis Charging and discharging efficiencies of the CES λmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' λmax Percentage coefficients of the CES capacity aj Binary variable to find the optimal CES location pRate j Optimal CES rated power- (kW) Ecap j Optimal CES capacity - (kWh) pRate min ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pRate max Minimum and maximum allowable CES rated power- (kW) ECES min ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' ECES max Minimum and maximum allowable CES capacity - (kWh) λp(t) Grid energy price at time t - (AUD/kWh) γCES Fixed part of the CES investment cost - (AUD) δCES Cost of the CES for a unit capacity - (AUD/kWh) ∆t Time difference between two adjacent time instances - (h) wi Weight coefficient of the ith objective func- tion Power Flows and Injections pL cj(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' qL cj(t) Real and reactive power consumption of the customer c at node j at time t - (kW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' kV AR) pP V cj (t) SPV generation of the customer c at node j at time t - (kW) Pij(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Qij(t) Real and reactive power flow from i to j node at time t - (kW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' kV AR) pj(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' qj(t) Real and reactive power absorption at node j at time t - (kW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' kV AR) pG cj(t)) Real power exchange with the grid by the customer c at node j at time t - (kW) pCES cj (t)) Real power exchange with the CES by the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='02372v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='SY] 6 Jan 2023 customer c at node j at time t - (kW) pG CES(t)) Real power exchange with the grid by the CES at time t- (kW) pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='ch j (t) Charging power of the CES at node j at time t - (kW) pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='dis j (t) Discharging power of the CES at node j at time t - (kW) Other Notations ECES j (t) Energy level of the CES at node j at time t - (kWh) Vj(t) Voltage magnitude of node j at time t - (V ) Uj(t) Squared voltage magnitude of node j at time t - (V 2) Iij(t) Current flow from node i to j at time t - (A) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' INTRODUCTION In the recent past, there has been a notable interest among the power systems research community and the industry for the uptake of community energy storage (CES) in low voltage (LV) power systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This trend is driven by the benefits gained from a CES such as providing the opportunity to increase the hosting capacity of the network, enhancing the solar energy self consumption of the customers, and increasing the community access to renewable energy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Additionally, CES devices can be deployed to gain technical merits such as real power loss minimization and economic benefits including the curtailment of energy purchase cost of the customers [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' As discussed in literature, a CES may be used in energy management problems to earn technical and monetary benefits together [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Those merits can be fully exploited if the CES planning aspects including its location, the rated power and the capacity are optimized simultaneously with the CES scheduling aspects namely, its charging and discharging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The existing literature on CES utilization in LV distribution networks can be divided into two categories as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' (i) optimiza- tion of CES scheduling only, (ii) optimization of both CES planning and scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' In the first category, the authors have presented optimization frameworks for CES scheduling without accounting for its planning aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' For instance, a method built up on game theory concepts to maximize the revenue for the CES provider and minimize energy costs for the customers is discussed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' A multi-objective framework to minimize the real energy loss and energy costs of the customers and the CES provider for trading energy with the grid is discussed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' A method based on model predictive control to optimize the CES scheduling is presented in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' In addition to the papers which have presented methods for optimizing only the CES scheduling, there are research work which have proposed methods for optimizing both planning and scheduling of CES simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' For instance, a method for maximizing the hosting capacity in a distribution network in the presence of a CES is proposed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Analytical meth- ods to minimize the real energy loss of a network by finding the optimal CES location and its capacity are discussed in [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' A common feature of these methods is that the optimal CES planning aspects are determined based on analytical (such as graphical or numerical methods) and sensitivity based approaches (methods which decide the optimal values based on a calculated sensitivity parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' These approaches can be computationally exhaustive as the optimal CES location and the capacity are found upon computing a sensitivity parameter for a large number of location-capacity combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Also, even after an exhaustive search, it is not always guaranteed to reach an optimal solution [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Thus, a robust formulation to optimize the capacity, the rated power and the location of a CES while generating the techno-economic benefits associated with such storage devices would be an effective alternative to overcome the challenges in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' In this paper, we study the extent to which the location, the capacity and the power rating of a CES in addition to its scheduling, affect network and economic benefits achievable from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' For this, we develop an optimization framework that optimizes both planning and scheduling of a CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The optimized planning and scheduling aspects are then leveraged to minimize the network power loss, cost incurred by the customers and the CES provider for trading energy with the grid and the investment cost of the CES simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' To the best of our knowledge, this problem has not been addressed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The contributions of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' A linearized power flow model is exploited with the CES operational constraints to develop a multi-objective optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' It is then solved as a mixed integer quadratic program according to the optimization algorithms in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The analytic hierarchy process (AHP) is used for fairly weighting the objective functions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The performance of the proposed optimization framework is evaluated on a real LV distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Here, we do a comparison between our proposed optimization framework, and the models that arbitrarily choose the CES planning aspects such as its location, to assess the impact of it on the objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Finally, a comprehensive analysis of the results is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Section II presents the mathematical models used in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The proposed CES planning and scheduling optimization frame- work is illustrated in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Section IV is about the numerical and graphical results along with their discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Eventually, the conclusion of the work and possible future developments are given in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' SYSTEM MATHEMATICAL MODELLING In this paper, the positive power absorption convention is considered for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Also, it is considered that there are multiple customers at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' All the real and reactive power quantities are measured in kW and kVAR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' It is assumed that power consumption (both real and reactive) and SPV generation of each customer are known ahead from their forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' A summary of the notations used in this paper, together with their definitions are given in the Nomenclature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 1: Possible power exchanges between a customer, the CES and the grid A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Power Flow Model The typical mutual power exchanges that can ensue between different entities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' customers, CES and grid) in the presence of a CES is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pCES cj (t) > 0 suggests a power import by the customer c at node j at time t from the CES, and pCES cj (t) < 0 occurs when that customer exports power to the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The same sign convention used for pCES cj (t) is valid for pG cj(t) and pG CES(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The mathematical relationships between the power flows shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 1 are described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The relationship between the line power flows and node absorptions which follows the LinDistflow model are given by (1), (2) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Pij(t) = pj(t) + � k:j→k Pjk(t) (1) Qij(t) = qj(t) + � k:j→k Qjk(t) (2) The nodal real and reactive power absorptions are illustrated by the equations (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The equation (3a) governs the real power absorption for the CES connected node and for the rest of the nodes (except the slack node), it is the equation (3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Additionally, we assume all the SPV units and the CES operate at unity power factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pj(t) = � c∈Cj pL cj(t) − � c∈Cj pP V cj (t) + pCES,ch j (t) −pCES,dis j (t) ∀j ∈ V\\ {0} , t∈ T (3a) pj(t) = � c∈Cj pL cj(t) − � c∈Cj pP V cj (t) ∀j ∈ V\\ {0} , t∈ T (3b) qj(t) = � c∈Cj qL cj(t) ∀j ∈ V\\ {0} , t∈ T (4) The equations (5) and (6) demonstrate how a customer exchanges power with the CES and the grid, when that customer encounters a mismatch of its real power consumption and SPV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' When a customer experiences a deficit of its SPV generation to supply its real power consumption, that deficit can be fulfilled in share by the CES and the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' On the other hand, if a customer has a surplus SPV generation, that customer exports the excess to both CES and the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The mathematical relationship between pG CES(t), pCES cj (t), pCES,ch j (t) and pCES,dis j (t) can be written as (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' If pL cj(t) ≥ pP V cj (t): 0 ≤ pG cj(t) + pCES cj (t) = pL cj(t) − pP V cj (t) (5a) 0 ≤ pG cj(t) ≤ pL cj(t) − pP V cj (t) ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' c ∈ Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' t∈ T (5b) Otherwise: pG cj(t) + pCES cj (t) = pL cj(t) − pP V cj (t) ≤ 0 (6a) pL cj(t) − pP V cj (t) ≤ pG cj(t) ≤ 0 ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' c ∈ Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' t∈ T (6b) pG CES(t) = N � j=1 � � � � c∈Cj pCES cj (t) + pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='ch j (t) − pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='dis j (t) � � � (7) The Lindistflow equations given in (1)-(4) can be written in matrix format as ((8) [11] U = U01 − 2˜Rp − 2˜Xq ∀t∈ T (8) where U = |V(t)|2 is the vector of squared voltage magni- tudes of nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 1 is a vector of all ones and U0 = |V0|2 is the squared voltage magnitude of the slack node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Also, p and q are the vectors of nodal real and reactive power absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The matrices ˜R and ˜X ∈ RN×N have the elements Rij = � (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='b) ∈ Li ∩ Ljrab and Xij = � (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='b) ∈ Li ∩ Ljxab, re- spectively where Li is the set of lines on the path connecting node 0 and “i” [3], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The squared voltage magnitudes at each node needs to be maintained within its allowable voltage magnitude limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This is guaranteed by the inequality given in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Here, Umin = Umin1 and Umax = Umax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Umin ≤ U ≤ Umax ∀t∈ T (9) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Community Energy Storage Model In this section we present the mathematical modelling of the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' We consider the CES is owned by a third party, and the owner is designated as the CES provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The set of constraints listed from (10) to (17) model the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The equations (10) and (11) imply that the CES charging and discharging power should not exceed the rated power pRate j of the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The temporal variation of the energy level of the CES is expressed by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Also, the CES energy level External Grid Grid pCEs(t) < 0 pCes(t) > 0 0 > ()号d pgj(t) > 0 田田 pCES(t) < 0 cth Customer CES Device at node jat any time should exist within its upper and lower state of charge (SoC) limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This is handled by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The continuity of the CES operation over the next day is guaranteed by the inequality given in (14) which is bounded by a small positive number ε [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Note that td in (14) represents the day num- ber of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Here td ∈ TD, where TD = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='., NT /24} and NT is the cardinality of set T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 0 ≤ pCES,ch j (t) ≤ pRate j ∀j ∈ V\\ {0} , t∈ T (10) 0 ≤ pCES,dis j (t) ≤ pRate j ∀j ∈ V\\ {0} , t∈ T (11) ECES j (t) = ECES j (t − 1) + (ηchpCES,ch j (t) − 1 ηdis pCES,dis j (t))∆t ∀j ∈ V\\ {0} , t∈ T (12) λminEcap j ≤ ECES j (t) ≤ λmaxEcap j ∀j ∈ V\\ {0} , t∈ T (13) ��ECES j (24td) − ECES j (0) �� ≤ ε ∀j ∈ V\\ {0} , td∈ T D (14) The equation (15) is used to find the optimal CES location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Also, (15) ensures that only one CES is installed in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' If aj = 0, it implies that there is no CES at node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' If aj = 1, then the CES is connected to node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' To determine the optimal CES capacity Ecap j , the inequality given in (16) is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' For a case aj = 0, (16) makes Ecap j also to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' When Ecap j = 0, the values ECES j (t), pCES,ch j (t) and pCES,dis j (t) in (12) and (13) also turn out to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The inequality in (17) guarantees the rated power of the CES is bounded by its minimum and maximum allowable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' N � j=1 aj = 1 ∀j ∈ V\\ {0} , aj ∈ {0, 1} (15) ajEcap min ≤ Ecap j ≤ ajEcap max ∀j ∈ V\\ {0} , aj ∈ {0, 1} (16) ajpRate min ≤ pRate j ≤ ajpRate max ∀j ∈ V\\ {0} , aj ∈ {0, 1} (17) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' OPTIMIZATION FRAMEWORK & PROBLEM FORMULATION In our paper, it is expected to minimize the real power loss of the network, energy trading costs of the customers and the CES provider with the grid, and to minimize the CES investment cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Therefore, a multi-objective function is obtained by combining those objectives functions, and its formulation is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Objective Functions 1) Minimizing the Real Power Loss of the Network: The real power loss in a network can be written as (19), in terms of (18), and by taking Ui(t) ≈ U0(t) ∀i ∈ V\\ {0} [4], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' |Iij(t)|2 = Pij(t)2 + Q2 ij(t) Ui(t) ∀(i, j)∈ E, t∈ T (18) fP loss = � t∈T � (i,j)∈E rij |Iij(t)|2 (19) 2) Minimizing the Energy Trading Cost of the Customers and the CES Provider with the Grid: The first term of the objective function given in (20) relates to the energy trading cost with the grid by customers, and latter for the CES provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' fEn,cost = � t∈T λp(t) � � � N � j=1 � c∈Cj pG cj(t) + pG CES(t) � � � ∆t (20) Here, it is considered a one-for-one non-dispatchable energy buyback scheme such that the same energy price for both imports and exports of energy from the grid by the customers and the CES is used [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This kind of an energy pricing scheme can effectively value the SPV power as being same as the power imported from the grid, which is usually generated by a conventional generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 3) Minimizing the Investment Cost of the CES: The third objective is to minimize the investment cost of the CES device which is given by (21) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' fInv,cost = γCES + δCESEcap j (21) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Problem Formulation The three objective functions are normalized and weighted to form the multi-objective function in (22), according to the techniques described in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The normalization guarantees the objective functions are converted into a form which can be added together (since fP loss is measured in kW, and fEn,cost, fInv,cost are measured in AUD ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' x∗ = argmin x∈X w1 � fP loss−f utopia P loss f Nadir P loss −f utopia P loss � + w2 � fEn,cost−f utopia En,cost f Nadir En,cost−f utopia En,cost � +w3 � fInv,cost−f utopia Inv,cost f Nadir Inv,cost−f utopia Inv,cost � (22) where X is the feasible set which is constrained by (1)-(17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The utopia values, individual minimum point values and nadir values of the multi-objective function are found by (23), (24) and (25), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Besides, the decision variable vector can be explicitly expressed as (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' f utopia P loss = fP loss(x∗ Ploss) (23a) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 2: 7-Node LV radial distribution network f utopia En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost = fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost(x∗ En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost) (23b) f utopia Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost = fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost(x∗ Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost) (23c) x∗ Ploss = argmin x∈X fP loss (24a) x∗ En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost = argmin x∈X fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost (24b) x∗ Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost = argmin x∈X fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost (24c) f Nadir P loss = Max � fP loss(x∗ Ploss),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' fP loss(x∗ En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' fP loss(x∗ Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost) � (25a) f Nadir En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost = Max � fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost(x∗ Ploss),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost(x∗ En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost(x∗ Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost) � (25b) f Nadir Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost = Max � fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost(x∗ Ploss),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost(x∗ En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost(x∗ Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='cost) � (25c) x = (aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pRate j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Ecap j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='ch j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='dis j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pG CES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pG cj) (26) In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' the optimization framework can be written as (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' subject to a set of constraints (1)-(17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Also, as (22) being a quadratically-constrained convex multi-objective function, it is solved as a mixed-integer quadratic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' NUMERICAL AND SIMULATION RESULTS In the simulations, a 7-node LV radial distribution network given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 2 is used and its line data can be found in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Also, real power consumption and SPV generation data of 30 customers in an Australian residential community were used for simulations [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' To be more practical, we randomly allocated multiple customers for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Hence, �N j=1 |Cj| = 30, and the number of customers at each node are marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Here, all the customers generate SPV power in addition to their real power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' However, reactive power consumption of the customers is not considered due to the lack of sufficient real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' As the optimization Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 3: Variation of grid energy price for 24 hours involves not only a scheduling problem but also a planning problem, the optimization is performed over a long time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Thus, we consider one year time period split in one hour time intervals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='|T | = 8760 ) for the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The voltage and power base are taken as 400V and 100 kVA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' In addition to that, V0 = 1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=', Umin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='9025p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=', Umax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='1025p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=', λmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='05, λmax = 1, ηch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='98, ηdis = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='02, Ecap min = 200kWh, Ecap max = 2000kWh, pRate min = 20kW, pRate max = 200kW, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='0001kWh and ∆t = 1h are used as the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The values of γCES and δCES are taken as 24000 AUD and 300 AUD/kWh as specified in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Additionally, the weighting factors w1, w2 and w3 were calculated according to the principles of AHP specified in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' We considered a moderate plus importance for both fEn,cost and fInv,cost compared to fP loss, and an equal importance for fEn,cost and fInv,cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Hence, based on the AHP method, the values of w1, w2 and w3 were calculated as 1/9, 4/9 and 4/9, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 3 depicts how the grid energy price varies with the time of the day following a time of use (ToU) tariff scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 3, the grid energy price is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='24871 AUD/kWh during T1(from 12am-7am) & T5(from 10pm-12am), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='31207 AUD/kWh during T2(from 7am-3pm) & T4 (from 9pm-10pm) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='52602 AUD/kWh during T3(from 3pm-9pm) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Case Study - Proposed Optimization Framework Vs Opti- mization Models With Arbitrary CES Locations We did a case study to compare the results of our model with four different cases by arbitrarily changing the CES location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' For this, we considered our optimization framework as Case I, while the rest as Case II-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The same optimization framework (except the constraint that finds the optimal CES location), and the model parameters as for Case I were used for Case II- V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' A synopsis of the results for the five cases are tabulated in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The Case I lists the planning results and the minimized objective function values for our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The Cases II and III suggest the same optimal CES capacity and the rated power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Nevertheless, due to their difference in CES location, Case II provides less real energy loss and energy trading cost compared with the Case III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' When the CES is at node 6, the optimization suggests the same optimal capacity as in Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' However, as node 6 is not the optimal location for CES, the real energy loss and energy trading cost for Case IV are higher 2 6 External Transformer IC2l = 4 IC6l = 4 Grid 22/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='4 kV 0 1 3 4 IC1l = 3 IC3l = 5 C4 = 6 ICzl = 5 5 ICsl = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='55 I (AUD/kWh) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='45 Signal ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='35 Price Energy I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='25 T4 T5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='20 0 5 10 15 20 25 Time Duration (24 HoursTABLE I: SUMMARY OF THE RESULTS FOR CASE STUDIES CES Location (Node) Optimal CES Capacity (kWh) Optimal CES Power Rating (kW) Real Energy Loss1 (kWh) Energy Trading Cost With Grid1 (AUD) CES Investment Cost (AUD) Base Case (Without CES) Not applicable Not applicable Not applicable 110116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='68 45585 Not applicable Case I (Proposed Model) 4 (optimal) 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='15 200 78200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='88 (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='02%) 38520 (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='50%) 168645 Case II 3 (chosen) 601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='32 200 80250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='48 (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='88%) 43362 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='12%) 204396 Case III 5 (chosen) 601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='32 200 81961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='28 (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='43%) 43840 (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='17%) 204396 Case IV 6 (chosen) 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='15 200 80761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='16 (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='34%) 44154 (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='86%) 168645 Case V 7 (chosen) 547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='69 200 86082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='52 (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='17%) 43625 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='70%) 188307 1 Percentage values are calculated with respect to their corresponding values without a CES Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 4: Total power exchange with the grid by the customers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 5: Total power exchange with the CES by the customers than in Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Also, our model has produced the highest cost reduction percentages for real energy loss (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='98%) and the energy trading cost with the grid (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='5%), compared to all the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Hence, it is clear that Case I yields the minimum values for all the three objective functions, and this justifies the effectiveness of our optimization framework compared to the models that optimize only the CES scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Analysis of the Results-Mutual Power Exchanges Between the customers, the CES and the grid In order to understand the CES scheduling and power exchanges between different entities, we select a single day (24 hours) for our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 4 shows the variation of total Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 6: Power exchange with the grid by the CES Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 7: CES charging and discharging power pattern power exchange that occurs with the grid by the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Since �N j=1 � c∈Cj pG cj(t) being a positive value approxi- mately during T1, T3, T4, T5 time intervals, it implies that the customers tend to import certain amount of power from the grid for satisfying their real power consumption during those time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' On the other hand, during T2 (time period of the day usually the SPV generation is high), the customers have a tendency to export a portion of their surplus SPV generation to the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This is evident as �N j=1 � c∈Cj pG cj(t) < 0 during T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This behavior guarantees a cost benefit for the customers for their exported power according to equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 5 depicts how the customers exchange power ) (kw) 50 ()53 0 50 100 0 5 10 15 20 2550 (kw) 0 50 100 150 200 0 5 10 15 20 25100 (kW) 50 0 CG 50 100 0 5 10 15 20 25 Time e Duration(24 Hours100 (kW) 50 0 and 50 100 p 0 5 10 15 20 25Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 8: Temporal variation of the CES energy level with the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' During T2, the customers export a part of their surplus SPV generation to the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' On the contrary, during rest of the time periods, the customers import a certain amount of power from the CES for satisfying their real power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This action results in reducing the cost for the customers as the amount of power imported from the grid is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 6 illustrates how the CES exchanges power with the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' As the grid energy price during T1 being the lowest, the CES tends to import power from the grid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pCES G (t) > 0) during T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This guarantees that the CES is charged with low priced energy from the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' However, during T2, T3 and T4, it is seen that the CES exports its power back to the grid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' pCES G (t) < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This happens as the CES provider can maximize its revenue by exporting power back to grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 7 and 8, it is observed that during T1, the CES charges (from the low priced grid energy) and partially dis- charges by the end of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' During T2, the CES continues to charge and by the end of this time period, it reaches its maximum energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The stored energy in the CES is fully utilized during T3 and T4 for partially supplying the real power consumption of the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' This facilitates monetary benefits for both the customers as the amount of expensive power imported from the grid is lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' Additionally, when observing the temporal variation of the CES energy level, it is visualized that it is the peak value of the CES energy level which was obtained as the optimal CES capacity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content='15 kWh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' CONCLUSION & FUTURE WORK In this work, we have explored how the optimization of the planning and scheduling aspects of a community energy storage (CES) can benefit both the network and the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' To this end, we developed a multi-objective mixed-integer quadratic optimization framework to minimize three objec- tives: (i) network real power loss, (ii) energy trading cost of the customers and the CES provider with the grid, and (iii) the CES investment cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' The simulation results highlighted our optimization framework is competent in acquiring the expected merits compared with the case without a CES, and optimization models that optimize only the scheduling of CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} +page_content=' As future work, we expect to develop the work considering a stochastic model taking into account the uncertainties of real power consumption 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'} diff --git a/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf b/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1ff5e5923a8e28c6fe7990ec60092f398b877f67 --- /dev/null +++ b/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:29ce0e0a870abf6e51ebd6c2efdcd90bdf0609912538d1eb649ee4bdaa339606 +size 3591594 diff --git a/19AzT4oBgHgl3EQfuP0B/vector_store/index.pkl b/19AzT4oBgHgl3EQfuP0B/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..9a8685dd282fe0d63909d7e2d1e67796c24c832a --- /dev/null +++ b/19AzT4oBgHgl3EQfuP0B/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dedd1a1b4a8c46909911a24822fb393f7ac17fad180b3c3f5f5e9580cdf66b15 +size 2386836 diff --git a/19E1T4oBgHgl3EQf5QWX/content/tmp_files/2301.03510v1.pdf.txt b/19E1T4oBgHgl3EQf5QWX/content/tmp_files/2301.03510v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..51d06b02782387dd3deecc9956296ff4fd07fc73 --- /dev/null +++ b/19E1T4oBgHgl3EQf5QWX/content/tmp_files/2301.03510v1.pdf.txt @@ -0,0 +1,1224 @@ +Parallel Reasoning Network for Human-Object Interaction Detection +Huan Peng1,2, Fenggang Liu2, Yangguang Li2, Bin Huang2, Jing Shao2, Nong Sang1, Changxin Gao1 +1Huazhong University of Science and Technology +2SenseTime Group +{nsang,cgao}@hust.edu.cn; liyangguang@sensetime.com; +{penghuan,liufenggang,huangbin1,shaojing}@senseauto.com +Abstract +Human-Object Interaction (HOI) detection aims to learn +how human interacts with surrounding objects. Previous +HOI detection frameworks simultaneously detect human, +objects and their corresponding interactions by using a +predictor. Using only one shared predictor cannot differ- +entiate the attentive field of instance-level prediction and +relation-level prediction. To solve this problem, we pro- +pose a new transformer-based method named Parallel Rea- +soning Network(PR-Net), which constructs two indepen- +dent predictors for instance-level localization and relation- +level understanding. The former predictor concentrates on +instance-level localization by perceiving instances’ extrem- +ity regions. The latter broadens the scope of relation region +to reach a better relation-level semantic understanding. Ex- +tensive experiments and analysis on HICO-DET benchmark +exhibit that our PR-Net effectively alleviated this problem. +Our PR-Net has achieved competitive results on HICO-DET +and V-COCO benchmarks. +1. Introduction +The real world contains large amounts of complex +human-centric activities, which are mainly composed of +various human-object interactions (HOIs). In order for ma- +chines to better understand these complex activities, we +need to detect all these HOIs accurately. To be specific, +HOI detection can be defined as detecting the human-object +pair and their corresponding interactions in an image. It +can be divided into two sub-tasks, instance detection, and +interaction understanding. Only if these two sub-tasks are +completed can we build a good HOI detector. +Previously, different methods were taken to process +these two sub-tasks. Traditional methods like [4,11,23,28] +first locates all instances and then extracts their correspond- +ing features with an off-the-shelf object detector like [12, +29]. After that, instance matching and feature fusing ap- +proaches are used to construct human-object pairs which +Figure 1. The attention fields for two different level predictors +in our PR-Net. The first column shows these input images. The +second column exhibits the attention fields of instance-level pre- +dictor, in which the model concentrates on the extremity region of +human and object. The third column exhibits the attention fields +of interaction-level predictor, in which the model spreads its scope +of attention to the relation-level region. +are more likely to have interactive relations. These pairs are +then sent into the intention parsing network as inputs, and +HOI is classified and outpus, so as to obtain the humain- +object position and corresponding interactive relation cate- +gory. In summary, these traditional two-stage approaches +suffer from the isolated training process of instance local- +ization and interaction understanding, so they cannot lo- +calize interactive human-object pairs and understand those +complex HOI instances. +To alleviate the above problems, multitask learning man- +ners [5, 17, 18, 24, 30, 35, 40, 42] are proposed to com- +plete these two sub-tasks simultaneously. Among these ap- +proaches, they [5,18,24,35,40] process these two sub-tasks +concurrently. +Whereas they need an additional complex +group composition procedure to match the predictions of +these two sub-tasks, which reduces the computation effi- +ciency. In addition, other one-stage methods [30, 42] pre- +dict human-object pairs and corresponding interactions us- +ing one shared prediction head, without needing matching +or gathering processes. However, they accomplish instance +1 +arXiv:2301.03510v1 [cs.CV] 9 Jan 2023 + +localization and interaction understanding in a mixed and +tied manner. This naive mixed prediction manner can cause +inconsistent focus in attentive fields between the instance- +level and the relation-level prediction. This inconsistent fo- +cus has caused limited interaction understanding for those +hard-negative HOIs, which leads to dissatisfactory HOI de- +tection performance. +To sum up, we propose a new transformer-based ap- +proach named Parallel Reasoning Network (PR-Net) to alle- +viate inconsistent focus of attentive fields for different level +prediction. Specificly, two parallel predictos, instance-level +predictor and relation-level predictor,are concluded in PR- +Net. The former focuses on instance-level localization, and +the latter keeps a watchful eye on relation-level semantic +understanding. As can be seen from the two examples in +the second columns of Figure 1, PR-Net’s attention to in- +stances is focused on the endpoints of human skeleton and +the particular edge regions of objects, indicating that the +instance-level predictor can accurately locate the localiza- +tion of human and objects by focusing on these critical ex- +tremity regions of instances. From the two examples in the +third column of Figure 1, it can be seen that PR-Net’s at- +tention to relational areas is focused on the interaction con- +tact areas between human and objects and some contextual +areas containing helpful understanding of the interaction, +which indicates that the relational level predictor spreads +its vision to relation areas to better understand the subtle +relationships between human and objects. In addition, the +instance-level queries of our instance-level predictor strictly +correspond to the relation-level queries of our relationship- +level predictors. So there is no need for any instance-level +queries between them, which greatly reduces the computa- +tional cost [30]. +Our contribution can be concluded in the following three +aspects: +• We propose PR-Net, which leverages a parallel reason- +ing architecture to effectively alleviate the problem of +inconsistent focus in attention fields between instance- +level and relation-level prediction. PR-Net achieves a +better trade-off between two contradictory sub-tasks of +HOI detection. The former needs more local informa- +tion from the extremity region of instances, the latter is +eager for more context information from the relation- +level area. +• With a decoupled prediction manner, PR-Net can de- +tect various HOIs simultaneously without any match- +ing or recomposition process to link the instance-level +prediction and relation-level prediction. +• Equipped with additional techniques, including Con- +sistency Loss for better training and Trident-NMS for +better post-processing, PR-Net achieves competitive +results on both HICO-DET and V-COCO benchmark +datasets in HOI detection. +2. Related Works +2.1. Two-stage Approaches in HOI Detection +Most two-stage HOI detectors firstly detect all the hu- +man and object instances with a modern object detection +framework such as Faster R-CNN, Mask R-CNN [12, 29]. +After instance-level feature extraction and contextual infor- +mation collection, these approaches pair the human and ob- +ject instances for interaction recognition. In the process of +interaction recognition, various contextual features are ag- +gregated to acquire a better relation-level semantic repre- +sentation. InteractNet [9] introduces an additional branch +for interaction prediction, iCAN [8] captures contextual in- +formation using attention mechanisms for interaction pre- +diction. TIN [23] further extends HOI detection models +with a transferable knowledge learner. In-GraphNet [37] +presents a novel graph-based interactive reasoning model to +infer HOIs. VSGNet [31] utilizes relative spatial reasoning +and structual connections to analyze HOIs. IDN [22] repre- +sents the implicit interaction in the transformation function +space to learn a better HOI semantic. Hou proposes fabri- +cating object representations in feature space for few-shot +learning [16] and learning to transfer object affordance for +HOI detection [15]. Zhang [38] proposes to merge multi- +modal features using a graphical model to generate a more +discriminative feature. +2.2. One-stage Approaches in HOI Detection +One-stage approaches directly detect Human-Object In- +teractions without complicated coarse-to-fine bounding box +regression [5, 17, 18, 24, 30, 35, 40, 42]. Among these ap- +proaches, [24, 36] introduced a keypoint-style interaction +detection method which performs inference at each interac- +tion key point. [17] introduced a real-time method to pre- +dict the interactions for each human-object union box. Re- +cently, transformer-based detection approach was proposed +to handle HOI detection as a sparse set prediction prob- +lem [5, 30, 42]. Specifically, [30] designed a transformer +encoder-decoder architecture to predict Human-Object In- +teractions in an end-to-end manner directly and introduced +additional cost terms for interaction prediction. On the other +hand, Kim et al. [19] and Chen et al. [6] propose an in- +teraction decoder to be used alongside the DETR instance +decoder. It is equally important for predicting interactions +and matching related human-object pairs. These aforemen- +tioned one-stage approaches have enormously boosted the +performance of Human-Object Interaction Detectors. +2 + +Pairwise +Instance +Decoder +Instance-level +Queries +Instance-level +Feature +Relation +Decoder +Relation-Level Predictor +Instance-level Predictor +Relation-level +Queries +Relation-level +Feature +Convolutional +Neural +Network +…… +Transformer +Encoder +… +… +Positional Encoding +Input Feature +Visual Memory +Image Feature Extractor +Classification Loss +Regression Loss +Consistency Loss +Training +Trident-NMS +Testing +Object Class +Human Box +Object Box +Relation Box +Relation class +…… +…… +…… +…… +Figure 2. The framework of our PR-Net. It is comprised of four components:Image Feature Extractor, Pairwise Instance Predictor, +Relation-level Predictor, Training and Post-processing Techniques. +3. Proposed Method +In this section, we present our Parallel Reasoning +Network(PR-Net) for HOI detection, which is illustrated in +the Figure 2. We can know that our PR-Net includes an +Image Feature Extractor(CNN backbone and transformer +encoder) and two parallel predictors (i.e., Instance-level +Predictor and Relation-level Predictor). The two parallel +predictors are designed to decode instance information(i.e. +human-box, object-box, object-class) and relation informa- +tion(i.e. relation-box, relation-class) respectively. Next, we +introduce the proposed instance-level and relation-level loss +functions to learn the location of instances and the interac- +tions within each human-object pair. At last, we introduce +the proposed Trident-NMS which is utilized to filter those +duplicated HOI predictions effectively. +3.1. Image Feature Extractor +The overall Image Feature Extractor architecture con- +sists of a standard CNN backbone fc and transformer en- +coder fe. The conventional CNN backbone is used to pro- +cess the input image xϵR3×H×W to a global context feature +map zϵRc×H′×W ′, in which typically images are down- +sampled to (H′, W ′) spatial shape with a dimension of c. +Then, the global context feature map is serialized as to- +kens, in which the spatial dimensions of the feature map +are collapsed into one dimension, resulting in H′ × W ′ +tokens. +Then, the tokens are linearly mapped to T = +{ti|tiϵRc′}Nq +i=1, where Nq = H′ × W ′. Afterward, these +tokens are shaped as a sequence to feed into the transformer +encoder. +For the transformer encoder, each encoder layer fol- +lows standard architecture of transformer, which con- +sists of a multi-head self-attention module and a feed +forward network (FFN). Additional position embedding +qeϵRc′×H′×W ′ is also added to the serialized token to +supplement the positional information. +With the mech- +anism of self-attention, the encoder can map the former +global context feature map from CNN to richer contex- +tual information. Finally, the set of encoded image fea- +tures {di|diϵRc′}Nq +i=1 can be formulated as visual memory +E = fe(T, qe). The visual memory E contains richer con- +textual information. +3.2. Instance-level Predictor +The Instance-level Predictor includes a standard trans- +former decoder fip with just three layers. +The decoder +response for above visual memory E, according to a set +of learnable instance query vectors Qp = {qi|qiϵRc′}Nq +i=1 +which is added with position embedding plϵRc′×H′×W ′. +The instance-level queries vectors are trained to learn a +more precise location of instances, which focuses more +on those local information about location of instances. +The independent predictors are composed of three feed- +forward networks (FFNs), including human-bounding-box +FFN φhb, object-bounding-box FFN φob, and object-class +FFN φoc, each of which response for decoding instance fea- +ture to human-box ˆbh, object-box ˆbo and object-class ˆco re- +spectively. The formulation can be denoted as: +ˆbh = φhb(fip(Qp, pl, E)), +ˆbo = φob(fip(Qp, pl, E)), +ˆco = φoc(fip(Qp, pl, E)). +(1) +3.3. Relation-level Predictor +We decouple the relation problems from HOI and use a +Relation-level Predictor to reason relationships from larger- +scale semantics. We propose a relation box to guide the +predictor to percept the human-object relationship in the +3 + +human blow cakeRelation-level Predictor. +The Relation-level Predictor consists of a standard trans- +former decoder frd and two independent predictors(FFNs). +Another relation-level queries Qr and position embedding +pr are randomly initialed and fed into the Relation-level +Predictor. One of the predictors φub predicts relation boxes +ˆbu, the other predictor φac decodes the relation class in- +formation ˆca. The relation boxes ˆbu and the relation class +information ˆca can be formulated as Eq. 2. +ˆbu = φub(fdr(Qr, pr, E)), +ˆca = φac(fdr(Qr, pr, E)). +(2) +Attributed to the relation boxes, the decoder of Interaction- +level Predictor is guided to enlarge the receptive field (as +shown in Figure 1). The relation queries Qr can pay at- +tention to the entire area where human and object interact. +Thus, the predictor φac can predict a more accurate relation +class. +In addiction, to match the relation class information ˆca +with the aforementioned human-box ˆbh, object-box ˆbo and +object-class ˆco from the Instance-level Predictor, we ditch +the complex matching method like HO pointer in HOTR. +Instead, we just match the relation class information ˆca +and the instances information ˆbh etc. one by one in order. +Specifically, for a pair of instances {ˆbh +i ,ˆbo +, ˆco +i , iϵNq}, ˆca +i is +the corresponding relation class. In this way, the instance- +level query vectors Qp and the relation-level query vectors +Qr represent the same human-object interaction, but have +the ability to focus on different receptive field. +3.4. Loss Functions +The overall loss functions consist of the instance-level +loss and relation-level loss, applied to Instance-level Predic- +tor and Relation-level Predictor, respectively. The instance- +level loss supervises the Instance-level Predictor to pre- +dict instance-level target, i.e., human-box, object-box, and +object-class. The relation-level loss assists the Relation- +level Predictor to predict relation-class and relation-box +from the larger receptive field. +3.4.1 +The Instance-level loss function +LIL supervises the instance information, including human- +box ˆbh, object-box ˆbo and object-class ˆco. The instance- +level loss function consists of human-box regression Lhr, +object-box regression Lor and object-class classification +Loc. Lhr and Lor are standard bounding-box regression +loss, i.e. L1 loss, to locate the position of human and ob- +ject. Loc is a classification loss to classify the categories of +the object. The loss functions can be defined as Eq. 5. +Lhr = 1 +N +N +� +i +||ˆbh +i − bh +i ||, +Lor = 1 +N +N +� +i +||ˆbo +i − bo +i ||, +Loc = 1 +N +N +� +i +CE(ˆco +i , co +i ), +(3) +where CE is cross entropy loss, co +i is the ground truth of +object class. +The instance-level loss function LIL can be defined as: +LIL = Whr ∗ Lhr + Wor ∗ Lor + Woc ∗ Loc. +(4) +3.4.2 +The Relation-level loss function +LRL supervises the relationship information, i.e., the rela- +tion class ˆca, primarily. In addition, auxiliary relation boxes +are also supervised to pay attention to the entire area where +the interaction happens. Thus, the Relation-level loss func- +tion consists of relation-box regression Lur, relation-box +consistency loss Luc and relation-class loss Lac. The Lac is +a classification loss to classify the categories of the interac- +tion. The relation-box regression loss function Lur is a L1 +loss to resemble the predicted relation boxes and its ground- +ing truth. The grounding truth of relation boxes is the outer +bounding box of human and object boxes. The relation-box +regression loss function helps the Relation-level Predictor +to be aware of the relation feature of human and object. The +consistency loss Luc are used to keep the consistency of ˆbh, +ˆbo and ˆbu. Specifically, a pseudo relation box ˆbho is gener- +ated by taking the outer bounding box of ˆbh and ˆbo. Then, +an L1 loss resemble ˆbu and ˆbho. With the relation box, the +relation-class loss can supervise better relation semantics. +Lur = 1 +N +N +� +i +||ˆbu +i − bu +i ||, +Luc = 1 +N +N +� +i +||ˆbu +i − ˆbho +i ||, +Lac = 1 +N +N +� +i +SigmoidCE(ˆca +i , ca +i ). +(5) +The relation-level loss function LRL can be defined as: +LRL = Wur ∗ Lur + Wuc ∗ Luc + Wac ∗ Lac. +(6) +In all, the overall loss fucntion L can be denoted as: +L = LIL + LRL. +(7) +4 + +3.5. Inference for HOI Detection +The inference process of our PR-Net can be divided into +two parts: the calculation of the HOI predictions and the +Trident-NMS post-processing technique. +HOI Prediction To acquire the final HOI detection results, +we need to predict human bounding box, object bounding +box, and object class using both instance-level predictions +and relation class and relation box using relation-level pre- +diction. Based on the above predictions, we can calculate +the final HOI prediction score as below: +shoi +i += {maxkso +i (k)} ∗ srel +i +(8) +Where maxkso +i (k) means the most probable class score of +the i-th output object from instance-level predictor; srel +i +means the multi-class scores of the i-th output interaction +from relation-level predictor. Note that each human-object +pair can only have one object with certain class, but there +maybe exist multiple human-object interactions within one +pair. +Trident-NMS For each predicted HOI class in one image, +we choose to filter its duplicated predictions according to +the above calculated HOI prediction scores with our pro- +posed Trident Non Maximal Suppression(Trident-NMS). In +detail, if the TriIoU(i, j) between the i-th and the j-th HOI +prediction is higher than the threshold Thresnms, we will +filter the prediction which has a lower HOI score. And the +calculation of TriIoU(i, j) is as below: +TriIoU(i, j) =IoU(bh +i , bh +j )Wh +× IoU(bo +i , bo +j)Wo +× IoU(brel +i , brel +j )Wrel +(9) +Where IoU(bh +i , bh +j ), IoU(bo +i , bo +j), IoU(brel +i , brel +j ) repre- +sent the Interaction over Union between the i-th and the j- +th human boxes, object boxes and relation boxes; Wh, Wo, +Wrel represent the weights of Human IoU, Object IoU and +Relation IoU. +4. Experiment +4.1. Datasets and Evaluation Metrics +We evaluate our method on two large-scale benchmarks, +including V-COCO [10] and HICO-DET [3] datasets. V- +COCO includes 10,346 images, which contains 16,199 hu- +man instances in total and provides 26 common verb cate- +gories. HICO-DET contains 47,776 images, where 80 ob- +ject categories and 117 verb categories compose of 600 HOI +categories. There are three different HOI category sets in +HICO-DET, which are: (a) all 600 HOI categories (Full), +(b) 138 HOI categories with less than 10 training instances +(Rare), and (c) 462 HOI categories with 10 or more training +instances (Non-Rare). Following the standard protocols, we +use mean average precision (mAP) in HICO-DET [4] and +role average precision (AProle) in V-COCO [10] to report +evaluation results. +4.2. Implementation Details +We use ResNet-50 and ResNet-101 [13] as a backbone +feature extractor. +The transformer encoder consist of 6 +transformer layers with multi-head attention of 8 heads. +The number of transformer layers in Instance-level Predic- +tor and Interaction-level Predictor is both set to be 3. The +reduced dimension size of visual memory is set to 256. The +number of instance-level and relation-level queries is set +to 100 for both HICO-Det and V-COCO benchmark. The +human, object and relation box FFNs both have 3 linear +layers with ReLU, while the object and relation category +FFNs have one linear layer. During training, we initial- +ize the network with the parameters of DETR [2] trained +on the MS-COCO dataset. We set the weight coefficients +of bounding box regression, Generalized IoU, object class, +relation class and consistency loss to 2.5, 1, 1, 1 and 0.5, +respectively, which follows QPIC [30]. We optimize the +network by AdamW [26] with the weight decay 10−4. We +train the model for 150 epochs with a learning rate of 10−5 +for the backbone and 10−4 for the other parts decreased by +10 times at the 100th and the 130th epoch respectively. All +experiments are conducted on the 8 Tesla A100 GPUs and +CUDA11.2, with a batch size of 16. +We select 100 detection results with the highest scores +for validation and then adopt Trident-NMS to filter results +further. +4.3. Overall Performance +We summarize the performance comparisons in this sub- +section. +Performance on HICO-DET. Table 1 shows the per- +formance comparison on HICO-DET. Firstly, the detection +results of our PR-Net are the best among all approaches un- +der the Full and Non-Rare settings, demonstrating that our +method is more competitive than the others in detecting the +most common HOIs. It is noted that PR-Net is also pre- +eminent in detecting rare HOIs (HOI categories with less +than 10 training instances), because our parallel reasoning +network can migrate the non-rare knowledge into a rare do- +main. Besides, our PR-Net obtains 32.86 mAP on HICO- +DET (Default Full), which achieves a relative gain of 9.8% +compared with the baseline. These results quantitatively +show the efficacy of our method. +Performance on V-COCO. Comparison results on V- +COCO in terms of mAProle are shown in Table 2. It can +be seen that our proposed PR-Net has a mAP(%) of 62.4, +obtaining the best performance among all approaches. Al- +though we do not adopt previous region-based feature learn- +ing (e.g., RPNN [41], Contextual Att [34]), or employ ad- +5 + +Table 1. Results on HICO-DET [4]. “COCO” is the COCO pre- +trained detector, “HICO-DET” means that the detector is further +fine-tuned on HICO-DET. +Default Full +Method +Detector +Backbone +Full +Rare Non-Rare +CNN-based +VCL [14] +COCO +ResNet-50 +19.43 16.55 +20.29 +VSGNet [31] +COCO +ResNet-152 +19.80 16.05 +20.91 +DJ-RN [21] +COCO +ResNet-50 +21.34 18.53 +22.18 +PPDM [24] +HICO-DET Hourglass-104 21.73 13.78 +24.10 +Bansal et al. [1] +HICO-DET +ResNet-101 +21.96 16.43 +23.62 +TIN [23]DRG +HICO-DET +ResNet-50 +23.17 15.02 +25.61 +VCL [14] +HICO-DET +ResNet-50 +23.63 17.21 +25.55 +GG-Net [40] +HICO-DET Hourglass-104 23.47 16.48 +25.60 +IDNDRG [22] +HICO-DET +ResNet-50 +26.29 22.61 +27.39 +Transformer-based +HOI-Trans [42] +HICO-DET +ResNet-50 +23.46 16.91 +25.41 +HOTR [18] +HICO-DET +ResNet-50 +25.10 17.34 +27.42 +AS-Net [5] +HICO-DET +ResNet-50 +28.87 24.25 +30.25 +QPIC [30] +HICO-DET +ResNet-50 +29.07 21.85 +31.23 +PR-Net (Ours) +HICO-DET +ResNet-50 +31.17 25.66 +32.82 +PR-Net (Ours) +HICO-DET +ResNet-101 +32.86 28.03 +34.30 +ditional human pose (e.g., PMFNet [32], TIN [23]), our +method outperforms these approaches with sizable gains. +Besides, our method achieves an absolute gain of 3.6 points, +a relative improvement of 6.1% compared with the baseline, +validating its efficacy in the HOI detection task. +Table 2. Performance comparison on V-COCO dataset. +Method +Backbone Network +APS1 +role +APS2 +role +CNN-based +VSGNet [31] +ResNet-152 +51.8 +57.0 +PMFNet [32] +ResNet-50-FPN +52.0 +- +PD-Net [39] +ResNet-152-FPN +52.6 +- +CHGNet [33] +ResNet-50-FPN +52.7 +- +FCMNet [25] +ResNet-50 +53.1 +- +ACP [20] +ResNet-152 +53.23 +- +IDN [22] +ResNet-50 +53.3 +60.3 +GG-Net [40] +Hourglass-104 +54.7 +- +DIRV [7] +EfficientDet-d3 +56.1 +- +Transformer-based +HOI-Trans [42] +ResNet-101 +52.9 +- +AS-Net [5] +ResNet-50 +53.9 +- +HOTR [18] +ResNet-50 +55.2 +64.4 +QPIC [30] +ResNet-50 +58.8 +61.0 +PR-Net (Ours) +ResNet-50 +61.4 +62.5 +PR-Net (Ours) +ResNet-101 +62.9 +64.2 +4.4. Ablation Analysis +To evaluate the contribution of different components +in our PR-Net, we first conduct a comprehensive ablation +analysis on the HICO-DET dataset. Next, we analyze the +impact of the number of different-level predictors. At last, +we analyze the effects of different post-processing manners. +Contribution of different components. +Compared +Table 3. Ablation analysis of the proposed PR-Net with the back- +bone of ResNet-101 on HICO-DET test set. Parallel Predictor +means we parallelly predict instance-level locations and relation- +level semantics. Consistency Loss means we constrain the union +box of the human-object pair and the relation box to be consis- +tent. Trident-NMS means duplicate filtering through human, ob- +ject, and relation bounding boxes. +Parallel Predictor +Consistency Loss +Trident-NMS +HICO-DET +Full +Rare +NonRare +- +- +- +29.90 +23.92 +31.69 +✓ +- +- +31.62 +25.43 +33.47 +✓ +✓ +- +31.87 +27.59 +33.14 +✓ +✓ +✓ +32.86 +28.03 +34.30 +with our baseline [30], the performance improvements of +our PR-Net are from three components: Parallel Predictor, +Consistency Loss, and Trident-NMS. From Table 3, we can +know the contribution of different components. +Among +these components, Parallel Predictor is our core approach. +With that, we can observe a noticeable gain of mAP in +HICO-DET by 1.72. It proves that the parallel reasoning +structure can significantly improve instance localization +and interaction understanding for an HOI detection model. +Additionally, we design a consistency loss between the +union box of the human-object pair and the relation box, +which can contribute about 0.25 mAP gain in the HICO- +DET test set. It shows that it is meaningful and helpful +to constrain the union region of instance-level predictions +and the relation region of relation-level predictions. +At +last, we design a more effective post-processing technique +named Trident-NMS, which brings about 1.0 mAP gain in +the HICO-DET test set. It reveals that the set-prediction +method can also benefit from duplicate filtering technique +and post-processing technique like NMS is essential for +HOI detection. +Impacts of different numbers of parallel predictors. +In our PR-Net, two parallel predictors are significant for +HOI detection, and we detailedly analyze the impact of +different numbers of parallel predictors. From Table 4, we +can know that equipped with three layers of instance-level +predictor and relation-level predictor, our PR-Net can +acquire the best mAP performance in the HICO-DET test +set. It reveals that our PR-Net can significantly outperform +the baseline QPIC [30] without additional computational +cost. +Interestingly, we can also observe that even with +only one layer of parallel predictors, our PR-Net can also +outperform the baseline equipped with a six-layer predictor. +Effects of different implements of Trident-NMS. In +Table 5, we analyze the effects of different implements +of Trident-NMS. We find that Product-based Trident- +NMS performs better than Sum-based Trident-NMS. +6 + +Table 4. Ablation analysis of the number of instance-level predic- +tor Ndec and the number of relation-level predictor Nreldec. +approaches +Backbone Ndec Nreldec Full Rare Non-Rare +QPIC(Baseline) [30] ResNet50 +6 +- +29.07 21.85 +31.23 +PR-Net(Ours) +ResNet50 +1 +1 +29.64 24.18 +31.27 +PR-Net(Ours) +ResNet50 +3 +3 +31.17 25.66 +32.82 +PR-Net(Ours) +ResNet50 +6 +6 +31.04 24.87 +32.89 +QPIC(Baseline) [30] ResNet101 +6 +- +29.90 23.92 +31.69 +PR-Net(Ours) +ResNet101 +1 +1 +30.26 23.27 +32.34 +PR-Net(Ours) +ResNet101 +3 +3 +32.86 28.03 +34.30 +PR-Net(Ours) +ResNet101 +6 +6 +32.52 27.04 +34.16 +Additionally, we can also observe that when the weight +of Human-IoU in TriIoU increases, the HOI detection +performance will be better. This reveals that human box +duplication is more frequent than that of object box or +relation box. In summary, with either the Product-based +or Sum-based TriIoU calculation, we should pay more +attention to the non-maximal suppression of the human box. +Table 5. Ablation analysis of the Trident-NMS module on HICO- +DET test set. Product means we calculate TriIoU by multiply- +ing these weighted Human-IoU, Object-IoU, and Relation-IoU. +Sum means we calculate TriIoU by adding all these weighted +Human-IoU, Object-IoU, and Relation-IoU. Wh, Wo, Wrel rep- +resent the weights of Human-IoU, Object-IoU and Relation-IoU +respectively. Thresnms means the threshold of non-maximum +suppression. +Product +Sum +Wh +Wo +Wrel +Thresnms +HICO-DET +Full +Rare +NonRare +- +- +- +- +- +- +31.87 +27.59 +33.14 +- +✓ +0.33 +0.33 +0.33 +0.5 +30.61 +27.00 +31.69 +- +✓ +0.33 +0.33 +0.33 +0.7 +32.53 +27.88 +33.91 +- +✓ +0.4 +0.4 +0.2 +0.7 +32.63 +27.96 +34.02 +- +✓ +0.5 +0.4 +0.1 +0.7 +32.66 +27.91 +34.00 +- +✓ +0.6 +0.3 +0.1 +0.7 +32.56 +27.70 +34.01 +✓ +- +1.0 +1.0 +1.0 +0.5 +32.77 +27.98 +34.20 +✓ +- +1.0 +1.0 +0.5 +0.5 +32.81 +28.02 +34.25 +✓ +- +0.5 +0.5 +0.5 +0.5 +32.61 +27.65 +34.08 +✓ +- +0.5 +1.0 +0.5 +0.5 +32.61 +27.67 +34.09 +✓ +- +1.0 +0.5 +0.5 +0.5 +32.86 +28.03 +34.30 +4.5. Visualization of features +Using the t-SNE visualization technique [27], we visual- +ize 20000 samples of output feature. These object and inter- +action features are extracted from the last layer of Instance- +level Predictor and Relation-level Predictor in our PR-Net, +respectively. From the Figure 3, we can observe that our +PR-Net can obviously distinguish different class of objects +and interactions. Interestingly, from this visualization of +features, our PR-Net can even learn better the complex +interaction representations then the object representations +which benefits from our advantageous parallel reasoning ar- +chitecture. +Figure 3. Visualization of object features and relation features on +HICO-DET dataset via t-SNE technique. Left is object features +and right is relation features. +4.6. Qualitative Examples +From Figure 4, we can observe that our PR-Net can accu- +rately detect both human box, object box, and relation box +as well as their corresponding interactions. From the first +row and second column of Figure 4, we can know that our +PR-Net can precisely distinguish which man is riding the +horse in the image. From the second row and third column +of Figure 4, our PR-Net can precisely detect those subtle +and indiscernible HOIs. In summary, our PR-Net can cor- +rectly detect those complex and hard HOIs. +Figure 4. Visualization of some HOI detection examples (Top 1 +result) detected by the proposed Parallel Reasoning Network on +the HICO-DET test set. +5. Conclusion +In this paper, we propose a new Human-Object Inter- +action Detector named Parallel Reasoning Network(PR- +Net), which consists of an instance-level predictor and +a relation-level predictor, to alleviate the problem of in- +consistent focus in attentive fields between instance-level +and interaction-level predictions. +In addition, our PR- +Net achieves a better trade-off between instance localiza- +tion and interaction understanding. 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In CVPR, 2021. 1, 2, 6 +9 + diff --git a/19E1T4oBgHgl3EQf5QWX/content/tmp_files/load_file.txt b/19E1T4oBgHgl3EQf5QWX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..26dfc04694b7e21efaa3da8b0acb395bc4ba0e37 --- /dev/null +++ b/19E1T4oBgHgl3EQf5QWX/content/tmp_files/load_file.txt @@ -0,0 +1,647 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf,len=646 +page_content='Parallel Reasoning Network for Human-Object Interaction Detection Huan Peng1,2, Fenggang Liu2, Yangguang Li2, Bin Huang2, Jing Shao2, Nong Sang1, Changxin Gao1 1Huazhong University of Science and Technology 2SenseTime Group {nsang,cgao}@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' liyangguang@sensetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' {penghuan,liufenggang,huangbin1,shaojing}@senseauto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='com Abstract Human-Object Interaction (HOI) detection aims to learn how human interacts with surrounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Previous HOI detection frameworks simultaneously detect human, objects and their corresponding interactions by using a predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Using only one shared predictor cannot differ- entiate the attentive field of instance-level prediction and relation-level prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' To solve this problem, we pro- pose a new transformer-based method named Parallel Rea- soning Network(PR-Net), which constructs two indepen- dent predictors for instance-level localization and relation- level understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The former predictor concentrates on instance-level localization by perceiving instances’ extrem- ity regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The latter broadens the scope of relation region to reach a better relation-level semantic understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Ex- tensive experiments and analysis on HICO-DET benchmark exhibit that our PR-Net effectively alleviated this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Our PR-Net has achieved competitive results on HICO-DET and V-COCO benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Introduction The real world contains large amounts of complex human-centric activities, which are mainly composed of various human-object interactions (HOIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In order for ma- chines to better understand these complex activities, we need to detect all these HOIs accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' To be specific, HOI detection can be defined as detecting the human-object pair and their corresponding interactions in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' It can be divided into two sub-tasks, instance detection, and interaction understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Only if these two sub-tasks are completed can we build a good HOI detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Previously, different methods were taken to process these two sub-tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Traditional methods like [4,11,23,28] first locates all instances and then extracts their correspond- ing features with an off-the-shelf object detector like [12, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' After that, instance matching and feature fusing ap- proaches are used to construct human-object pairs which Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The attention fields for two different level predictors in our PR-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The first column shows these input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The second column exhibits the attention fields of instance-level pre- dictor, in which the model concentrates on the extremity region of human and object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The third column exhibits the attention fields of interaction-level predictor, in which the model spreads its scope of attention to the relation-level region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' are more likely to have interactive relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' These pairs are then sent into the intention parsing network as inputs, and HOI is classified and outpus, so as to obtain the humain- object position and corresponding interactive relation cate- gory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In summary, these traditional two-stage approaches suffer from the isolated training process of instance local- ization and interaction understanding, so they cannot lo- calize interactive human-object pairs and understand those complex HOI instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' To alleviate the above problems, multitask learning man- ners [5, 17, 18, 24, 30, 35, 40, 42] are proposed to com- plete these two sub-tasks simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Among these ap- proaches, they [5,18,24,35,40] process these two sub-tasks concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Whereas they need an additional complex group composition procedure to match the predictions of these two sub-tasks, which reduces the computation effi- ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In addition, other one-stage methods [30, 42] pre- dict human-object pairs and corresponding interactions us- ing one shared prediction head, without needing matching or gathering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' However, they accomplish instance 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='03510v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='CV] 9 Jan 2023 localization and interaction understanding in a mixed and tied manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' This naive mixed prediction manner can cause inconsistent focus in attentive fields between the instance- level and the relation-level prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' This inconsistent fo- cus has caused limited interaction understanding for those hard-negative HOIs, which leads to dissatisfactory HOI de- tection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' To sum up, we propose a new transformer-based ap- proach named Parallel Reasoning Network (PR-Net) to alle- viate inconsistent focus of attentive fields for different level prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Specificly, two parallel predictos, instance-level predictor and relation-level predictor,are concluded in PR- Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The former focuses on instance-level localization, and the latter keeps a watchful eye on relation-level semantic understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' As can be seen from the two examples in the second columns of Figure 1, PR-Net’s attention to in- stances is focused on the endpoints of human skeleton and the particular edge regions of objects, indicating that the instance-level predictor can accurately locate the localiza- tion of human and objects by focusing on these critical ex- tremity regions of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' From the two examples in the third column of Figure 1, it can be seen that PR-Net’s at- tention to relational areas is focused on the interaction con- tact areas between human and objects and some contextual areas containing helpful understanding of the interaction, which indicates that the relational level predictor spreads its vision to relation areas to better understand the subtle relationships between human and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In addition, the instance-level queries of our instance-level predictor strictly correspond to the relation-level queries of our relationship- level predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' So there is no need for any instance-level queries between them, which greatly reduces the computa- tional cost [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Our contribution can be concluded in the following three aspects: We propose PR-Net, which leverages a parallel reason- ing architecture to effectively alleviate the problem of inconsistent focus in attention fields between instance- level and relation-level prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' PR-Net achieves a better trade-off between two contradictory sub-tasks of HOI detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The former needs more local informa- tion from the extremity region of instances, the latter is eager for more context information from the relation- level area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' With a decoupled prediction manner, PR-Net can de- tect various HOIs simultaneously without any match- ing or recomposition process to link the instance-level prediction and relation-level prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Equipped with additional techniques, including Con- sistency Loss for better training and Trident-NMS for better post-processing, PR-Net achieves competitive results on both HICO-DET and V-COCO benchmark datasets in HOI detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Related Works 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Two-stage Approaches in HOI Detection Most two-stage HOI detectors firstly detect all the hu- man and object instances with a modern object detection framework such as Faster R-CNN, Mask R-CNN [12, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' After instance-level feature extraction and contextual infor- mation collection, these approaches pair the human and ob- ject instances for interaction recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In the process of interaction recognition, various contextual features are ag- gregated to acquire a better relation-level semantic repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' InteractNet [9] introduces an additional branch for interaction prediction, iCAN [8] captures contextual in- formation using attention mechanisms for interaction pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' TIN [23] further extends HOI detection models with a transferable knowledge learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In-GraphNet [37] presents a novel graph-based interactive reasoning model to infer HOIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' VSGNet [31] utilizes relative spatial reasoning and structual connections to analyze HOIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' IDN [22] repre- sents the implicit interaction in the transformation function space to learn a better HOI semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Hou proposes fabri- cating object representations in feature space for few-shot learning [16] and learning to transfer object affordance for HOI detection [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Zhang [38] proposes to merge multi- modal features using a graphical model to generate a more discriminative feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' One-stage Approaches in HOI Detection One-stage approaches directly detect Human-Object In- teractions without complicated coarse-to-fine bounding box regression [5, 17, 18, 24, 30, 35, 40, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Among these ap- proaches, [24, 36] introduced a keypoint-style interaction detection method which performs inference at each interac- tion key point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' [17] introduced a real-time method to pre- dict the interactions for each human-object union box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Re- cently, transformer-based detection approach was proposed to handle HOI detection as a sparse set prediction prob- lem [5, 30, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Specifically, [30] designed a transformer encoder-decoder architecture to predict Human-Object In- teractions in an end-to-end manner directly and introduced additional cost terms for interaction prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' On the other hand, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' [19] and Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' [6] propose an in- teraction decoder to be used alongside the DETR instance decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' It is equally important for predicting interactions and matching related human-object pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' These aforemen- tioned one-stage approaches have enormously boosted the performance of Human-Object Interaction Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 2 Pairwise Instance Decoder Instance-level Queries Instance-level Feature Relation Decoder Relation-Level Predictor Instance-level Predictor Relation-level Queries Relation-level Feature Convolutional Neural Network ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Transformer Encoder … … Positional Encoding Input Feature Visual Memory Image Feature Extractor Classification Loss Regression Loss Consistency Loss Training Trident-NMS Testing Object Class Human Box Object Box Relation Box Relation class ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The framework of our PR-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' It is comprised of four components:Image Feature Extractor, Pairwise Instance Predictor, Relation-level Predictor, Training and Post-processing Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Proposed Method In this section, we present our Parallel Reasoning Network(PR-Net) for HOI detection, which is illustrated in the Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' We can know that our PR-Net includes an Image Feature Extractor(CNN backbone and transformer encoder) and two parallel predictors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=', Instance-level Predictor and Relation-level Predictor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The two parallel predictors are designed to decode instance information(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' human-box, object-box, object-class) and relation informa- tion(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' relation-box, relation-class) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Next, we introduce the proposed instance-level and relation-level loss functions to learn the location of instances and the interac- tions within each human-object pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' At last, we introduce the proposed Trident-NMS which is utilized to filter those duplicated HOI predictions effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Image Feature Extractor The overall Image Feature Extractor architecture con- sists of a standard CNN backbone fc and transformer en- coder fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The conventional CNN backbone is used to pro- cess the input image xϵR3×H×W to a global context feature map zϵRc×H′×W ′, in which typically images are down- sampled to (H′, W ′) spatial shape with a dimension of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Then, the global context feature map is serialized as to- kens, in which the spatial dimensions of the feature map are collapsed into one dimension, resulting in H′ × W ′ tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Then, the tokens are linearly mapped to T = {ti|tiϵRc′}Nq i=1, where Nq = H′ × W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Afterward, these tokens are shaped as a sequence to feed into the transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' For the transformer encoder, each encoder layer fol- lows standard architecture of transformer, which con- sists of a multi-head self-attention module and a feed forward network (FFN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Additional position embedding qeϵRc′×H′×W ′ is also added to the serialized token to supplement the positional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' With the mech- anism of self-attention, the encoder can map the former global context feature map from CNN to richer contex- tual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Finally, the set of encoded image fea- tures {di|diϵRc′}Nq i=1 can be formulated as visual memory E = fe(T, qe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The visual memory E contains richer con- textual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Instance-level Predictor The Instance-level Predictor includes a standard trans- former decoder fip with just three layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The decoder response for above visual memory E, according to a set of learnable instance query vectors Qp = {qi|qiϵRc′}Nq i=1 which is added with position embedding plϵRc′×H′×W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The instance-level queries vectors are trained to learn a more precise location of instances, which focuses more on those local information about location of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The independent predictors are composed of three feed- forward networks (FFNs), including human-bounding-box FFN φhb, object-bounding-box FFN φob, and object-class FFN φoc, each of which response for decoding instance fea- ture to human-box ˆbh, object-box ˆbo and object-class ˆco re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The formulation can be denoted as: ˆbh = φhb(fip(Qp, pl, E)), ˆbo = φob(fip(Qp, pl, E)), ˆco = φoc(fip(Qp, pl, E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' (1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Relation-level Predictor We decouple the relation problems from HOI and use a Relation-level Predictor to reason relationships from larger- scale semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' We propose a relation box to guide the predictor to percept the human-object relationship in the 3 human blow cakeRelation-level Predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The Relation-level Predictor consists of a standard trans- former decoder frd and two independent predictors(FFNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Another relation-level queries Qr and position embedding pr are randomly initialed and fed into the Relation-level Predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' One of the predictors φub predicts relation boxes ˆbu, the other predictor φac decodes the relation class in- formation ˆca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The relation boxes ˆbu and the relation class information ˆca can be formulated as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' ˆbu = φub(fdr(Qr, pr, E)), ˆca = φac(fdr(Qr, pr, E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' (2) Attributed to the relation boxes, the decoder of Interaction- level Predictor is guided to enlarge the receptive field (as shown in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The relation queries Qr can pay at- tention to the entire area where human and object interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Thus, the predictor φac can predict a more accurate relation class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In addiction, to match the relation class information ˆca with the aforementioned human-box ˆbh, object-box ˆbo and object-class ˆco from the Instance-level Predictor, we ditch the complex matching method like HO pointer in HOTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Instead, we just match the relation class information ˆca and the instances information ˆbh etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' one by one in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Specifically, for a pair of instances {ˆbh i ,ˆbo , ˆco i , iϵNq}, ˆca i is the corresponding relation class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In this way, the instance- level query vectors Qp and the relation-level query vectors Qr represent the same human-object interaction, but have the ability to focus on different receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Loss Functions The overall loss functions consist of the instance-level loss and relation-level loss, applied to Instance-level Predic- tor and Relation-level Predictor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The instance- level loss supervises the Instance-level Predictor to pre- dict instance-level target, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=', human-box, object-box, and object-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The relation-level loss assists the Relation- level Predictor to predict relation-class and relation-box from the larger receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='1 The Instance-level loss function LIL supervises the instance information, including human- box ˆbh, object-box ˆbo and object-class ˆco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The instance- level loss function consists of human-box regression Lhr, object-box regression Lor and object-class classification Loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Lhr and Lor are standard bounding-box regression loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' L1 loss, to locate the position of human and ob- ject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Loc is a classification loss to classify the categories of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The loss functions can be defined as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Lhr = 1 N N � i ||ˆbh i − bh i ||, Lor = 1 N N � i ||ˆbo i − bo i ||, Loc = 1 N N � i CE(ˆco i , co i ), (3) where CE is cross entropy loss, co i is the ground truth of object class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The instance-level loss function LIL can be defined as: LIL = Whr ∗ Lhr + Wor ∗ Lor + Woc ∗ Loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' (4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='2 The Relation-level loss function LRL supervises the relationship information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=', the rela- tion class ˆca, primarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In addition, auxiliary relation boxes are also supervised to pay attention to the entire area where the interaction happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Thus, the Relation-level loss func- tion consists of relation-box regression Lur, relation-box consistency loss Luc and relation-class loss Lac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The Lac is a classification loss to classify the categories of the interac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The relation-box regression loss function Lur is a L1 loss to resemble the predicted relation boxes and its ground- ing truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The grounding truth of relation boxes is the outer bounding box of human and object boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The relation-box regression loss function helps the Relation-level Predictor to be aware of the relation feature of human and object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The consistency loss Luc are used to keep the consistency of ˆbh, ˆbo and ˆbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Specifically, a pseudo relation box ˆbho is gener- ated by taking the outer bounding box of ˆbh and ˆbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Then, an L1 loss resemble ˆbu and ˆbho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' With the relation box, the relation-class loss can supervise better relation semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Lur = 1 N N � i ||ˆbu i − bu i ||, Luc = 1 N N � i ||ˆbu i − ˆbho i ||, Lac = 1 N N � i SigmoidCE(ˆca i , ca i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' (5) The relation-level loss function LRL can be defined as: LRL = Wur ∗ Lur + Wuc ∗ Luc + Wac ∗ Lac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' (6) In all, the overall loss fucntion L can be denoted as: L = LIL + LRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' (7) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Inference for HOI Detection The inference process of our PR-Net can be divided into two parts: the calculation of the HOI predictions and the Trident-NMS post-processing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' HOI Prediction To acquire the final HOI detection results, we need to predict human bounding box, object bounding box, and object class using both instance-level predictions and relation class and relation box using relation-level pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Based on the above predictions, we can calculate the final HOI prediction score as below: shoi i = {maxkso i (k)} ∗ srel i (8) Where maxkso i (k) means the most probable class score of the i-th output object from instance-level predictor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' srel i means the multi-class scores of the i-th output interaction from relation-level predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Note that each human-object pair can only have one object with certain class, but there maybe exist multiple human-object interactions within one pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Trident-NMS For each predicted HOI class in one image, we choose to filter its duplicated predictions according to the above calculated HOI prediction scores with our pro- posed Trident Non Maximal Suppression(Trident-NMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In detail, if the TriIoU(i, j) between the i-th and the j-th HOI prediction is higher than the threshold Thresnms, we will filter the prediction which has a lower HOI score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' And the calculation of TriIoU(i, j) is as below: TriIoU(i, j) =IoU(bh i , bh j )Wh × IoU(bo i , bo j)Wo × IoU(brel i , brel j )Wrel (9) Where IoU(bh i , bh j ), IoU(bo i , bo j), IoU(brel i , brel j ) repre- sent the Interaction over Union between the i-th and the j- th human boxes, object boxes and relation boxes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Wh, Wo, Wrel represent the weights of Human IoU, Object IoU and Relation IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Datasets and Evaluation Metrics We evaluate our method on two large-scale benchmarks, including V-COCO [10] and HICO-DET [3] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' V- COCO includes 10,346 images, which contains 16,199 hu- man instances in total and provides 26 common verb cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' HICO-DET contains 47,776 images, where 80 ob- ject categories and 117 verb categories compose of 600 HOI categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' There are three different HOI category sets in HICO-DET, which are: (a) all 600 HOI categories (Full), (b) 138 HOI categories with less than 10 training instances (Rare), and (c) 462 HOI categories with 10 or more training instances (Non-Rare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Following the standard protocols, we use mean average precision (mAP) in HICO-DET [4] and role average precision (AProle) in V-COCO [10] to report evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Implementation Details We use ResNet-50 and ResNet-101 [13] as a backbone feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The transformer encoder consist of 6 transformer layers with multi-head attention of 8 heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The number of transformer layers in Instance-level Predic- tor and Interaction-level Predictor is both set to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The reduced dimension size of visual memory is set to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The number of instance-level and relation-level queries is set to 100 for both HICO-Det and V-COCO benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' The human, object and relation box FFNs both have 3 linear layers with ReLU, while the object and relation category FFNs have one linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' During training, we initial- ize the network with the parameters of DETR [2] trained on the MS-COCO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' We set the weight coefficients of bounding box regression, Generalized IoU, object class, relation class and consistency loss to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='5, 1, 1, 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='5, respectively, which follows QPIC [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' We optimize the network by AdamW [26] with the weight decay 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' We train the model for 150 epochs with a learning rate of 10−5 for the backbone and 10−4 for the other parts decreased by 10 times at the 100th and the 130th epoch respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' All experiments are conducted on the 8 Tesla A100 GPUs and CUDA11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='2, with a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' We select 100 detection results with the highest scores for validation and then adopt Trident-NMS to filter results further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Overall Performance We summarize the performance comparisons in this sub- section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Performance on HICO-DET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Table 1 shows the per- formance comparison on HICO-DET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Firstly, the detection results of our PR-Net are the best among all approaches un- der the Full and Non-Rare settings, demonstrating that our method is more competitive than the others in detecting the most common HOIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' It is noted that PR-Net is also pre- eminent in detecting rare HOIs (HOI categories with less than 10 training instances), because our parallel reasoning network can migrate the non-rare knowledge into a rare do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Besides, our PR-Net obtains 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='86 mAP on HICO- DET (Default Full), which achieves a relative gain of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='8% compared with the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' These results quantitatively show the efficacy of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Performance on V-COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Comparison results on V- COCO in terms of mAProle are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' It can be seen that our proposed PR-Net has a mAP(%) of 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='4, obtaining the best performance among all approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Al- though we do not adopt previous region-based feature learn- ing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=', RPNN [41], Contextual Att [34]), or employ ad- 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Results on HICO-DET [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' “COCO” is the COCO pre- trained detector, “HICO-DET” means that the detector is further fine-tuned on HICO-DET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Default Full Method Detector Backbone Full Rare Non-Rare CNN-based VCL [14] COCO ResNet-50 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='43 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='55 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='29 VSGNet [31] COCO ResNet-152 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='80 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='05 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='91 DJ-RN [21] COCO ResNet-50 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='34 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='53 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='18 PPDM [24] HICO-DET Hourglass-104 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='73 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='78 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='10 Bansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' [1] HICO-DET ResNet-101 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='96 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='43 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='62 TIN [23]DRG HICO-DET ResNet-50 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='17 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='02 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='61 VCL [14] HICO-DET ResNet-50 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='63 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='21 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='55 GG-Net [40] HICO-DET Hourglass-104 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='47 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='48 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='60 IDNDRG [22] HICO-DET ResNet-50 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='29 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='61 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='39 Transformer-based HOI-Trans [42] HICO-DET ResNet-50 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='46 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='91 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='41 HOTR [18] HICO-DET ResNet-50 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='10 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='34 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='42 AS-Net [5] HICO-DET ResNet-50 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='87 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='25 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='25 QPIC [30] HICO-DET ResNet-50 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='07 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='85 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='23 PR-Net (Ours) HICO-DET ResNet-50 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='17 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='66 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='82 PR-Net (Ours) HICO-DET ResNet-101 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='86 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='03 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='30 ditional human pose (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=', PMFNet [32], TIN [23]), our method outperforms these approaches with sizable gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Besides, our method achieves an absolute gain of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='6 points, a relative improvement of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='1% compared with the baseline, validating its efficacy in the HOI detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Performance comparison on V-COCO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Method Backbone Network APS1 role APS2 role CNN-based VSGNet [31] ResNet-152 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='0 PMFNet [32] ResNet-50-FPN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='0 PD-Net [39] ResNet-152-FPN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='6 CHGNet [33] ResNet-50-FPN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='7 FCMNet [25] ResNet-50 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='1 ACP [20] ResNet-152 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='23 IDN [22] ResNet-50 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='3 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='3 GG-Net [40] Hourglass-104 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='7 DIRV [7] EfficientDet-d3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='1 Transformer-based HOI-Trans [42] ResNet-101 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='9 AS-Net [5] ResNet-50 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='9 HOTR [18] ResNet-50 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='4 QPIC [30] ResNet-50 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='0 PR-Net (Ours) ResNet-50 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='5 PR-Net (Ours) ResNet-101 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Ablation Analysis To evaluate the contribution of different components in our PR-Net, we first conduct a comprehensive ablation analysis on the HICO-DET dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Next, we analyze the impact of the number of different-level predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' At last, we analyze the effects of different post-processing manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Contribution of different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Compared Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Ablation analysis of the proposed PR-Net with the back- bone of ResNet-101 on HICO-DET test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Parallel Predictor means we parallelly predict instance-level locations and relation- level semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Consistency Loss means we constrain the union box of the human-object pair and the relation box to be consis- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Trident-NMS means duplicate filtering through human, ob- ject, and relation bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Parallel Predictor Consistency Loss Trident-NMS HICO-DET Full Rare NonRare 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='90 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='92 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='69 ✓ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='62 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='43 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='47 ✓ ✓ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='87 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='59 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='14 ✓ ✓ ✓ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='86 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='03 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='30 with our baseline [30], the performance improvements of our PR-Net are from three components: Parallel Predictor, Consistency Loss, and Trident-NMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' From Table 3, we can know the contribution of different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Among these components, Parallel Predictor is our core approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' With that, we can observe a noticeable gain of mAP in HICO-DET by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' It proves that the parallel reasoning structure can significantly improve instance localization and interaction understanding for an HOI detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Additionally, we design a consistency loss between the union box of the human-object pair and the relation box, which can contribute about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='25 mAP gain in the HICO- DET test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' It shows that it is meaningful and helpful to constrain the union region of instance-level predictions and the relation region of relation-level predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' At last, we design a more effective post-processing technique named Trident-NMS, which brings about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='0 mAP gain in the HICO-DET test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' It reveals that the set-prediction method can also benefit from duplicate filtering technique and post-processing technique like NMS is essential for HOI detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Impacts of different numbers of parallel predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In our PR-Net, two parallel predictors are significant for HOI detection, and we detailedly analyze the impact of different numbers of parallel predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' From Table 4, we can know that equipped with three layers of instance-level predictor and relation-level predictor, our PR-Net can acquire the best mAP performance in the HICO-DET test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' It reveals that our PR-Net can significantly outperform the baseline QPIC [30] without additional computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Interestingly, we can also observe that even with only one layer of parallel predictors, our PR-Net can also outperform the baseline equipped with a six-layer predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Effects of different implements of Trident-NMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In Table 5, we analyze the effects of different implements of Trident-NMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' We find that Product-based Trident- NMS performs better than Sum-based Trident-NMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 6 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Ablation analysis of the number of instance-level predic- tor Ndec and the number of relation-level predictor Nreldec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' approaches Backbone Ndec Nreldec Full Rare Non-Rare QPIC(Baseline) [30] ResNet50 6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='07 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='85 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='23 PR-Net(Ours) ResNet50 1 1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='64 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='18 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='27 PR-Net(Ours) ResNet50 3 3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='17 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='66 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='82 PR-Net(Ours) ResNet50 6 6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='04 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='87 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='89 QPIC(Baseline) [30] ResNet101 6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='90 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='92 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='69 PR-Net(Ours) ResNet101 1 1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='26 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='27 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='34 PR-Net(Ours) ResNet101 3 3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='86 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='03 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='30 PR-Net(Ours) ResNet101 6 6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='52 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='04 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='16 Additionally, we can also observe that when the weight of Human-IoU in TriIoU increases, the HOI detection performance will be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' This reveals that human box duplication is more frequent than that of object box or relation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In summary, with either the Product-based or Sum-based TriIoU calculation, we should pay more attention to the non-maximal suppression of the human box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Ablation analysis of the Trident-NMS module on HICO- DET test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Product means we calculate TriIoU by multiply- ing these weighted Human-IoU, Object-IoU, and Relation-IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Sum means we calculate TriIoU by adding all these weighted Human-IoU, Object-IoU, and Relation-IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Wh, Wo, Wrel rep- resent the weights of Human-IoU, Object-IoU and Relation-IoU respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Thresnms means the threshold of non-maximum suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Product Sum Wh Wo Wrel Thresnms HICO-DET Full Rare NonRare 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='87 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='59 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='14 ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='33 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Visualization of features Using the t-SNE visualization technique [27], we visual- ize 20000 samples of output feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' These object and inter- action features are extracted from the last layer of Instance- level Predictor and Relation-level Predictor in our PR-Net, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' From the Figure 3, we can observe that our PR-Net can obviously distinguish different class of objects and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Interestingly, from this visualization of features, our PR-Net can even learn better the complex interaction representations then the object representations which benefits from our advantageous parallel reasoning ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Visualization of object features and relation features on HICO-DET dataset via t-SNE technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Left is object features and right is relation features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Qualitative Examples From Figure 4, we can observe that our PR-Net can accu- rately detect both human box, object box, and relation box as well as their corresponding interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' From the first row and second column of Figure 4, we can know that our PR-Net can precisely distinguish which man is riding the horse in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' From the second row and third column of Figure 4, our PR-Net can precisely detect those subtle and indiscernible HOIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In summary, our PR-Net can cor- rectly detect those complex and hard HOIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Visualization of some HOI detection examples (Top 1 result) detected by the proposed Parallel Reasoning Network on the HICO-DET test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Conclusion In this paper, we propose a new Human-Object Inter- action Detector named Parallel Reasoning Network(PR- Net), which consists of an instance-level predictor and a relation-level predictor, to alleviate the problem of in- consistent focus in attentive fields between instance-level and interaction-level predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In addition, our PR- Net achieves a better trade-off between instance localiza- tion and interaction understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Furthermore, equipped with Consistency Loss and Trident-NMS, our PR-Net has achieved competitive results on two main HOI benchmarks, validating its efficacy in detecting Human-Object Interac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 7 100 100 75 75 50 50 25 25 0 0 25 25 50 50 75 75 100 100 100 75 50 25 0 25 50 75 100 100 75 50 25 0 25 50 75 100uman human lie_on chair ridehorse LOG numan sit on benchReferences [1] Ankan Bansal, Sai Saketh Rambhatla, Abhinav Shrivastava, and Rama Chellappa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Detecting human-object interactions via functional generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' AAAI, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 6 [2] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' End-to- end object detection with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In ECCV, 2020.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In Proceedings of the IEEE Winter Conference on Applications of Computer Vision, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 1, 5, 6 [5] Mingfei Chen, Yue Liao, Si Liu, Zhiyuan Chen, Fei Wang, and Chen Qian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' Reformulating hoi detection as adaptive set prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' In CVPR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQf5QWX/content/2301.03510v1.pdf'} +page_content=' 1, 2, 6 [6] Mingfei Chen, Yue Liao, Si Liu, Zhiyuan Chen, Fei Wang, and Chen Qian.' metadata={'source': 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b/3dAyT4oBgHgl3EQf1vn1/content/tmp_files/2301.00742v1.pdf.txt @@ -0,0 +1,3143 @@ +UDK 539.125.17; 539.126.17 +Ya. D. Krivenko-Emetov* +Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kyiv, Ukraine +National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Kyiv, Ukraine +*Corresponding author: y.kryvenko-emetov@kpi.ua; krivemet@ukr.net +MULTICOMPONENT VAN DER WAALS MODEL OF A NUCLEAR FIREBALL IN THE +FREEZE-OUT STAGE + +Abstract. A two-component van der Waals gas model is proposed to describe the hadronic stages of the +evolution of a nuclear fireball in the cooling stage. At the first stage of hadronization, when mesons +dominate, a two-component meson model ( +0 + - and + + -mesons) with an effective two-particle interaction +potential of a rectangular well is proposed. At the late-stage hadronization, when almost all mesons have +decayed, a two-component nucleon model of protons and neutrons is proposed with the corresponding +effective rectangular nucleon potential. The saddle point method has been applied for analytical calculations +of the partition function. This made it possible to uniformly obtain analytical expressions for both the +pressure and density, taking into account the finite dimensions of the system, and the analytical expressions +for chemical potentials. It is assumed that the proposed models and derived formulas can be used to analyze +experimental data connected to the quantitative characteristics of the particle yields of different types in the +final state from the hadronic stages of the evolution of a nuclear fireball, as well as to determine the critical +parameters of the system in high-energy nucleus-nucleus collisions. +Keywords: fireball, freeze-out, van der Waals equation, effective nuclear capability, Grand Canonical +Ensemble, pressure fluctuation, quark-gluon plasma, experimental data + +Introduction + +Experimental observations of an elliptical flow in non-central collisions of heavy nuclei at high +energies provide much evidence that in these collisions of nuclei a state of quark-gluon plasma +appears and thermalization occurs, which is associated with the fact that particles collide with each +other more than once. For this state, one can introduce the concept of temperature, viscosity, +density, and other thermodynamic quantities that characterize the substance. In these terms, one can +describe and study the phenomena that occur during the cooling of a hadron gas formed after a +phase transition from the state of a quark-gluon plasma. It is believed that at a critical temperature +( +150 + +T + MeV, the so-called Hagedorn temperature), hadrons "melt" and a phase transition of the +hadron gas (hadron matter) into the quark-gluon phase occurs. Therefore, in recent decades, +statistical models of hadron gas have been actively used to describe the data of the Large Hadron +Collider (LHC), the Relativistic Heavy Ion Collider (RHIC), and even earlier to describe the data of + +Alternating Gradient Synchrotron (AGS) and Super proton synchrotron (SPS), on the particle yields +in a relativistic nuclear-nuclear ( +A +A  +) collision at high energies [1, 2]. The van der Waals (vdW) +model, taking into account hadron-hadron interactions at short distances, is especially useful in this +description [3-10]. This is due to the fact that taking into account the effect of repulsion (off +volumes) leads to the prevention of an undesirably high density at high temperatures, which appears +in ideal gas models [11]. In addition, collisions of heavy high-energy ions in the LHC produce a +large number of different types of particles. The number of these particles is not fixed. Therefore, +the formalism of the Grand Canonical Ensemble (GCE) is one of the adequate mathematical +formalisms for these phenomena. In this case, the thermodynamic quantities do not depend on the +number of particles, but on the chemical potentials. For many years, researchers have proposed and +applied different versions of vdW models. These models have been mainly used to describe +experimental data on the number of particles at high energies, when tens or even hundreds of +hadrons of different types can be generated. Naturally, this generation process is limited only by the +energy of collisions. +Among these models, the model proposed in [11] should be noted. In this model, the +phenomenological parameters of the radii of the hard-core +ii +R and +ij +R are introduced, which +significantly changes the number of the yield particles with different types +i +N (i is the particle +sort) and is mainly confirmed by experimental data. In order to describe more subtle effects in the +dependence of the hadronic gas pressure on density, various authors (e.g. [12, 13]) proposed the +development of this model [11]. Here, the effects of attraction between hadrons at a large distance +have been taken into account, which leads to the appearance of a corresponding contribution to the +pressure as +2 +an +Pattr + +~ + ( n is the density). For a multicomponent gas, the parameter a , +corresponding to attraction, transforms into parameters +ij +a , and the repulsive parameter b +transforms into parameters +ij +b . At the same time, the parameters of the effective potential +corresponding to attraction and repulsion depend on the effective radii of repulsion +0 +iR and +attraction +iR as follows: + + +ij +ij +ij +ij +b +c +u +a + +0 +~ + , + + +3 +0 +0 +3 +2 +j +i +ij +R +R +b + + + +, + +3 +3 +2 +j +i +ij +R +R +c + + + +, +ij +u0 is the depth +of the effective potential well [12]. +However, even this vdW model cannot be developed properly when considering a finite nuclear +system. So, in the case of nuclear collisions, a nuclear fireball with dimensions +10 +7  + + +~ +r + Fm is +observed. In a fairly general case, this problem (without taking into account the effects of reflection +from the system wall) has been solved for a two-component system. In this case, the GCE +formalism leads to the use of a double sum, which, in turn, can be transformed into a + +multidimensional integral, which can be integrated by the saddle point method in the vicinity of a +saddle point with coordinates + + +2 +1 N +N , + [12]. +Of course, it would be good to apply this model to the analysis of experimental data obtained for +collisions of heavy nuclei at CERN. One of the variants of such a model was presented in [14] in a +concise form. It was believed that the collision energies were not high enough, and one could limit +oneself to only two varieties: protons and neutrons. It was assumed that the characteristic +temperatures do not exceed the temperatures at which new particles can be generated. The +temperatures of the nuclear fireball are of the order of +140 +135  +~ +T + MeV in units +1 + +B +k + (freeze- +out phase, see Fig. 1) at the stage after the transition to the hadronic gas phase. Therefore, the model +itself should have a transparent nonrelativistic limit, taking into account the law of conservation of +the total number of nucleons without the generation of new particles. +The successive stages of the evolution of a nuclear fireball are schematically shown in Fig. 1 +[15]. From left to right: two touching ultrarelativistic nuclei; the state of a hot and superdense +nuclear system; quark-gluon phase; hadronization and chemical freeze-out; kinetic freeze-out. + + +Fig. 1. The successive stages of the evolution of a nuclear fireball + +A more detailed and consistent description of the mathematical apparatus of the model [14] is +proposed in the article. Some more subtle effects are estimated (additional corrections for pressure, +density, and root-mean-square (RMS) fluctuations). A new two-component meson model [16] has +been proposed for the case of temperatures above the production threshold ( +135 + +T + MeV ), when +the number of mesons is not conserved. + +1 One-component vdW gas + +According to various estimates, the lifetime of a nuclear fireball is much longer than the +characteristic nuclear interaction time +23 +22 +10 +10 + + + +~ +'t + c. (see Fig. 1). It is of the order of the +relaxation time +23 +21 +10 +10 + + + + ~ + c for sufficiently small local fireball volumes (subsystem). + +0 +10.01 +1 +10 +1100 +t (fm/c)Therefore, we will assume that at each moment of time exceeding the relaxation time, a local +statistical equilibrium has time to be established in the subsystem. That is, such a local fireball +region is quasi-stationary, and therefore the methods of statistical physics can be applied to it. Since +all thermodynamic potentials, as well as entropy and volume, are additive (extensive) quantities, +therefore, the corresponding potentials (values) of the entire system (fireball) can be defined as the +sum of the corresponding thermodynamic potentials of quasi-closed subsystems [17]. Then, at each +moment of time, one can give a standard representation of the partition function of a rarefied +quasi-ideal van der Waals gas in the canonical ensemble (CE) for such quasi-closed subsystems. In +the approximation of pair interaction and condition   +1 + +V +N +T +B +, this quantity has the form [17]: + + + + +  + +N +N +N +T +B +V +m +T +N +N +T +V +Z + + + +, +! +, +, +1 +, + + + +(1) +where, respectively, N and m are the number and mass of particles, V and T are the volume and +temperature of the gas. +Formula (1) uses the notation [11]: + + + + +T +m +K +T +m +dp +T +p +m +p +m +T +2 +2 +2 +0 +2 +2 +2 +2 +2 + +exp +2 +1 + + + + + + + + + + + + + + + + + +, +, + +(2) +where + z +K2 + is the modified Bessel function, and the second virial coefficient in (1) has the form: +  + + + + dV +T +U +T +B + + + + + +0 +exp +1 +2 +1 + + + + +(3) +and includes pairwise interaction of particles, +  + +j +i +ij +U +U +. +In relativistic limit +T +m  + one can easy obtain, given the asymptotes of the Bessel function: + + + + + + + + + + + + + + + + +T +m +mT +m +T +exp +2 +2 +3 +~ +, +. +This formula further leads to the effect of exponential suppression of the particle yields with +large mass, which is important in the study of quark-gluon plasma. +The pressure in the system is easy to find from the partition function: + + + + + + + N +T +B +V +TN +N +T +V +Z +V +T +N +T +V + + + + + +, +, +, +, +ln +P +. (4) +Note that if the Stirling formula is used in the partition function for the factorial: + +N +e +N +N +N + + +2 +! +, then the final pressure formula (4) will not change. +*THE MODEL. According to the above, all calculations for subsystems will be carried out by +methods of statistical physics. This implies, in addition to the local statistical equilibrium, the + +fulfillment of the condition of the statistical (thermodynamic) boundary: +A +N +N  +, where +A +N is the +Avogadro constant. +In this case, the final formulas can be applied to the nuclear fireball due to the indicated +additivity of thermodynamic potentials and volume. Since the number of generated particles in a +fireball is about 4-6 thousand during high-energy nucleus-nucleus interactions, this assumption is +more or less justified at the first stages of its evolution. +Of course, at later stages of evolution, since the number of nucleons at the nonrelativistic +boundary is limited by the baryon number conservation law and equal to ( heavy element nuclei +collide with mass number ), this assumption is, in general, somewhat doubtful. +Of course, at later stages of evolution, this assumption is, in general, somewhat doubtful, since +the number of nucleons n the nonrelativistic limit is confined by the baryon number conservation +law and is equal to +300 +200  + +N + (the nuclei of heavy elements collide with the mass number +200 +~ +A +). However, the practical application of the van der Waals equation quite often goes +beyond the conditions under which the virial approximation has been obtained as experience +shows. Considering this fact, as well as the fact that we can always restrict ourselves to the first +stage (see Section 3), we believe that our approximation is sufficiently justified. +Although calculations by the saddle point method are made when +  +0 + +T +B +, however, for the +reasons stated above, the final formulas are extended to region where the second virial coefficient +  +T +B + is not necessarily negative. +From the partition function + + +N +T +V +Z +, +, + one can also get: free energy + +  +N +T +V +F +, +, + + + + + +N +T +V +Z +T +, +, +ln + + +, chemical potential + + + + + + + + +  + + + + + + + + + + + + + + + + +V +N +T +B +T,m +V +N +T +N +N +T +V +F +2 +ln +ln +, +, + + +(5) +and the derivative of the chemical potential which in the statistical limit has the form: + + + + + +  +  +V +T +T +B +V +T +T +B +N +T +N +V +V +P +N +2 +2 +lim +A +N +N +2 + + + + + + + + + + + + + + + +. + +(6) +Then, we obtain the Grand partition function (GPF) + + + +, +,T +V +Z + from the partition function + + +N +T +V +Z +, +, + taking into account the above physical considerations (see, e.g., [18, 19]): + + + + +N +T +V +Z +T +N +T +V +N +, +, +, +, + + + + + +  + + +exp +Z +. + + + + +(7) +At high temperatures (which, for example, are realized during collisions of heavy ions in the +GCE, and +' +dN +T +N + + +) one can turn from the sum to the integral using the Euler-Maclaurin + +formula. In this case, the first integral term remains and the logarithm of the statistical sum is +introduced into the exponent. Let's denote this indicator by + +' +N + +: + + +  + + + + + + + + + + + + + + + + + + + + +0 +0 +exp +ln +exp +' +' +' +, +' +' +, +, +N +dN +T +T +N +V +Z +N +dN +T +T +V +Z +. + +(8) +Further integration is performed by the saddle point method [16], since at high temperatures the +integrand has a strongly pronounced maximum. We obtain the following expression for finding the +maximum point ( + +N ) for the integrand from the extremum condition imposed on the saddle point: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +N +N +N +N +N +N +N +T +V +Z +N +T +N +, +, +ln +, + + + +(9) + + + + + + + + + + +m +T +V +N +T +N +, + + + + + + + +ln +ln +, + + + + +(10) +where + + is the chemical potential at the saddle point. +As a result, we obtain: + + + + + + +  + + + + + + + + + + + + + + + + + + + + + + + + +N +V +N +T +B +T +V +e +N +N +m +T +N +N +T +V +N +N +N +N +N +exp +2 +2 +2 +2 +, +, +, +Z +, (11) +where the second derivative of the exponent  at the saddle point is defined as follows: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +N +N +N +N +N +N +N +N +N +N +T +V +Z +N +T +N +T +N +N +2 +2 +2 +2 +2 +2 +ln +2 +, +, + +  +  +0 +2 +2 +1 +1 +1 +2 +2 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +V +T +B +V +T +B +N +N +N +T +N +T +N +N +N +N +N +, +because + + + + + + + + + + + + + + + +N +N +T +N +N +N +1 +2 +2 +, + + + + + + + + + + + + + + + + + + + + + + + +N +N +N +N +N +N +T +V +Z +N +T +2 +2ln +1 +, +, +. +The pressure in the GCE is defined as follows in terms of the temperature and the logarithm of +the GPF (see, e.g., [18]): + + + + +V +T +V +T +T +P + + + +, +, +, +Z +ln +. + + + + +(12) +It's easy to show that pressure (12), taking into account (5) and (11), can be rewritten as follows: + + +  + + +  +  + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +V +T +B +T +B +T +V +N +T +B +T +T +P +N +N +2 +ln +1 +2 +ln +1 +2 +2 +, +, (13) +where the saddle point + + V +T +V +N + + + + + +, +, + is defined according to (5) and (10) as +  + + +  + + +T +N +T +m +N + + + + + +exp +, +. The parameter  can be eliminated from equation (13) using the +definition of density, which in the thermodynamic limit turns into the well-known formula [1]: + + + +  + + +  + + + + + + + + + + + + + + + + +T +B +V +T +B +T +P +n +2 +1 +2 +1 +2 +1 +, +. + + +(14) +In the thermodynamic limit ( +A +N +N  +, + + +V +) the chemical potential of the saddle point + + +from (10) when +  + + +V +N +T +B +N +N +2 +1 + + + turns into the chemical potential  ( + + + +), which is +determined by the well-known thermodynamic equation (5). +Both equations (13) and (14) in parametric form (the saddle point  acts as a parameter) +determine the relationship between pressure P , temperature T , and density n . We obtain the state +equation in GCE by excluding explicitly this parameter from the system of equations (13) and (14): + + +  + + +dP +n +T +B +Tn +n +T +P + + + + +1 +, +, +. + + + + (15) +Of course, the resulting state equation is implicitly a parametric equation, since the saddle point + (and, hence, and n ) determines the chemical potential  according to (5) and (10). +It is important that the resulting formula takes into account the contribution to the pressure of the +finite volume of the system, +s +V . This contribution naturally vanishes in the thermodynamic limit, +where there is no difference between CE and GCE. +Only the last of the three terms remains for large but finite volumes, as in the nuclear fireball +model discussed in Sec. 4: +  +  + + + + +  + + +s +V +V +V +n +T +B +T +n +T +B +n +T +B +V +T +dP +s +2 +ln +ln +1 +2 + + + + + + +lim +. (16) +RMS fluctuations of pressure and density calculated by known formulas (see, e.g., [20]) give +estimates of the found corrections to the corresponding quantities: + + + + +  + + +n +T +B +V +n +T +n +P +V +Tn +P +s +2 +1 +2 +2 + + + + + + +~ +, + + + (17) + + + + +  + + +n +T +B +V +N +V +N +T +n +N +N +2 +1 +1 +2 +2 +2 + + + + + + + + + + +~ +. + + + + (18) + +2 Two-component vdW gas + +Let us consider the procedure for taking into account the excluded volume and attraction in the +vdW model for the case of a two-component hadron gas of two types of particles “i ” and “ j ”. +1 +N +and +2 +N are the number of particles of the first and second sorts. In this case, the partition function +has the form: + +  +2 +1 N +N +T +V +Z +, +, +, + + +  + + + + +  + + + + +  +  + + + + +T +U +r +d +r +d +T +m +T +m +N +N +N +k +k +N +l +l +N +N +12 +1 +2 +3 +1 +1 +3 +2 +1 +2 +1 +exp +! +! +1 +2 +1 +2 +1 + + + + + + + + + + + +, +, +, (19) +This expression for the pair-interactions approximation ( + + + + +12 +123 +U +U + +) and a weakly ideal gas +( +1 +2 + +V +NB +) can be rewritten as follows [11]: + + +  + + +  + + + + + +2 +1 +2 +1 +2 +1 +2 +1 +! +! +1 +N +N +T +m +T +m +N +N +N +N +T +V +Z +, +, +~ +, +, +, + + + + + +2 +1 +1 +12 +2 +22 +2 +21 +1 +11 +N +N +N +B +N +B +V +N +B +N +B +V +~ +~ + + + + + + +. + + + (20) +Here we have introduced the notation: +jj +ii +ij +ii +ij +B +B +B +B +B + + 2 +~ +. +The two-particle partition function + + +2 +1  + , +, +,T +V +Z + in GCE is expressed in terms of the +two-particle partition function  + +2 +1 N +N +T +V +Z +, +, +, + in CE [12, 17], as: + + + +  + + + + + + + + + + +2 +1 +2 +0 +1 +0 +2 +2 +1 +2 +2 +1 +1 +e +d +d +' +, +' +, +, +' +' +, +, +, +' +' +N +N +T +V +Z +N +N +T +T +V +N +N +Z + + + + + +2 +1 +2 +0 +1 +0 +2 +exp +d +d +' +, +' +' +' +N +N +N +N +T + + + + + + +. (21) +Integration of (21) by the saddle point method [21] leads us to the following result: + + + + + + + + + + + + + + + + + + + + + + + +2 +1 +2 +2 +1 +1 +2 +1 +exp +2 +N +N +T +V +Z +T +N +N +N +N +, +, +, +, +~ +Z +, where the coordinates of the saddle +point + +i +N ( +2 +1, + +i +) are found from the extremum conditions: + + +0 + + + + + + + +i +j +i +N +N +N , +, + + +22 +21 +12 +11 +2 +1 +c +c +c +c +N +N +det +, + + + + + + +, + + + + + + + + + + + + + + + + + + + + + + +N +N +j +i +j +i +ij +N +N +N +N +c +, +2 +. +Substituting the value of the partition function into the definition of pressure in the GCE [18], we +obtain the following expression [12]: + + + + + + + + + + + + + + + + + + + + + + + + + + + +V +C +B +B +B +B +T +V +T +V +T +T +P +2 +ln +ln +2 +1 +21 +12 +22 +2 +2 +11 +2 +1 +2 +1 +2 +1 +2 +1 +~ +~ +~ +, +, +, +, +, +Z +, (22) +where +21 +2 +12 +1 +22 +2 +11 +1 +B +B +B +B +C +~ +~  + + + + + +. +Using such a mathematical apparatus, one can organically introduce the law of conservation of +chemical potentials. The latter are related to the condition imposed on the integrand when finding +the saddle point. In the thermodynamic limit the chemical potential determined by the extremum +condition coincides with the definition of the chemical potential itself: + + +i +j +i +i +i +N +N +N +T +V +F + + + + + + +, +, +, +, + +where  + + + + + +2 +1 +2 +1 +ln +N +N +T +V +Z +T +N +N +T +V +F +, +, +, +, +, +, + + + is the definition of free energy (10). +We get from the definition of density + + + + + + + + +ji +ij +j +ii +i +i +i +j +i +i +B +B +B +T +P +n +~ +~ +~ +, +, + + + + + + + + + + + + +2 +1 +. (23) +The virial expansion (22) can be rewritten, taking into account (23), as a two-component vdW +equation in the approximation +1 + +V +N +b +i +ij + and + + +1 + +ij +ij +Tb +a +): + +  + + +2 +1 +2 +1 +n +n +T +P +, +, +, +, + + + + + +dP +n +a +n +a +n +n +a +n +a +n +n +b +n +b +Tn +n +b +n +b +Tn + + + + + + + + + + + +1 +12 +2 +22 +2 +2 +21 +1 +11 +1 +1 +12 +2 +22 +2 +2 +21 +1 +11 +1 +1 +1 +~ +~ +~ +~ +, (24) +where dP , according to (22), takes into account the finite size of the fireball. When formula (24) +was derived, the expression +T +a +b +B +ij +ij +ij +~ +~ +~ + + + was used (see, e.g., [12]), and for each type of particles +the corresponding parameters of attraction and repulsion were introduced [11]: + + +jj +ii +ii +ij +ij +a +a +a +a +a + + + + +2 +~ +, + + +jj +ii +ij +ii +ij +b +b +b +b +b + + +2 +~ +,  is a phenomenological parameter reflecting the +complexity of the problem. + +3 The asymmetric two-component freeze-out model with non-conservation of the number +of particles + +The considering nucleus-nucleus collisions  + +A +A + have very high energies, more than 1 GeV +per nucleon. At the same time, mesons of different sorts dominate in the initial freeze-out stages. +Therefore, to describe the nucleus-nucleus interactions at this stage of the freeze-out above the +production threshold of new particles ( +135 + +T + MeV), we propose a generalization of the vdW +model to a medium-sized nuclear fireball [16]: + + + + + + + + + + + + + + + + + + + +A +r +b +a +V +V +V +f +f +f +3 +0 +2 +4 +3 +4 +3 +2 +~ +~ +~ +max +min +. +Here +2 +1 +1 +1 +0 +. +.  + +r + Fm, + + a +, + + b + are the mean semiaxes of the ellipsoid, and + + A + is the +mass number of nuclei left in the fireball after the collision. In our considerations we assume that +the fireball consists, mainly, of mesons, given that the number of nucleons is much less than the +number of mesons ( + + 300 +200 +~ +pn +N +5000 +4000  + ~ +N +). We neglect the contribution of other +particles. Therefore, we introduce the following additional natural assumptions. +1. The average internucleon energies do not exceed the production threshold of the heavy +mesons. Therefore, we restrict ourselves to two sorts of particles ("0" is the +0 + -meson, "+" is the + + -meson). + +2. Since + + -meson production reactions are twice as likely as +0 + -meson production reactions, +we assume that +n +kn +n + + + +0 +, where, +1 + +k +, +0 +n is the +0 + -meson density, and + +n is the + + -meson +density. This corresponds to a more probable production of the + + -mesons in reactions + + + + + + +n +d +d +p +, +0 + + + + + +p +d +d +p + than production of the +0 + -mesons. +3. We introduce the effective potential of the meson interaction + +j +i +U +, where  +  + +0, +, + + +j +i +. That +is, "(0+)" is the interaction of +0 + -mesons with + + -mesons, "(++)" is the interaction of + + -mesons + + + + + + + + + + + + + + + + + + + +r +R +R +if +R +R +r +R +R +if +u +R +R +r +if +U +j +i +j +i +j +i +j +i +j +i +j +i +0 +0 +0 +0 +0 +0 +, +. (25) +Since the effective rectangular potential a well leads to approximately the same values of +pressure and density as the real potential (see Fig. 2, where + + +U +U + +0 +, + + + + +0 +0 +u +). Therefore, the +real meson-meson potential (a) can be replaced by a similar effective rectangular potential (b). + +a b +Fig. 2. Meson-meson potential + +4. We accept that the +0 + -meson hard-core radius is much smaller than the + + -meson hard-core +radius: +0 +0 +0 + + R +R +. The radius of the hard-core of the + + -meson is assumed to be known. +Average pressure and density fluctuations are easily found within the framework of the proposed +model, similarly to formulas (18) and (19): + + + + + + +Tn +B +V +n +T +P +f + + + + + + +1 +~ +, + + + + + +(26) + + + + + + +Tn +B +V +n +V +n +f +f + + + + + + + + +1 +1 +~ +. + + + + +(27) +The following results are obtained (Fig. 3, Fig. 4). Such data have been used (Fig. 3): +3 +142, + +T + MeV, the effective radius of the + + -meson, +45 +0 +0 +, + + +R + Fm, and +0 + -meson, +01 +0 +0 +0 +, + +R + Fm, the average value of the volume of the meson fireball is taken as the value +600 +~ + + +f +V + Fm 3 , +5 +0. + +k +, the parameter of the potential depth, + + +100 +80 +0 +0 + + +~ +, +u + MeV. One can + +U来 +U*clearly see (Fig. 4) an increase in the correction + + P +dP + at low densities, which is typical in the +final stages of the freeze-out. + + +Fig. 3. Dependence of the meson pressure P (24) on the meson density +n +kn +n + + + +0 + for the +two-component asymmetric vdW model with correction (upper isotherm) and without correction (lower +isotherm) + + +Fig. 4. Ratio of the correction to pressure dP from the size of the meson fireball to the value of the RMS +pressure fluctuation + + P + (26) as a function of the meson density +n +kn +n + + + +0 + + +4 Two-component model of a nucleon fireball at the final stage of the freeze-out + +The average lifetime of mesons dominating in the initial stages of the freeze-out is relatively +short ( +16 +8 +10 +10 + +  + ~ + c). That's why they decay pretty quickly. Accordingly, baryons, namely +protons and neutrons, begin to dominate at the final stage of freezing. In addition, as shown above, +the effects of the finite volume size become noticeable at sufficiently low density values. This +formally corresponds to just such final stages of the fireball evolution. Therefore, despite a certain +doubt about the existence of a fireball at such late stages, when the boundary between the gas and +the aggregate of individual nucleons gradually disappears, to describe the nucleus-nucleus + +P(T=142.3, n), MeV/Fm-3 +10 +n, Fm-3 +0.05 +0.10 +0.20 +0.25 +0.30 +-10 +-20 +-30 +-40 +-50 FdP/

+2.0 +1.5 +1.0 +n, Fm-3 +0.05 +0.10 +0.15 +9.20 +0.25 +0.30interactions at the last stage of the freeze-out, which is below the production threshold of new +particles ( +135 + +T + MeV), in [14] the following generalization of the vdW model to the nucleon +fireball was proposed. We accept the following simplifications by analogy with the previous +section. +1. The average energies of internucleon collisions do not exceed the production threshold of +other hadrons. Therefore, we restrict ourselves to two varieties (“ p ” is the proton, “ n ” is the +neutron). +2. We take the relation between the density of protons and neutrons in the form +n +n +n +n +p + + + +following from the law of conservation of the baryon number, +A +N +Z + + +. +3. We assume that the nucleon composition of colliding nuclei is known as such +n +p +kn +n  +, where +1 + +k +, since heavy nuclei have an excess of neutrons. +4. The effective potential of the proton-neutron, proton-proton and neutron-neutron interactions, +which leads to approximately the same values of pressure and density as the real potential (Fig. 3), +can be represented by analogy to (25) as + +j +i +U +, where + + +n +p +j +i +, +, + +. +5. The hard-core radius of the proton is assumed to be known, +5 +0 +0 +, + +p +R + Fm. We accept that the +radius of the neutron is much less than the radius of the proton: +0 +0 +p +n +R +R  +. +We get from equation (27): + + + + +  +  +dP +n +a +n +n +n +k +Tn +T +P + + + + + + + + + + + + + + + + + + +2 +2 +2 +1 +1 +1 +, +, +, (28) +where + + +k +n +n + + + +1 +, k is a dimensionless quantity, +k +b +k +b +b +k +b +22 +2 +21 +12 +11 + + + + + +~ +~ +, +22 +21 +12 +11 +b +k +b +b +k +b + + + + + +~ +~ +, +k +b +b +b +b +b +b +k +b +kb +21 +12 +21 +22 +12 +11 +2 +22 +11 +~ +~ +~ +~ + + + + + +, + + +22 +21 +12 +2 +11 +a +k +a +a +k +a +a + + + + + +~ +~ +. +It follows from the condition +0 +1 +0 +2 +R +R  + that +11 +22 +b +b + +, + + + +. By analogy with Eqs. (18) and +(19), we find the corresponding average fluctuations of pressure and density. +Functional dependences for pressure, obtained by Eq. (28), and the ratio of dP to RMS pressure +fluctuations are shown in Figs. 5 and 6. + + + +Fig. 5. Dependence of nucleon pressure P (28) on nucleon density, +n +kn +n +n +p + + +, in the two-component +asymmetric vdW model with correction (upper isotherm) and without correction (lower isotherm) + + + +Fig. 6. The ratio of correction from the size of the nucleon fireball to pressure dP to the value of the +RMS pressure fluctuation + + P + depending on the density of nucleons, +n +kn +n +n +p + + + + +It can be seen that the correction dP makes a nonzero contribution to the total pressure also in +this case. On the other hand, it is negligibly small almost everywhere in comparison with the +contribution from fluctuations. The correction makes a contribution comparable to fluctuations only +in the region near zero density that is nonphysical for a nuclear fireball. But it can be neglected in +this region, as can be seen from Fig. 6. + +Summary + +The effect of taking into account the excluded volume and attraction is analyzed in the case of a +two-component gas: (i) +0 + - and + + -mesons; (ii) protons and neutrons. The calculations have been +performed in the Canonical and Grand Canonical ensembles by the saddle point method for a two- +component system. The particles interact with the potentials of the hard-core at short distances and +with relatively high potentials at large distances (effective attraction radii). For effective + +P(T=142.3. n), MeV/Fm-3 +10 +n, Fm-3 +0.05 +0.10 +N15 +0.20 +0.25 +0.30 +-10 +-20 +-30 +-40 +-50 FdP/

+1.0 +0.9 +0.8 +F +0.7 +n, Fm-3 +0.05 +0.10 +0.15 +0.20 +0.23 +0.30interparticle interactions of this type, an equation of state has been obtained with corrections that +take into account the finite dimensions of the nuclear fireball, as well as RMS fluctuations of +pressure and density. +The pressure correction disappears in the thermodynamic limit, when, according to statistical +physics, there is no difference between various statistical ensembles. The formulas for pressure and +density obtained by the saddle point method can be used to analyze experimental data concerning +the relative number of the yield particleshe of various sorts and critical parameters in high-energy +nuclear-nucleus collisions. A generalization of the presented vdW model to the asymmetric case of +a two-component model ( +0 + - and + + -mesons) with realistic parameters of the hard-core and +attraction has been proposed as an example of such a use. The ratio of the pressure correction to the +RMS value of pressure fluctuation is estimated for the case of an asymmetric two-component +meson fireball model. An increase in the correction has been found at low density values +corresponding to the final stages of freezing. +It is found that the contribution to pressure and relative fluctuations, taking into account different +radii and the finiteness of the nuclear fireball, is noticeable in the case of the meson model with +nonconservation of the number of particles. However, this correction can be neglected for the final +stages of the freeze-out, when nucleons begin to dominate (the model of Sec. 4). Therefore, the +developed model is applicable in the analysis of experimental data on the study of the initial meson +phase of a nuclear fireball (the model of Sec. 3), which occurs, in particular, in experiments on the +study of quark-gluon plasma. +The research was carried out within the framework of the initiative scientific topic 0122U200549 +(“National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, +Ukraine is the customer). + +REFERENCES + +1. J. Stachel, U. Heidelberg. Tests of thermalization in relativistic nucleus nucleus collisions. Nucl. Phys. A +610 (1996) 509C. +2. P. Braun-Munzinger, J. Stachel. Dynamics of ultra-relativistic nuclear collisions with heavy beams: An +experimental overview. Nucl. Phys. A 638 (1998) 3C. +3. J. Cleymans, H. Satz. Thermal Hadron Production in High Energy Heavy Ion Collisions. Z. Phys. C 57 +(1993) 135. +4. J. Cleymans et al. The hadronisation of a quark-gluon plasma. Z. Phys. C 58 (1993) 347. +5. K. Redlich et al. Hadronisation of quark-gluon plasma. Nucl. Phys. A 566 (1994) 391. + +6. P. Braun-Munzinger et al. Thermal equilibration and expansion in nucleus-nucleus collisions at the AGS. +Phys. Lett. B 344 (1995) 43. +7. P. Braun-Munzinger et al. Thermal and hadrochemical equilibration in nucleus-nucleus collisions at the +SPS. Phys. Lett. B 365 (1996) 1. +8. R.A. Ritchie, M.I. Gorenstein, H.G. Miller. The excluded volume hadron gas model and pion production +at the SPS. Z. Phys. C 75 (1997) 535. +9. G.D. Yen et al. Excluded volume hadron gas model for particle number ratios in +A +A  + collisions. Phys. +Rev. C 56 (1997) 2210. +10. G.D. Yen at al. Chemical freezeout in relativistic +A +A  + collisions: is it close to the quark-gluon plasma? +J. Phys. G 24 (1998) 1777. +11. M.I. Gorenstein, A.P. Kostyuk, Ya.D Krivenko. Van der Waals excluded-volume model of +multicomponent hadron gas. J. Phys. G 25 (1999) 75. +12. Ya.D. Krivenko-Emetov. Attractive inter-particle force in van der Waals model of multicomponent +hadron gas in the grand canonical ensemble. 2019 arXiv:1909.08441v1 [hep-ph]; Ya.D. Krivenko- +Emetov. Interparticle attractive forces account of the multicomponent hadron gas in the grand canonical +ensenble. Book of abstract of the 24th Annual Scientific Conf. of Inst. for Nucl. Research, Kyiv, Ukraine, +April 10-13, 2017 (Kyiv, 2017) p. 36. +13. V. Vovchenko at al. Multicomponent van der Waals equation of state: Applications in nuclear and +hadronic physics. Phys. Rev. C 96 (2017) 045202. +14. Ya.D. Krivenko-Emetov. Finite volume effects in the two-component van der Waals model in +relativistic nucleus-nucleus collisions of heavy ions. Book of abstract of the 28th Annual Scientific +Conf. of Inst. for Nucl. Research, Kyiv, Ukraine, Sept. 27 – Oct. 01, 2021 (Kyiv, 2021) p. 27. +15. Quark-Gluon +Plasma +(QGP) +Physics +with +ALICE +at +the +CERN +LHC. +URL: +https://indico.cern.ch/event/1013634/contributions/4255256/attachments/2227069/3772748/IoP- +April2021.pdf. +16. Krivenko-Emetov, Ya.D. Pressure corrections for one-component and two-component van der Waals +nuclear fireball models at the freezeout stage. Book of abstract of 29th Annual Scientific Conf. of Inst. for +Nucl. Research, Kyiv, Sept. 26 – 30, 2022, p.21-22. (Ukr). D. Sokolyuk, Ya. Krivenko-Emetov. Two- +component van der Waals model of a nuclear fireball in the cooling stage (freezeout). Mat. of XX All- +Ukrainian science and practice conf. students, postgraduates and young scientists “Theoretical and +applied problems of physics, mathematics and informatics”, Kyiv, June 15, 2022 (Igor Sikorsky Kyiv +Polytechnic Institute, 2022) p. 88. (Ukr). +17. L.D. Landau, E.M. Lifshitz. Statistical Physics Vol. 5 of Course of Theoretical Physics. (2 ed. Addison +Wesley, 1969) 484 p. +18. R. Kubo. Statistical mechanics (Moskva: Mir, 1967) 452 p. (Rus). +19. R.P. Feynman. Statistical Mechanics: a set of lectures. Advanced Book Classics (2 ed. Perseus Books, +Reading, Mass., 1998) 354 p. + +20. A.M. Fedorchenko. Theoretical physics. T.2. Quantum mechanics, thermodynamics and statistical +physics (Kyiv: Vyshcha shkola, 1993) 415 p. (Ukr). +21. M.V. Fedoruk. Saddle point method (Moskva, 1977) 368 p. (Rus). + +Я. Д. Кривенко-Еметов* +Інститут ядерних досліджень НАН України, Київ, Україна +Національний технічний університет України «Київський політехнічний інститут +імені Ігоря Сікорського», Київ, Україна +*Відповідальний автор: y.kryvenko-emetov@kpi.ua; krivemet@ukr.net +БАГАТОКОМПОНЕНТНА МОДЕЛЬ ВАН ДЕР ВАЛЬСА +ЯДЕРНОГО ФАЄРБОЛУ НА СТАДІЇ ФРІЗАУТУ + +Двокомпонентна газова модель Ван-дер-Ваальсу запропонована для опису адронних етапів +еволюції ядерного фаєрболу у стадії охолодження. Для першого етапу адронізації, коли домінують +мезони, запропонована двокомпонентна мезонна модель( +0 + - та + + -мезонів) з ефективним +двочастинковим потенціалом взаємодії прямокутної ями. Для останнього етапу, коли майже усі +мезони розпались, запропонована двокомпонентна нуклонна модель протонів та нейтронів з +відповідним ефективним прямокутним нуклонним потенціалом. При аналітичних розрахунках +статистичної суми використовувався методу перевалу, що дозволило єдиним чином отримати +аналітичні вирази, як для тиску та щільності з урахуванням скінченних розмірів системи, так і вирази +для хімічних потенціалів. Очікується, що запропоновані моделі й отримані формули можуть бути +використані для аналізу експериментальних даних щодо кількісних характеристик виходу частинок +різних сортів у кінцевому стані від адронних стадій еволюції ядерного фаєрболу, а також для +визначення критичних параметрів системи у ядро-ядерних зіткненнях за високих енергій. +Ключові слова: фаєрбол, фрізаут, рівняння Ван-дер Ваальса, ефективний ядерний потенціал, +Великий канонічний ансамбль, флуктуація тиску, кварк-глюонна плазма, експериментальні дані. + diff --git a/3dAyT4oBgHgl3EQf1vn1/content/tmp_files/load_file.txt b/3dAyT4oBgHgl3EQf1vn1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c892d96931baad9c24fde2e3bee2d343a34a3e7 --- /dev/null +++ b/3dAyT4oBgHgl3EQf1vn1/content/tmp_files/load_file.txt @@ -0,0 +1,525 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf,len=524 +page_content='UDK 539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='17 Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Krivenko-Emetov* Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kyiv, Ukraine National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Kyiv, Ukraine Corresponding author: y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='kryvenko-emetov@kpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='ua;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' krivemet@ukr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='net MULTICOMPONENT VAN DER WAALS MODEL OF A NUCLEAR FIREBALL IN THE FREEZE-OUT STAGE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' A two-component van der Waals gas model is proposed to describe the hadronic stages of the evolution of a nuclear fireball in the cooling stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' At the first stage of hadronization, when mesons dominate, a two-component meson model ( 0 \uf070 - and \uf02b \uf070 -mesons) with an effective two-particle interaction potential of a rectangular well is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' At the late-stage hadronization, when almost all mesons have decayed, a two-component nucleon model of protons and neutrons is proposed with the corresponding effective rectangular nucleon potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The saddle point method has been applied for analytical calculations of the partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This made it possible to uniformly obtain analytical expressions for both the pressure and density, taking into account the finite dimensions of the system, and the analytical expressions for chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It is assumed that the proposed models and derived formulas can be used to analyze experimental data connected to the quantitative characteristics of the particle yields of different types in the final state from the hadronic stages of the evolution of a nuclear fireball, as well as to determine the critical parameters of the system in high-energy nucleus-nucleus collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Keywords: fireball,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' freeze-out,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' van der Waals equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' effective nuclear capability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Grand Canonical Ensemble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' pressure fluctuation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' quark-gluon plasma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' experimental data Introduction Experimental observations of an elliptical flow in non-central collisions of heavy nuclei at high energies provide much evidence that in these collisions of nuclei a state of quark-gluon plasma appears and thermalization occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' which is associated with the fact that particles collide with each other more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' For this state, one can introduce the concept of temperature, viscosity, density, and other thermodynamic quantities that characterize the substance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In these terms, one can describe and study the phenomena that occur during the cooling of a hadron gas formed after a phase transition from the state of a quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It is believed that at a critical temperature ( 150 \uf03e T MeV, the so-called Hagedorn temperature), hadrons "melt" and a phase transition of the hadron gas (hadron matter) into the quark-gluon phase occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, in recent decades, statistical models of hadron gas have been actively used to describe the data of the Large Hadron Collider (LHC), the Relativistic Heavy Ion Collider (RHIC), and even earlier to describe the data of Alternating Gradient Synchrotron (AGS) and Super proton synchrotron (SPS), on the particle yields in a relativistic nuclear-nuclear ( A A \uf02b ) collision at high energies [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The van der Waals (vdW) model, taking into account hadron-hadron interactions at short distances, is especially useful in this description [3-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This is due to the fact that taking into account the effect of repulsion (off volumes) leads to the prevention of an undesirably high density at high temperatures, which appears in ideal gas models [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In addition, collisions of heavy high-energy ions in the LHC produce a large number of different types of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The number of these particles is not fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, the formalism of the Grand Canonical Ensemble (GCE) is one of the adequate mathematical formalisms for these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the thermodynamic quantities do not depend on the number of particles, but on the chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' For many years, researchers have proposed and applied different versions of vdW models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' These models have been mainly used to describe experimental data on the number of particles at high energies, when tens or even hundreds of hadrons of different types can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Naturally, this generation process is limited only by the energy of collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Among these models, the model proposed in [11] should be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this model, the phenomenological parameters of the radii of the hard-core ii R and ij R are introduced, which significantly changes the number of the yield particles with different types i N (i is the particle sort) and is mainly confirmed by experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In order to describe more subtle effects in the dependence of the hadronic gas pressure on density, various authors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' [12, 13]) proposed the development of this model [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Here, the effects of attraction between hadrons at a large distance have been taken into account, which leads to the appearance of a corresponding contribution to the pressure as 2 an Pattr \uf02d ~ ( n is the density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' For a multicomponent gas, the parameter a , corresponding to attraction, transforms into parameters ij a , and the repulsive parameter b transforms into parameters ij b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' At the same time, the parameters of the effective potential corresponding to attraction and repulsion depend on the effective radii of repulsion 0 iR and attraction iR as follows: \uf028 \uf029 ij ij ij ij b c u a \uf02d 0 ~ , \uf028 \uf029 3 0 0 3 2 j i ij R R b \uf02b \uf070 \uf03d , \uf028 \uf0293 3 2 j i ij R R c \uf02b \uf070 \uf03d , ij u0 is the depth of the effective potential well [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' However, even this vdW model cannot be developed properly when considering a finite nuclear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' So, in the case of nuclear collisions, a nuclear fireball with dimensions 10 7 \uf0b8 \uf03e \uf03c ~ r Fm is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In a fairly general case, this problem (without taking into account the effects of reflection from the system wall) has been solved for a two-component system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the GCE formalism leads to the use of a double sum, which, in turn, can be transformed into a multidimensional integral, which can be integrated by the saddle point method in the vicinity of a saddle point with coordinates \uf02a \uf02a 2 1 N N , [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Of course, it would be good to apply this model to the analysis of experimental data obtained for collisions of heavy nuclei at CERN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' One of the variants of such a model was presented in [14] in a concise form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It was believed that the collision energies were not high enough, and one could limit oneself to only two varieties: protons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It was assumed that the characteristic temperatures do not exceed the temperatures at which new particles can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The temperatures of the nuclear fireball are of the order of 140 135 \uf0b8 ~ T MeV in units 1 \uf03d B k (freeze- out phase, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1) at the stage after the transition to the hadronic gas phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, the model itself should have a transparent nonrelativistic limit, taking into account the law of conservation of the total number of nucleons without the generation of new particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The successive stages of the evolution of a nuclear fireball are schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' From left to right: two touching ultrarelativistic nuclei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' the state of a hot and superdense nuclear system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' quark-gluon phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' hadronization and chemical freeze-out;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' kinetic freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The successive stages of the evolution of a nuclear fireball A more detailed and consistent description of the mathematical apparatus of the model [14] is proposed in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Some more subtle effects are estimated (additional corrections for pressure, density, and root-mean-square (RMS) fluctuations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' A new two-component meson model [16] has been proposed for the case of temperatures above the production threshold ( 135 \uf03e T MeV ), when the number of mesons is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" 1 One-component vdW gas According to various estimates, the lifetime of a nuclear fireball is much longer than the characteristic nuclear interaction time 23 22 10 10 \uf02d \uf02d \uf0b8 ~ 't c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It is of the order of the relaxation time 23 21 10 10 \uf02d \uf02d \uf0b8 \uf074 ~ c for sufficiently small local fireball volumes (subsystem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='01 1 10 1100 t (fm/c)Therefore, we will assume that at each moment of time exceeding the relaxation time, a local statistical equilibrium has time to be established in the subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' That is, such a local fireball region is quasi-stationary, and therefore the methods of statistical physics can be applied to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Since all thermodynamic potentials, as well as entropy and volume, are additive (extensive) quantities, therefore, the corresponding potentials (values) of the entire system (fireball) can be defined as the sum of the corresponding thermodynamic potentials of quasi-closed subsystems [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Then, at each moment of time, one can give a standard representation of the partition function of a rarefied quasi-ideal van der Waals gas in the canonical ensemble (CE) for such quasi-closed subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In the approximation of pair interaction and condition \uf028 \uf029 1 \uf03c\uf03c V N T B , this quantity has the form [17]: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029N N N T B V m T N N T V Z \uf02d \uf06a \uf03d , !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' , , 1 , (1) where, respectively, N and m are the number and mass of particles, V and T are the volume and temperature of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Formula (1) uses the notation [11]: \uf028 \uf029 \uf028 \uf029 T m K T m dp T p m p m T 2 2 2 0 2 2 2 2 2 exp 2 1 \uf070 \uf03d \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02b \uf02d \uf070 \uf03d \uf06a \uf0f2 \uf0a5 , , (2) where \uf028 \uf029z K2 is the modified Bessel function, and the second virial coefficient in (1) has the form: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 dV T U T B \uf0f2 \uf0a5 \uf02d \uf02d \uf03d 0 exp 1 2 1 (3) and includes pairwise interaction of particles, \uf0e5 \uf03c \uf03d j i ij U U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In relativistic limit T m \uf03e\uf03e one can easy obtain, given the asymptotes of the Bessel function: \uf028 \uf029 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6\uf02d \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6 \uf070 \uf06a T m mT m T exp 2 2 3 ~ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This formula further leads to the effect of exponential suppression of the particle yields with large mass, which is important in the study of quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The pressure in the system is easy to find from the partition function: \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029N T B V TN N T V Z V T N T V \uf02d \uf03d \uf0b6 \uf0b6 \uf03d , , , , ln P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (4) Note that if the Stirling formula is used in the partition function for the factorial: \uf028 \uf029N e N N N \uf070 \uf0bb 2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' , then the final pressure formula (4) will not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' THE MODEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' According to the above, all calculations for subsystems will be carried out by methods of statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This implies, in addition to the local statistical equilibrium, the fulfillment of the condition of the statistical (thermodynamic) boundary: A N N \uf0ae , where A N is the Avogadro constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the final formulas can be applied to the nuclear fireball due to the indicated additivity of thermodynamic potentials and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Since the number of generated particles in a fireball is about 4-6 thousand during high-energy nucleus-nucleus interactions, this assumption is more or less justified at the first stages of its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Of course, at later stages of evolution, since the number of nucleons at the nonrelativistic boundary is limited by the baryon number conservation law and equal to ( heavy element nuclei collide with mass number ), this assumption is, in general, somewhat doubtful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Of course, at later stages of evolution, this assumption is, in general, somewhat doubtful, since the number of nucleons n the nonrelativistic limit is confined by the baryon number conservation law and is equal to 300 200 \uf0b8 \uf03d N (the nuclei of heavy elements collide with the mass number 200 ~ A ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' However, the practical application of the van der Waals equation quite often goes beyond the conditions under which the virial approximation has been obtained as experience shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Considering this fact, as well as the fact that we can always restrict ourselves to the first stage (see Section 3), we believe that our approximation is sufficiently justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Although calculations by the saddle point method are made when \uf028 \uf029 0 \uf03c T B , however, for the reasons stated above, the final formulas are extended to region where the second virial coefficient \uf028 \uf029 T B is not necessarily negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' From the partition function \uf028 \uf029 N T V Z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' one can also get: free energy \uf028 \uf029 \uf03d N T V F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf05b \uf05d N T V Z T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ln \uf02d \uf03d ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' chemical potential \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0fa\uf0fb \uf0f9 \uf0ea\uf0eb \uf0e9 \uf02b \uf06a \uf02d \uf03d \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf03d \uf06d V N T B T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='m V N T N N T V F 2 ln ln ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (5) and the derivative of the chemical potential which in the statistical limit has the form: \uf028 \uf029 \uf028 \uf029\uf028 \uf029 \uf028 \uf029 \uf028 \uf029 V T T B V T T B N T N V V P N 2 2 lim A N N 2 \uf0ae \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf02b \uf03d \uf0b6 \uf0b6 \uf02d \uf03d \uf0b6 \uf06d \uf0b6 \uf0ae .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (6) Then, we obtain the Grand partition function (GPF) \uf028 \uf029 \uf06d , ,T V Z from the partition function \uf028 \uf029 N T V Z , , taking into account the above physical considerations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=', [18, 19]): \uf028 \uf029 \uf028 \uf029 N T V Z T N T V N , , , , \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6 \uf06d \uf03d \uf06d \uf0e5exp Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" (7) At high temperatures (which, for example, are realized during collisions of heavy ions in the GCE, and ' dN T N \uf0ae \uf044 ) one can turn from the sum to the integral using the Euler-Maclaurin formula." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the first integral term remains and the logarithm of the statistical sum is introduced into the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" Let's denote this indicator by \uf028 \uf029' N \uf046 : \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0f2 \uf0f2 \uf0a5 \uf0a5 \uf046 \uf03d \uf02b \uf06d \uf03d \uf06d 0 0 exp ln exp ' ' ' , ' ' , , N dN T T N V Z N dN T T V Z ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (8) Further integration is performed by the saddle point method [16], since at high temperatures the integrand has a strongly pronounced maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We obtain the following expression for finding the maximum point ( \uf02a N ) for the integrand from the extremum condition imposed on the saddle point: \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf02a \uf02a \uf03d \uf02a \uf03d \uf02a \uf02a \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf02d \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf02d \uf03d \uf06d N N N N N N N T V Z N T N , , ln , (9) \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 m T V N T N , \uf06a \uf02d \uf0bb \uf06d \uf02a \uf02a \uf02a ln ln , (10) where \uf02a \uf06d is the chemical potential at the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' As a result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' we obtain: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf02a \uf02a \uf02a \uf02a \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf02d \uf06d \uf070 \uf06a \uf0b4 \uf0b6 \uf046 \uf0b6 \uf070 \uf03d \uf02a \uf02a \uf02a N V N T B T V e N N m T N N T V N N N N N exp 2 2 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (11) where the second derivative of the exponent \uf046 at the saddle point is defined as follows: \uf028 \uf029 \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf02b \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf02b \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf046 \uf0b6 \uf02a \uf02a \uf02a \uf02a \uf03d \uf03d \uf03d \uf02a \uf03d N N N N N N N N N N T V Z N T N T N N 2 2 2 2 2 2 ln 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 0 2 2 1 1 1 2 2 \uf03c \uf03d \uf02b \uf02b \uf02d \uf03d \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf02b \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf03d \uf02a \uf02a \uf03d \uf03d \uf02a \uf02a \uf02a V T B V T B N N N T N T N N N N N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' because \uf02a \uf03d \uf02a \uf02d \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf02a N N T N N N 1 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf02a \uf02a \uf03d \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf02d \uf03d \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 N N N N N N T V Z N T 2 2ln 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The pressure in the GCE is defined as follows in terms of the temperature and the logarithm of the GPF (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=', [18]): \uf028 \uf029 \uf028 \uf029 V T V T T P \uf06d \uf03d \uf06d , , , Z ln .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" (12) It's easy to show that pressure (12)," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' taking into account (5) and (11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' can be rewritten as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0fa\uf0fb \uf0f9 \uf0ea\uf0eb \uf0e9 \uf078 \uf078 \uf02d \uf078 \uf02d \uf078 \uf0bb \uf0fa \uf0fa \uf0fb \uf0f9 \uf0ea \uf0ea \uf0eb \uf0e9 \uf078 \uf0b6 \uf046 \uf0b6 \uf02d \uf078 \uf02d \uf078 \uf0bb \uf06d \uf02a \uf03d \uf02a \uf02a V T B T B T V N T B T T P N N 2 ln 1 2 ln 1 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (13) where the saddle point \uf028 \uf029 V T V N \uf02a \uf02a \uf06d \uf03d \uf078 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' is defined according to (5) and (10) as \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 T N T m N \uf02a \uf06d \uf06a \uf0bb \uf078 exp ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The parameter \uf078 can be eliminated from equation (13) using the definition of density, which in the thermodynamic limit turns into the well-known formula [1]: \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029 \uf05b \uf05d \uf078 \uf02d \uf078 \uf0ae \uf02d \uf078 \uf02d \uf078 \uf03d \uf06d \uf0b6 \uf06d \uf0b6 \uf03d T B V T B T P n 2 1 2 1 2 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (14) In the thermodynamic limit ( A N N \uf0ae , \uf0a5 \uf0ae V ) the chemical potential of the saddle point \uf02a \uf06d from (10) when \uf028 \uf029 \uf028 \uf029 V N T B N N 2 1\uf02d \uf03d \uf02a turns into the chemical potential \uf06d ( \uf06d \uf0ae \uf06d\uf02a ), which is determined by the well-known thermodynamic equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Both equations (13) and (14) in parametric form (the saddle point \uf078 acts as a parameter) determine the relationship between pressure P , temperature T , and density n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We obtain the state equation in GCE by excluding explicitly this parameter from the system of equations (13) and (14): \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d dP n T B Tn n T P \uf02b \uf02b \uf0bb \uf06d 1 , , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (15) Of course, the resulting state equation is implicitly a parametric equation, since the saddle point \uf078 (and, hence, and n ) determines the chemical potential \uf06d according to (5) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It is important that the resulting formula takes into account the contribution to the pressure of the finite volume of the system, s V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This contribution naturally vanishes in the thermodynamic limit, where there is no difference between CE and GCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Only the last of the three terms remains for large but finite volumes, as in the nuclear fireball model discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4: \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf05b \uf05d \uf028 \uf029 \uf05b \uf05d s V V V n T B T n T B n T B V T dP s 2 ln ln 1 2 \uf02d \uf0ae \uf02d \uf02b \uf03d \uf0ae lim .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (16) RMS fluctuations of pressure and density calculated by known formulas (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=', [20]) give estimates of the found corrections to the corresponding quantities: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d n T B V n T n P V Tn P s 2 1 2 2 \uf02b \uf0b6 \uf0b6 \uf03e\uf03d \uf044 \uf03c ~ , (17) \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d n T B V N V N T n N N 2 1 1 2 2 2 \uf02d \uf0b6 \uf06d \uf0b6 \uf03e\uf03d \uf044 \uf03c \uf02a \uf03d \uf02a ~ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (18) 2 Two-component vdW gas Let us consider the procedure for taking into account the excluded volume and attraction in the vdW model for the case of a two-component hadron gas of two types of particles “i ” and “ j ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1 N and 2 N are the number of particles of the first and second sorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the partition function has the form: \uf028 \uf029 \uf03d 2 1 N N T V Z , , , \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 T U r d r d T m T m N N N k k N l l N N 12 1 2 3 1 1 3 2 1 2 1 exp !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1 2 1 2 1 \uf02d \uf0b4 \uf0b4 \uf0b4 \uf06a \uf06a \uf03d \uf0f2\uf0d5 \uf0f2\uf0d5 \uf03d \uf03d , , , (19) This expression for the pair-interactions approximation ( \uf028 \uf029 \uf028 \uf029 12 123 U U \uf03c\uf03c ) and a weakly ideal gas ( 1 2 \uf03c\uf03c V NB ) can be rewritten as follows [11]: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0b4 \uf06a \uf06a 2 1 2 1 2 1 2 1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1 N N T m T m N N N N T V Z , , ~ , , , \uf028 \uf029 \uf028 \uf029 2 1 1 12 2 22 2 21 1 11 N N N B N B V N B N B V ~ ~ \uf02d \uf02d \uf0b4 \uf02d \uf02d \uf0b4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (20) Here we have introduced the notation: jj ii ij ii ij B B B B B \uf02b \uf03d 2 ~ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" The two-particle partition function \uf028 \uf029 2 1 \uf06d \uf06d , , ,T V Z in GCE is expressed in terms of the two-particle partition function \uf028 \uf029 2 1 N N T V Z , , , in CE [12, 17], as: \uf028 \uf029 \uf028 \uf029 \uf03d \uf03d \uf06d \uf06d \uf06d \uf02b \uf06d \uf0a5 \uf0a5 \uf0f2 \uf0f2 2 1 2 0 1 0 2 2 1 2 2 1 1 e d d ' , ' , , ' ' , , , ' ' N N T V Z N N T T V N N Z \uf028 \uf029 \uf028 \uf029 2 1 2 0 1 0 2 exp d d ' , ' ' ' N N N N T \uf046 \uf03d \uf0f2 \uf0f2 \uf0a5 \uf0a5 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (21) Integration of (21) by the saddle point method [21] leads us to the following result: \uf028 \uf029 \uf028 \uf029 \uf02a \uf02a \uf02a \uf02a \uf02a \uf02a \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf06d \uf02b \uf06d \uf0a2 \uf0a2 \uf046\uf0a2\uf0a2 \uf070 2 1 2 2 1 1 2 1 exp 2 N N T V Z T N N N N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ~ Z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' where the coordinates of the saddle point \uf02a i N ( 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf03d i ) are found from the extremum conditions: \uf028 \uf029 0 \uf03d \uf0a2 \uf0b6 \uf0a2 \uf0a2 \uf046 \uf0b6 i j i N N N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 22 21 12 11 2 1 c c c c N N det ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf03d \uf0a2 \uf0a2 \uf046\uf0a2\uf0a2 \uf02a \uf02a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf02a\uf0a2 \uf03d \uf0a2 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf0a2 \uf0b6 \uf0a2 \uf0b6 \uf0a2 \uf0a2 \uf046 \uf0b6 \uf03d N N j i j i ij N N N N c ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Substituting the value of the partition function into the definition of pressure in the GCE [18], we obtain the following expression [12]: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0fa\uf0fb \uf0f9 \uf0ea\uf0eb \uf0e9 \uf02d \uf078 \uf078 \uf02b \uf02d \uf078 \uf02d \uf078 \uf02d \uf078 \uf02b \uf078 \uf06d \uf06d \uf03d \uf06d \uf06d V C B B B B T V T V T T P 2 ln ln 2 1 21 12 22 2 2 11 2 1 2 1 2 1 2 1 ~ ~ ~ , , , , , Z , (22) where 21 2 12 1 22 2 11 1 B B B B C ~ ~ \uf078 \uf078 \uf02d \uf078 \uf078 \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Using such a mathematical apparatus, one can organically introduce the law of conservation of chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The latter are related to the condition imposed on the integrand when finding the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In the thermodynamic limit the chemical potential determined by the extremum condition coincides with the definition of the chemical potential itself: \uf028 \uf029 i j i i i N N N T V F \uf0b6 \uf0b6 \uf03d \uf06d \uf0ae \uf06d\uf02a , , , , where \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d 2 1 2 1 ln N N T V Z T N N T V F , , , , , , \uf02d \uf03d is the definition of free energy (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We get from the definition of density \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d ji ij j ii i i i j i i B B B T P n ~ ~ ~ , , \uf02b \uf078 \uf02b \uf078 \uf02d \uf078 \uf06d \uf0b6 \uf06d \uf06d \uf0b6 \uf03d 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (23) The virial expansion (22) can be rewritten,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' taking into account (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' as a two-component vdW equation in the approximation 1 \uf03c\uf03c V N b i ij and \uf028 \uf029 1 \uf03c\uf03c ij ij Tb a ): \uf028 \uf029 \uf03d \uf06d \uf06d 2 1 2 1 n n T P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 dP n a n a n n a n a n n b n b Tn n b n b Tn \uf02b \uf02b \uf02d \uf02b \uf02d \uf02d \uf02d \uf02b \uf02d \uf02d \uf03d 1 12 2 22 2 2 21 1 11 1 1 12 2 22 2 2 21 1 11 1 1 1 ~ ~ ~ ~ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (24) where dP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' according to (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' takes into account the finite size of the fireball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' When formula (24) was derived, the expression T a b B ij ij ij ~ ~ ~ \uf02d \uf0bb was used (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=', [12]), and for each type of particles the corresponding parameters of attraction and repulsion were introduced [11]: \uf028 \uf029 jj ii ii ij ij a a a a a \uf02b \uf0b4 \uf067 \uf0bb 2 ~ , \uf028 \uf029 jj ii ij ii ij b b b b b \uf02b \uf03d 2 ~ , \uf067 is a phenomenological parameter reflecting the complexity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3 The asymmetric two-component freeze-out model with non-conservation of the number of particles The considering nucleus-nucleus collisions \uf028 \uf029 A A\uf02b have very high energies, more than 1 GeV per nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' At the same time, mesons of different sorts dominate in the initial freeze-out stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, to describe the nucleus-nucleus interactions at this stage of the freeze-out above the production threshold of new particles ( 135 \uf03e T MeV), we propose a generalization of the vdW model to a medium-sized nuclear fireball [16]: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf03e\uf03e \uf03c\uf03c \uf070 \uf03e \uf03c \uf03e \uf03c \uf070 \uf03e \uf02b \uf03c \uf03e \uf03c A r b a V V V f f f 3 0 2 4 3 4 3 2 ~ ~ ~ max min .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Here 2 1 1 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf0b8 \uf03d r Fm, \uf03e \uf03c a , \uf03e \uf03c b are the mean semiaxes of the ellipsoid, and \uf03e\uf03e \uf03c\uf03c A is the mass number of nuclei left in the fireball after the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In our considerations we assume that the fireball consists, mainly, of mesons, given that the number of nucleons is much less than the number of mesons ( \uf03c\uf03c \uf0b8 300 200 ~ pn N 5000 4000 \uf0b8 \uf070\uf072\uf077 ~ N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We neglect the contribution of other particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, we introduce the following additional natural assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The average internucleon energies do not exceed the production threshold of the heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, we restrict ourselves to two sorts of particles ("0" is the 0 \uf070 -meson, "+" is the \uf02b \uf070 -meson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Since \uf02b \uf070 -meson production reactions are twice as likely as 0 \uf070 -meson production reactions, we assume that n kn n \uf03d \uf03d \uf02b 0 , where, 1 \uf03c k , 0 n is the 0 \uf070 -meson density, and \uf02b n is the \uf02b \uf070 -meson density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This corresponds to a more probable production of the \uf02b \uf070 -mesons in reactions \uf02b \uf070 \uf02b \uf02b \uf0ae \uf02b n d d p , 0 \uf070 \uf02b \uf02b \uf0ae \uf02b p d d p than production of the 0 \uf070 -mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We introduce the effective potential of the meson interaction \uf028 \uf029j i U , where \uf028 \uf029 \uf07b \uf07d 0, , \uf02b \uf03d j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' That is, "(0+)" is the interaction of 0 \uf070 -mesons with \uf02b \uf070 -mesons, "(++)" is the interaction of \uf02b \uf070 -mesons \uf028 \uf029 \uf028 \uf029 \uf0ef \uf0ee \uf0ef \uf0ed \uf0ec \uf0b3 \uf02b \uf02b \uf03c \uf0a3 \uf02b \uf02b \uf03c \uf0a5 \uf03d r R R if R R r R R if u R R r if U j i j i j i j i j i j i 0 0 0 0 0 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (25) Since the effective rectangular potential a well leads to approximately the same values of pressure and density as the real potential (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2, where \uf028 \uf029 U U \uf03d \uf02b0 , \uf028 \uf029 \uf065 \uf03d \uf02b0 0 u ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, the real meson-meson potential (a) can be replaced by a similar effective rectangular potential (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Meson-meson potential 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We accept that the 0 \uf070 -meson hard-core radius is much smaller than the \uf02b \uf070 -meson hard-core radius: 0 0 0 \uf02b \uf03c\uf03c R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The radius of the hard-core of the \uf02b \uf070 -meson is assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Average pressure and density fluctuations are easily found within the framework of the proposed model, similarly to formulas (18) and (19): \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d Tn B V n T P f \uf02b \uf03e \uf03c \uf03e \uf044 \uf03c 1 ~ , (26) \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d Tn B V n V n f f \uf02d \uf03e \uf03c \uf03e \uf03c \uf03e \uf044 \uf03c 1 1 ~ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (27) The following results are obtained (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Such data have been used (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3): 3 142, \uf03d T MeV, the effective radius of the \uf02b \uf070 -meson, 45 0 0 , \uf03d \uf02b R Fm, and 0 \uf070 -meson, 01 0 0 0 , \uf03d R Fm, the average value of the volume of the meson fireball is taken as the value 600 ~ \uf03e \uf03c f V Fm 3 , 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf03d k , the parameter of the potential depth, \uf028 \uf029 100 80 0 0 \uf0b8 \uf02b ~ , u MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' One can U来 U*clearly see (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4) an increase in the correction \uf03e \uf03c P dP at low densities, which is typical in the final stages of the freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Dependence of the meson pressure P (24) on the meson density n kn n \uf03d \uf03d \uf02b 0 for the two-component asymmetric vdW model with correction (upper isotherm) and without correction (lower isotherm) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Ratio of the correction to pressure dP from the size of the meson fireball to the value of the RMS pressure fluctuation \uf03e \uf03c P (26) as a function of the meson density n kn n \uf03d \uf03d \uf02b 0 4 Two-component model of a nucleon fireball at the final stage of the freeze-out The average lifetime of mesons dominating in the initial stages of the freeze-out is relatively short ( 16 8 10 10 \uf02d \uf02d \uf0b8 \uf074 ~ c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" That's why they decay pretty quickly." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Accordingly, baryons, namely protons and neutrons, begin to dominate at the final stage of freezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In addition, as shown above, the effects of the finite volume size become noticeable at sufficiently low density values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This formally corresponds to just such final stages of the fireball evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, despite a certain doubt about the existence of a fireball at such late stages, when the boundary between the gas and the aggregate of individual nucleons gradually disappears, to describe the nucleus-nucleus P(T=142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='3, n), MeV/Fm-3 10 n, Fm-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='30 10 20 30 40 50 FdP/

2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='0 n, Fm-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='15 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='30interactions at the last stage of the freeze-out, which is below the production threshold of new particles ( 135 \uf03c T MeV), in [14] the following generalization of the vdW model to the nucleon fireball was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We accept the following simplifications by analogy with the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The average energies of internucleon collisions do not exceed the production threshold of other hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, we restrict ourselves to two varieties (“ p ” is the proton, “ n ” is the neutron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We take the relation between the density of protons and neutrons in the form n n n n p \uf03d \uf02b following from the law of conservation of the baryon number, A N Z \uf03d \uf02b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We assume that the nucleon composition of colliding nuclei is known as such n p kn n \uf03d , where 1 \uf03c k , since heavy nuclei have an excess of neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The effective potential of the proton-neutron, proton-proton and neutron-neutron interactions, which leads to approximately the same values of pressure and density as the real potential (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3), can be represented by analogy to (25) as \uf028 \uf029j i U , where \uf028 \uf029 n p j i , , \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The hard-core radius of the proton is assumed to be known, 5 0 0 , \uf03d p R Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We accept that the radius of the neutron is much less than the radius of the proton: 0 0 p n R R \uf03c\uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We get from equation (27): \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 dP n a n n n k Tn T P \uf02b \uf02d \uf064 \uf02b \uf062 \uf02d \uf061 \uf02d \uf02b \uf03d \uf06d \uf06d \uf02a \uf02a \uf02a \uf02a \uf02a \uf02a 2 2 2 1 1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (28) where \uf028 \uf029 k n n \uf02b \uf03d \uf02a 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' k is a dimensionless quantity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' k b k b b k b 22 2 21 12 11 \uf02b \uf02b \uf02b \uf03d \uf061 ~ ~ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 22 21 12 11 b k b b k b \uf02b \uf02b \uf02b \uf03d \uf062 ~ ~ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' k b b b b b b k b kb 21 12 21 22 12 11 2 22 11 ~ ~ ~ ~ \uf02b \uf02b \uf02b \uf03d \uf064 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 22 21 12 2 11 a k a a k a a \uf02b \uf02b \uf02b \uf03d \uf02a ~ ~ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It follows from the condition 0 1 0 2 R R \uf03c\uf03c that 11 22 b b \uf03c\uf03c , \uf062 \uf040 \uf061 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' By analogy with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (18) and (19), we find the corresponding average fluctuations of pressure and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Functional dependences for pressure, obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (28), and the ratio of dP to RMS pressure fluctuations are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Dependence of nucleon pressure P (28) on nucleon density, n kn n n p \uf03d \uf03d , in the two-component asymmetric vdW model with correction (upper isotherm) and without correction (lower isotherm) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The ratio of correction from the size of the nucleon fireball to pressure dP to the value of the RMS pressure fluctuation \uf03e \uf03c P depending on the density of nucleons, n kn n n p \uf03d \uf03d It can be seen that the correction dP makes a nonzero contribution to the total pressure also in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' On the other hand, it is negligibly small almost everywhere in comparison with the contribution from fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The correction makes a contribution comparable to fluctuations only in the region near zero density that is nonphysical for a nuclear fireball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' But it can be neglected in this region, as can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Summary The effect of taking into account the excluded volume and attraction is analyzed in the case of a two-component gas: (i) 0 \uf070 - and \uf02b \uf070 -mesons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (ii) protons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The calculations have been performed in the Canonical and Grand Canonical ensembles by the saddle point method for a two- component system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The particles interact with the potentials of the hard-core at short distances and with relatively high potentials at large distances (effective attraction radii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' For effective P(T=142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' n), MeV/Fm-3 10 n, Fm-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='10 N15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='30 10 20 30 40 50 FdP/

1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='8 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='7 n, Fm-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='30interparticle interactions of this type, an equation of state has been obtained with corrections that take into account the finite dimensions of the nuclear fireball, as well as RMS fluctuations of pressure and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The pressure correction disappears in the thermodynamic limit, when, according to statistical physics, there is no difference between various statistical ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The formulas for pressure and density obtained by the saddle point method can be used to analyze experimental data concerning the relative number of the yield particleshe of various sorts and critical parameters in high-energy nuclear-nucleus collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' A generalization of the presented vdW model to the asymmetric case of a two-component model ( 0 \uf070 - and \uf02b \uf070 -mesons) with realistic parameters of the hard-core and attraction has been proposed as an example of such a use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The ratio of the pressure correction to the RMS value of pressure fluctuation is estimated for the case of an asymmetric two-component meson fireball model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' An increase in the correction has been found at low density values corresponding to the final stages of freezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It is found that the contribution to pressure and relative fluctuations, taking into account different radii and the finiteness of the nuclear fireball, is noticeable in the case of the meson model with nonconservation of the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' However, this correction can be neglected for the final stages of the freeze-out, when nucleons begin to dominate (the model of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, the developed model is applicable in the analysis of experimental data on the study of the initial meson phase of a nuclear fireball (the model of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3), which occurs, in particular, in experiments on the study of quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The research was carried out within the framework of the initiative scientific topic 0122U200549 (“National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine is the customer).' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' of Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' for Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Research, Kyiv, Ukraine, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 27 – Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 01, 2021 (Kyiv, 2021) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 27.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Quantum mechanics, thermodynamics and statistical physics (Kyiv: Vyshcha shkola, 1993) 415 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (Ukr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Fedoruk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Saddle point method (Moskva, 1977) 368 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (Rus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Я.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Кривенко-Еметов* Інститут ядерних досліджень НАН України, Київ, Україна Національний технічний університет України «Київський політехнічний інститут імені Ігоря Сікорського», Київ, Україна Відповідальний автор: y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='kryvenko-emetov@kpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='ua;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' krivemet@ukr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='net БАГАТОКОМПОНЕНТНА МОДЕЛЬ ВАН ДЕР ВАЛЬСА ЯДЕРНОГО ФАЄРБОЛУ НА СТАДІЇ ФРІЗАУТУ Двокомпонентна газова модель Ван-дер-Ваальсу запропонована для опису адронних етапів еволюції ядерного фаєрболу у стадії охолодження.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Для першого етапу адронізації, коли домінують мезони, запропонована двокомпонентна мезонна модель( 0 \uf070 - та \uf02b \uf070 -мезонів) з ефективним двочастинковим потенціалом взаємодії прямокутної ями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Для останнього етапу, коли майже усі мезони розпались, запропонована двокомпонентна нуклонна модель протонів та нейтронів з відповідним ефективним прямокутним нуклонним потенціалом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' При аналітичних розрахунках статистичної суми використовувався методу перевалу, що дозволило єдиним чином отримати аналітичні вирази, як для тиску та щільності з урахуванням скінченних розмірів системи, так і вирази для хімічних потенціалів.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Очікується, що запропоновані моделі й отримані формули можуть бути використані для аналізу експериментальних даних щодо кількісних характеристик виходу частинок різних сортів у кінцевому стані від адронних стадій еволюції ядерного фаєрболу, а також для визначення критичних параметрів системи у ядро-ядерних зіткненнях за високих енергій.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Ключові слова: фаєрбол, фрізаут, рівняння Ван-дер Ваальса, ефективний ядерний потенціал, Великий канонічний ансамбль, флуктуація тиску, кварк-глюонна плазма, експериментальні дані.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} diff --git a/3dAzT4oBgHgl3EQffPzi/content/tmp_files/2301.01451v1.pdf.txt b/3dAzT4oBgHgl3EQffPzi/content/tmp_files/2301.01451v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..22101d962a969d07441a8a505ed533b25437106d --- /dev/null +++ b/3dAzT4oBgHgl3EQffPzi/content/tmp_files/2301.01451v1.pdf.txt @@ -0,0 +1,1824 @@ +arXiv:2301.01451v1 [quant-ph] 4 Jan 2023 +Reduced dynamics with Poincar´e symmetry in open quantum system +Akira Matsumura∗ +Department of Physics, Kyushu University, Fukuoka, 819-0395, Japan +Abstract +We consider how the reduced dynamics of an open quantum system coupled to an environment admits +the Poincar´e symmetry. The reduced dynamics is described by a dynamical map, which is given by tracing +out the environment from the total dynamics. Introducing the notion of covariant map, we investigate the +dynamical map which is symmetric under the Poincar´e group. Based on the representation theory of the +Poincar´e group, we develop a systematic way to give the dynamical map with the Poincar´e symmetry. Using +this way, we derive the dynamical map for a massive particle with a finite spin and a massless particle with +a finite spin and a nonzero momentum. We show that the derived map gives the unitary evolution of a +particle when its energy is conserved. We also find that the dynamical map for a particle does not have the +Poincar´e symmetry when the superposition state of the particle decoheres into a mixed state. +∗Electronic address: matsumura.akira@phys.kyushu-u.ac.jp +1 + +Contents +I. Introduction +2 +II. Quantum dynamical map and its symmetry +4 +III. Dynamical map with Poincar´e symmetry +5 +IV. A model of the dynamical map for a single particle +10 +V. Conclusion +13 +Acknowledgments +14 +A. Derivation of Eqs.(54),(55),(56),(57),(58) and (59) +14 +B. Analysis of a massive particle +15 +C. Analysis on a massless particle +21 +References +27 +I. +INTRODUCTION +It is difficult to isolate a quantum system perfectly, which is affected by the inevitable influence +of a surrounding environment. Such a quantum system is called an open quantum system. Since +we encounter open quantum systems in a wide range of fields such as quantum information science +[1, 2], condensed matter physics [3, 4] and high energy physics [5], it is important to understand +their dynamics. +In general, the dynamics of an open quantum system, the so-called reduced +dynamics, is very complicated. This is because the environment may have infinitely many degrees +of freedom and they are uncontrollable. One needs the effective theory with relevant degrees of +freedom to describe the reduced dynamics of an open quantum system [2]. +As is well-known, symmetry gives a powerful tool for capturing relevant degrees of freedom in +the dynamics of interest. For example, let us focus on the symmetry in the Minkowski spacetime, +which is called the Poincar´e symmetry. Imposing the Poincar´e symmetry on a quantum theory, +one finds that quantum dynamics in the theory is described by the fundamental degrees of freedom +such as a massive particle and a massless particle [6]. The approach based on symmetries provides +2 + +a way to get the effective theory of open quantum systems. +In this paper, we discuss the consequences of the Poincar´e symmetry on the reduced dynamics +of an open quantum system. This may give the understanding of the relativistic theories of open +quantum systems (for example, [7–14]). +The present paper is also motivated by the theory of +quantum gravity. Since the unification of quantum mechanics and gravity has not been completed +yet, we do not exactly know how gravity is incorporated in quantum mechanics. This situation +has prompted to propose many models on the gravity of quantum systems. In the previous work +[15], the model with a classical gravitational interaction between quantum systems was proposed, +which is called the Kafri-Taylor-Milburn model. In addition, the Diosi-Penrose model [16–18] and +the Tilloy-Diosi model [19] were advocated, for which the gravity of a quantum system intrinsically +induces decoherence. +They are formulated by the theory of open quantum systems in a non- +relativistic regime. +One may concern how the models are consistent with a relativistic theory. +Our analysis on reduced dynamics with the Poincar´e symmetry would help to obtain a relativistic +extension of the above proposed models. +For our analysis, we assume that the reduced dynamics of an open quantum system is described +by a dynamical map. The dynamical map is obtained by tracing out the environment from the total +unitary evolution with an initial product state. It is known that the dynamical map is represented +by using the Kraus operators [2, 20–22]. The notion of covariant map is adopted for incorporating +a symmetry into a dynamical map. We derive the condition of a dynamical map with the Poincar´e +symmetry in terms of the Kraus operators. With the help of the representation theory of the +Poincar´e group, we obtain a systematic way to deduce those Kraus operators. +Applying the way, we exemplify the dynamical map with the Poincar´e symmetry. +To get +the concrete Kraus operators, we focus on the dynamics of a single particle, which is possible to +decay into the vacuum state. Assuming that the particle is a massive particle with a finite spin or a +massless particle with a finite spin and a nonzero momentum, we get a model of the dynamical map +with the Poincar´e symmetry. In the model, we find the following consequences: (i) if the particle +is stable or the energy of particle is conserved, the obtained map turns out to be the unitary map +given by the Hamiltonian of particle. (ii) If the superposition state of a particle decoheres into +a mixed state, the dynamical map for the particle does not have the Poincar´e symmetry. These +consequences imply that the Poincar´e symmetry can strongly constraint the reduced dynamics of +an open quantum system. +The structure of this paper is as follows. In Sec.II, we discuss a dynamical map describing the +reduced dynamics of an open quantum system and consider how symmetries are introduced in the +3 + +dynamical map. In Sec.III, we derive the condition that the dynamical map is symmetric under the +Poincar´e group. In Sec.IV, focusing on the dynamics of a single particle, we present a model of the +dynamical map with the Poincar´e symmetry. We then investigate the properties of the dynamical +map in details. Sec.V is devoted as the conclusion. We use the unit ℏ = c = 1 in this paper. +II. +QUANTUM DYNAMICAL MAP AND ITS SYMMETRY +In this section, we consider the reduced dynamics of an open quantum system and discuss the +symmetry of the dynamics. The reduced dynamics is given as the time evolution of the density +operator of the system. The time evolution from a time τ = s to τ = t is assumed to be given by +ρ(t) = Φt,s[ρ(s)] = TrE[ ˆU(t, s)ρ(s) ⊗ ρE ˆU †(t, s)], +(1) +where ρ(τ) is the system density operator, ρE is the density operator of an environment and ˆU(t, s) +is the unitary evolution operator of the total system. +In this paper, the map Φt,s is called a +dynamical map, which has the property called completely positive and trace-preserving (CPTP) +[2, 20–22]. The dynamical map Φt,s is rewritten in the operator-sum representation, +Φt,s[ρ(s)] = +� +λ +ˆF t,s +λ ρ(s) ˆF t,s † +λ +, +(2) +where ˆF t,s +λ +called the Kraus operators satisfy the completeness condition, +� +λ +ˆF t,s † +λ +ˆF t,s +λ += ˆI. +(3) +In this notation, λ takes discrete values. When the label λ is continuous, we should replace the +summation � +λ with the integration +� +dµ(λ) with an appropriate measure µ(λ). It is known that +two dynamical maps Φ and Φ′ with +Φ[ρ] = +� +λ +ˆFλ ρ ˆF † +λ, +Φ′[ρ] = +� +λ +ˆF ′ +λ ρ ˆF +′† +λ , +(4) +are equivalent to each other (i.e. Φ[ρ] = Φ′[ρ] for any density operator ρ) if and only if there is a +unitary matrix Uλλ′ satisfying � +λ Uλ1λU∗ +λ2λ = δλ1λ2 = � +λ Uλλ1U∗ +λλ2 and +ˆF +′ +λ = +� +λ′ +Uλλ′ ˆFλ′. +(5) +This is the uniqueness of a dynamical map [2, 20–22]. +We introduce the notion of covariant map [22–24] to impose symmetry on dynamical maps. A +dynamical map Φt,s is covariant under a group G if +Φt,s[ ˆUs(g) ρ(s) ˆU † +s (g)] = ˆUt(g)Φt,s[ρ(s)] ˆU † +t (g), +(6) +4 + +where ˆUs(g) and ˆUt(g) with g ∈ G are the unitary representations of G. In this paper, the dynamical +map Φt,s satisfying (6) is called symmetric under the group G. In the next section, we will discuss +the dynamical map which is symmetric under the Poincar´e group. +III. +DYNAMICAL MAP WITH POINCAR´E SYMMETRY +In this section, we consider a quantum theory with the Poincar´e symmetry and discuss the +general conditions on a dynamical map with the Poincare symmetry. The generators of the unitary +representation of the Poincar´e group in the Schr¨odinger picture [6] are given by +ˆPµ = +� +d3x ˆT 0 +µ, +ˆJµν = +� +d3x ˆ +Mµν0, +(7) +where ˆTµν is the energy-momentum tensor satisfying +∂µ ˆT µ +ν = 0, +ˆTµν = ˆTνµ +(8) +and ˆ +Mµνρ with +ˆ +Mµνρ = xµ ˆT ρ +ν − xν ˆT ρ +µ +(9) +is the Noether current associated with the Lorentz transformations. From Eq.(8), we can show +that ∂ρ ˆ +Mµνρ = 0. Focusing on each component of the generators, we have +ˆH = ˆP 0 = +� +d3x ˆT 00, +ˆP i = +� +d3x ˆT 0i, +(10) +ˆJk = 1 +2ǫijk ˆJij = +� +d3x ǫijkxi ˆT 0 +j , +ˆKk = +� +d3(xk ˆT 00 − t ˆT 0k), +(11) +where note that the boost generator ˆKk explicitly depends on a time t. These operators satisfy +the commutation relations, +[ ˆPi, ˆPj] = 0, +(12) +[ ˆPi, ˆH] = 0, +(13) +[ ˆJi, ˆH] = 0, +(14) +[ ˆJi, ˆJj] = iǫijk ˆJk, +(15) +[ ˆJi, ˆPj] = iǫijk ˆP k, +(16) +[ ˆJi, ˆKj] = iǫijk ˆKk, +(17) +[ ˆKi, ˆPj] = iδij ˆH, +(18) +[ ˆKi, ˆH] = i ˆPi, +(19) +[ ˆKi, ˆKj] = −iǫijk ˆJk, +(20) +5 + +which correspond to the Poincar´e algebra. +We consider a dynamical map Φt,s from ρ(s) to ρ(t) = Φt,s[ρ(s)]. The Poincar´e symmetry of +the dynamical map requires that +ˆUt(Λ, a)Φt,s[ρ(s)] ˆU † +t (Λ, a) = Φt,s[ ˆUs(Λ, a)ρ(s) ˆU † +s (Λ, a)], +(21) +where the unitary operator ˆUt(Λ, a) depends on the proper (detΛ = 1) orthochronous (Λ00 ≥ 1) +Lorentz transformation matrix Λµν and the real parameters aµ for the spacetime translations. The +unitary operator ˆUt(Λ, a) generated by ˆH, ˆPi, ˆJi and ˆKi has the group multiplication rule +ˆUt(Λ′, a′) ˆUt(Λ, a) = ˆUt(Λ′Λ, a′ + Λ′a), +(22) +where we used the fact that we can always adopt the non-projective unitary representation of the +Poincar´e group [6]. The explicit time dependence of ˆUt comes from the boost generator ˆKi. Using +the operator-sum representation, we have +ˆUt(Λ, a) +� +λ +ˆF t,s +λ ρ(s) ˆF t,s† +λ +ˆU † +t (Λ, a) = +� +λ +ˆF t,s +λ +ˆUs(Λ, a)ρ(s) ˆU † +s (Λ, a) ˆF t,s† +λ +. +From the uniqueness of the Kraus operators ˆF t,s +λ +(see Eq.(5)), we obtain +ˆU † +t (Λ, a) ˆF t,s +λ ˆUs(Λ, a) = +� +λ′ +Uλλ′(Λ, a) ˆF t,s +λ′ . +(23) +We can always choose ˆF t,s +λ +so that { ˆF t,s +λ }λ is the set of linearly independent operators. This linear +independence and the group multiplication rule of ˆUt(Λ, a) given in (22) lead to the fact that the +unitary matrix Uλλ′(Λ, a) satisfies the group multiplication rule +� +λ′ +Uλλ′(Λ′, a′)Uλ′λ′′(Λ, a) = Uλλ′′(Λ′Λ, Λa + a′). +(24) +Hence, the unitary matrix Uλλ′(Λ, a) is a representation of the Poincar´e group. +Before discussing the condition of symmetry, Eq.(23), we present the useful relation +ˆKi = e−i ˆ +Ht ˆKi +0 ei ˆ +Ht, +(25) +where +ˆKi +0 = +� +d3x xi ˆT 00. +(26) +According to the Poincar´e algebra, we have +ˆUt(Λ, a) = e−i ˆ +Ht ˆU0(Λ, a)ei ˆ +Ht, +(27) +6 + +where ˆU0(Λ, a) is the unitary representation of the Poincar´e group with the genrators ˆH, ˆP i, ˆKi +0 and +ˆJi. In the scattering theory, Eq. (27) is consistent with the Poincar´e invariance of the S-operator +ˆS(∞, −∞), where ˆS(tf, ti) = ei ˆ +H0tfe−i ˆ +H(tf−ti)e−i ˆ +H0ti and ˆH = ˆH0 + ˆV . This is because +ˆU I† +tf (Λ, a) ˆS(tf, ti) ˆU I +ti(Λ, a) = ei ˆ +H0tf ˆU † +tf(Λ, a)e−i ˆ +H(tf−ti) ˆUti(Λ, a)e−i ˆ +H0ti += ei ˆ +H0tfe−i ˆ +Htf ˆU † +0(Λ, a) ˆU0(Λ, a)ei ˆ +Htie−i ˆ +H0ti += ˆS(tf, ti), +(28) +where ˆU I +t(Λ, a) = ei ˆ +H0t ˆUt(Λ, a)e−i ˆ +H0t. Eq.(27) also implies that the unitary evolution generated by +ˆH is symmetric under the Poincar´e group. Indeed, we can show that the unitary map +Ut,s[ρ(s)] = e−i ˆ +H(t−s) ρ(s) ei ˆ +H(t−s) +(29) +satisfies the condition of symmetry (21) as +Ut,s[ ˆUs(Λ, a)ρ(s) ˆU † +s (Λ, a)] = e−i ˆ +H(t−s) ˆUs(Λ, a) ρ(s) ˆU † +s (Λ, a)ei ˆ +H(t−s) += e−i ˆ +H(t−s) ˆUs(Λ, a)ei ˆ +H(t−s) Ut,s[ρ(s)] e−i ˆ +H(t−s) ˆU † +s(Λ, a)ei ˆ +H(t−s) += ˆUt(Λ, a) Ut,s[ρ(s)] ˆU † +t (Λ, a). +Eq. +(27) helps us to simplify the condition of symmetry, Eq.(23), on the Kraus operators. +Defining the Kraus operators ˆEt,s +λ +as +ˆEt,s +λ = ei ˆ +Ht ˆF t,s +λ e−i ˆ +Hs +(30) +which have the completeness condition, +� +λ +ˆEt,s† +λ +ˆEt,s +λ = ˆI, +(31) +we can rewrite Eq.(23) as +ˆU † +0(Λ, a) ˆE ˆU0(Λ, a) = U(Λ, a) ˆE. +(32) +Here, we introduced the vector ˆE with the λ component given by ˆEt,s +λ +and the matrix U(Λ, a) with +the (λ, λ′) component given by Uλλ′(Λ, a). We define the dynamical map Et,s as +Et,s[ρ] = +� +λ +ˆEt,s +λ ρ ˆEt,s† +λ +. +(33) +The condition (32) gives the fact that the map Et,s is symmetric under the Poincar`e group in the +sense that +ˆU0(Λ, a)Et,s[ρ] ˆU † +0(Λ, a) = Et,s[ ˆU0(Λ, a) ρ ˆU † +0(Λ, a)]. +(34) +7 + +The dynamical map Φt,s is written by the unitary map Ut,s and the dynamical map Et,s as +Φt,s[ρ] = +� +λ +ˆF t,s +λ ρ ˆF t,s† +λ += e−i ˆ +Ht � +λ +ˆEt,s +λ ei ˆ +Hsρe−i ˆ +Hs ˆEt,s† +λ +ei ˆ +Ht += e−i ˆ +HtEt,s[ei ˆ +Hsρe−i ˆ +Hs]ei ˆ +Ht += e−i ˆ +H(t−s)Et,s[ρ]ei ˆ +H(t−s) += Ut,s ◦ Et,s[ρ], +(35) +where in the fourth equality we used the symmetric condition (34) noticing that ei ˆ +Hs is the unitary +transformation of the time translation. +Our task is to determine ˆE satisfying Eq.(32) (or Et,s +satisfying Eq.(34)). Since Eq. (32) is decomposed into equations for each irreducible representation +subspace, the irreducible unitary representations of the Poincar´e group is useful for our analysis. +Let us present how to classify the unitary representations of the Poincar´e group [6]. We consider +the standard momentum ℓµ and the Lorentz transformation matrix (Sq)µν with +qµ = (Sq)µνℓν. +(36) +The unitary matrix U(Λ, a) is written as +U(Λ, a) = U(I, a)U(Λ, 0) = T (a)V(Λ), +(37) +where I is the identity matrix, U(I, a) = T (a) = e−iPµaµ and U(Λ, 0) = V(Λ). We define the vector +vq,ξ as +vq,ξ = NqV(Sq)vℓ,ξ, +(38) +where Pµvℓ,ξ = ℓµvℓ,ξ, Nq is the normalization and the label ξ describes the degrees of freedom +other than them determined by ℓµ. We obtain the following transformation rules for the vector +vq,ξ: +T (a)vq,ξ = Nqe−iP µaµV(Sq)vℓ,ξ += NqV(Sq)e−i(Sq)µνP νaµvℓ,ξ += NqV(Sq)e−i(Sq)µνℓνaµvℓ,ξ += NqV(Sq)e−iqµaµvℓ,ξ += e−iqµaµvq,ξ +(39) +8 + +and +V(Λ)vq,ξ = NqV(Λ)V(Sq)vℓ,ξ += NqV(ΛSq)vℓ,ξ += NqV(SΛq)V(S−1 +Λq ΛSq)vℓ,ξ += NqV(SΛq) +� +ξ′ +Dξ′ξ(Q(Λ, q))vℓ,s′ += Nq +NΛq +� +ξ′ +Dξ′ξ(Q(Λ, q))vΛq,s′, +(40) +where Q(Λ, q) = S−1 +Λq ΛSq is an element of the little group, which satisfies Qµνℓν = ℓµ, and Dξ′ξ(Q) +is the unitary representation of the little group. The irreducible unitary representations of the +Poincar´e group are classified by the standard momentum ℓµ and the irreducible unitary represen- +tations of the little group which does not change ℓµ. +standard momentum ℓµ +little group composed of Qµν with Qµνℓν = ℓµ +(a) +ℓµ = [M, 0, 0, 0], M > 0 +SO(3) +(b) +ℓµ = [−M, 0, 0, 0], M > 0 +SO(3) +(c) +ℓµ = [κ, 0, 0, κ], κ > 0 +ISO(2) +(d) +ℓµ = [−κ, 0, 0, κ], κ > 0 +ISO(2) +(e) +ℓµ = [0, 0, 0, N], N 2 > 0 +SO(2,1) +(f) +ℓµ = [0, 0, 0, 0] +SO(3,1) +TABLE I: Classification of the standard momentum ℓµ and the little group associated with ℓµ. +For simplicity, ξ is regarded as the label of basis vectors of the irreducible representation sub- +spaces of the little group. Other degeneracies not represented by q and ξ will be reintroduced in +the form of the dynamical map Φt,s, which we will see in the next section. We investigate Eq.(32) +restricted on each irreducible representation. For convenience, we separately focus on the Lorentz +transformation and the spacetime translation in Eq.(32). The unitary operator ˆU0(Λ, a) is written +as +ˆU0(Λ, a) = ˆU0(I, a) ˆU0(Λ, 0) = ˆT(a) ˆV (Λ), +(41) +where ˆU0(I, a) = ˆT(a) = e−i ˆPµaµ with ˆP µ = [ ˆH, ˆP 1, ˆP 2, ˆP 3] and ˆU0(Λ, 0) = ˆV (Λ) with the genera- +tors ˆJi and ˆKi +0. From Eq.(32) for Λ = I, we have +ˆT †(a) ˆE ˆT(a) = T (a) ˆE. +(42) +9 + +Eq.(32) for aµ = 0 gives +ˆV †(Λ) ˆE ˆV (Λ) = V(Λ) ˆE. +(43) +Introducing ˆEq,ξ = v† +q,ξ ˆE, we obtain the following equations from Eqs.(42) and (43): +ˆT †(a) ˆEq,ξ ˆT(a) = e−iqµaµ ˆEq,ξ +(44) +and +ˆV †(Λ) ˆEq,ξ ˆV (Λ) = +N ∗ +q +N ∗ +Λ−1q +� +ξ′ +D∗ +ξ′ξ(Q(Λ−1, q)) ˆEΛ−1q,ξ′, +(45) +where we used Eqs.(39) and (40), and Q(Λ, q) = S−1 +Λq ΛSq. The label ξ can take discrete or contin- +uous values. For the continous case, the summation � +ξ is replaced with the integration +� +dµ(ξ) +with a measure µ(ξ). Focusing on Eq.(45) for Λ = Sq, we get +ˆV †(Sq) ˆEq,ξ ˆV (Sq) = N ∗ +q ˆEℓ,ξ, +(46) +where note that Nℓ = 1 and Q(S−1 +q , q) = S−1 +S−1 +q +qS−1 +q Sq = S−1 +ℓ += I hold by the definition of vq,ξ. +Eq.(46) tells us that the Kraus operators ˆEq,ξ is determined from the Kraus operators ˆEℓ,ξ with +the standard momentum ℓµ. All we have to do is to give the form of the Kraus operators ˆEℓ,ξ. +To this end, we present the following equations given by Eq.(44) for qµ = ℓµ and by Eq.(45) for +qµ = ℓµ and Λ = W with W µνℓν = ℓµ, respectively: +ˆT †(a) ˆEℓ,ξ ˆT(a) = e−iℓµaµ ˆEℓ,ξ, +(47) +ˆV †(W) ˆEℓ,ξ ˆV (W) = +� +ξ′ +D∗ +ξ′ξ(W −1) ˆEℓ,ξ′, +(48) +where Q(Λ−1, q) = Q(W −1, ℓ) = S−1 +W −1ℓW −1Sℓ = W −1. In the next section, we construct a model +of the dynamical map with the Poincar´e symmetry to describe the reduced dynamics of a single +particle. +IV. +A MODEL OF THE DYNAMICAL MAP FOR A SINGLE PARTICLE +In this section, based on Eqs.(47) and (48), we give a model of the dynamical map with the +Poincar´e symmetry. To simplify the analysis, we consider the Hilbert space H0 ⊗ H1, where H0 is +the one-dimensional Hilbert space with a vacuum state |0⟩ and H1 is the irreducible subspace with +one-particle states. Any state vector |Ψ⟩ in H1 ( |Ψ⟩ ∈ H1 ) is given by +|Ψ⟩ = +� +d3q +� +σ +Ψ(p, σ) ˆa†(p, σ)|0⟩, +(49) +10 + +where |0⟩ is the vacuum state satisfying ˆa(p, σ)|0⟩ = 0, Ψ(p, σ) with the momentum p and the spin +σ is the wave function, ˆa(p, σ) and ˆa†(p, σ) are the annihilation and creation operators satisfying +[ˆa(p, σ), ˆa(p′, σ′)]± = 0 = [ˆa†(p, σ), ˆa†(p′, σ′)]±, +[ˆa(p, σ), ˆa†(p′, σ′)]± = δ3(p − p′)δσσ′. +(50) +In the above notation, [ ˆA, ˆB]± is defined as [ ˆA, ˆB]± = ˆA ˆB ± ˆB ˆA, in which the signs − and + apply +bosons and fermions, respectively. In Ref.[6, 26], the transformation rules of ˆa†(p, σ) are given by +ˆT(a)ˆa†(p, σ) ˆT †(a) = e−ipµaµˆa†(p, σ), +(51) +ˆV (Λ)ˆa†(p, σ) ˆV †(Λ) = +� +EpΛ +Ep +� +σ′ +Dσ′σ(Q(Λ, p))ˆa†(pΛ, σ′), +(52) +where Ep = p0, EpΛ = (Λp)0 and pΛ is the vector with the component (pΛ)i = (Λp)i. The matrix +Q(Λ, p) = S−1 +Λp ΛSp is the element of the little group which satisfies Q(Λ, p)µνkν = kµ, where kµ +is the standard momentum for a massive particle (kµ = [m, 0, 0, 0], m > 0) or a massless particle +(kµ = [k, 0, 0, k], k > 0). The momentum pµ and the standard momentum kµ are connected with +(Sp)µνkν = pµ, and Dσ′σ(Q(Λ, p)) is the irreducible unitary representation of the little group. +We consider the Kraus operators ˆEℓ,ξ acting on the Hilbert space H0 +� H1, that is, ˆEℓ,ξ : +H0 +� H1 → H0 +� H1, which have the following form +ˆEℓ,ξ = Aℓ,ξˆI + +� +d3p +� +σ +Bℓ,ξ(p, σ)ˆa(p, σ) + +� +d3p′d3p +� +σ′,σ +Cℓ,ξ(p′, σ′, p, σ)ˆa†(p′, σ′)ˆa(p, σ). +(53) +The dynamical map given by these operators describes the reduced dynamics of a single particle, +which can possibly decay into the vacuum state. Substituting the above operators into Eq.(47) +and Eq.(48), we obtain the equations +Aℓ,ξ = e−iℓµaµAℓ,ξ, +(54) +Bℓ,ξ(p, σ)e−ipµaµ = Bℓ,ξ(p, σ)e−iℓµaµ, +(55) +Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cℓ,ξ(p′, σ′, p, σ)e−iℓµaµ, +(56) +and +Aℓ,ξ = +� +ξ′ +D∗ +ξ′ξ(W −1)Aℓ,ξ′, +(57) +� +EpW +Ep +� +σ +Bℓ,ξ(pW , σ)D∗ +σ′σ(Q) = +� +ξ′ +D∗ +ξ′ξ(W −1)Bℓ,ξ′(p, σ′), +(58) +� +Ep′ +W EpW +Ep′Ep +� +σ′,σ +Cℓ,ξ(p′ +W, σ′, pW , σ)D¯σ′σ′(Q′)D∗ +¯σσ(Q) = +� +ξ′ +D∗ +ξ′ξ(W −1)Cℓ,ξ′(p′, ¯σ′, p, ¯σ), +(59) +11 + +where Q = Q(W −1, Wp) and Q′ = Q(W −1, Wp′). The derivation of these equations is devoted in +Appendix A. +We can analyze the explicit form of Aℓ,ξ, Bℓ,ξ(p, σ) and Cℓ,ξ(p′, σ′, p, σ) for a massive particle +and a massless particle, respectively. For the analysis, we assume that the massive particle has +a finite spin and the massless particle has a finite spin and a nonzero momentum. Through the +long computations presented in Appendices B and C, we get the following dynamical map with +the Poincar´e symmetry, +Φt,s[ρ(s)] = Ut,s◦Et,s[ρ(s)], +Et,s[ρ] = +� +j +� +β(j) +t,s +� +d3p +� +σ +ˆa(p, σ)ρˆa†(p, σ)+α(j) +t,s +�ˆI+γ(j) +t,s ˆN +� +ρ +�ˆI+γ(j)∗ +t,s +ˆN +�� +, +(60) +where α(j) +t,s , β(j) +t,s and γ(j) +t,s are the parameters depending on time, ˆN is the number operator defined +as +ˆN = +� +d3p +� +σ +ˆa†(p, σ)ˆa(p, σ), +(61) +and Ut,s is the unitary map given in (29). In the form of the dynamical map Φt,s, we recovered +the labels j which represent the degeneracies other than the labels q and ξ appearing in the Kraus +operators ˆEq,ξ defined around (44). The parameters α(j) +t,s, β(j) +t,s and γ(j) +t,s in Eq.(60) satisfy +0 ≤ α(j) +t,s ≤ 1, +� +j +α(j) +t,s = 1, +0 ≤ β(j) +t,s , +0 ≤ +� +j +β(j) +t,s ≤ 1 +� +j +� +β(j) +t,s + α(j) +t,s +� +γ(j) +t,s + γ(j)∗ +t,s ++ |γ(j) +t,s |2�� += 0. +(62) +These conditions come from the completeness condition of the Kraus operators (3). For the com- +putation of the completeness condition, note that the number operator ˆN satisfies ˆN 2 = ˆN on the +Hilbert space H0 ⊗ H1, since we assume that the dynamical map describes the reduced dynamics +of a single particle. +From the transformation rules of the creation and the annihilation operators, Eqs.(51) and (52), +it is easy to check that the map Et,s satisfies the condition of symmetry given in (34). Since the +unitary map Ut,s is symmetric under the Poincar´e group, which is checked around Eq.(29), we can +confirm that Φt,s is also symmetric. +Let us consider the case where there is no decay under the dynamical map Φt,s and focus on the +dynamics of one-particle states. In this case, the parameter � +j β(j) +t,s vanishes. Since the density +operator ρ given by one-particle states satisfies ˆNρ = ρ = ρ ˆN, we have +Φt,s[ρ(s)] = Ut,s ◦ Et,s[ρ(s)] = +� +j +α(j) +t,s |1 + γ(j) +t,s |2Ut,s[ρ(s)] = Ut,s[ρ(s)], +(63) +12 + +where we used the condition (62) with � +j β(j) +t,s = 0 in the third equality. This means that the +dynamical map with the Poincar´e symmetry for a non-decaying particle is the unitary map. The +result corresponds to a non-perturbative extension of the analysis in [25], which gives an implication +on the particle dynamics. For example, if the superposition state of a particle decoheres under +a non-unitary evolution, the Poincar´e symmetry breaks in the particle dynamics described by a +dynamical map. +We discuss the energy conservation. The expectation value of ˆHn at a time t, where n is a +natural number, is computed as +Tr[ ˆHnρ(t)] = +� +j +� +β(j) +t,s Tr[ ˆHn +� +d3p +� +σ +ˆa(p, σ)ρ(s)ˆa†(p, σ)] + α(j) +t,s Tr[ ˆHn(ˆI + γ(j) +t,s ˆN)ρ(s)(ˆI + γ(j)∗ +t,s +ˆN)] +� += (1 − +� +j +β(j) +t,s )Tr[ ˆHnρ(s)]. +(64) +In the reduced dynamics by the dynamical map Φt,s, the energy of a single particle is not conserved +unless � +j β(j) +t,s is a constant, even when the map is symmetric under the Poincar´e group. Such +a deviation between symmetry and conservation law was discussed in, for example, Refs [23] and +[24]. If the parameter � +j β(j) +t,s is a constant, then � +j β(j) +t,s = � +j β(j) +s,s = 0 and the dynamical map +Φt,s is reduced to the unitary map Ut,s as discussed above. +V. +CONCLUSION +We discussed what a dynamical map describing the reduced dynamics of an open quantum +system is realized under the Poincar´e symmetry. The unitary representation theory of the Poincar´e +group refines the condition for the dynamical map with the Poincar´e symmetry. For a massive +particle and a massless particle, we derived a concrete model of the dynamical map. In the model, +the particle can decay into the vacuum state. If there is no decay process, the dynamical map +describes the unitary evolution generated by the Hamiltonian of the particle. This means that +the non-decaying single particle does not decohere as long as the dynamical map for the particle +has the Poincar´e symmetry. In this way, it was exemplified that the Poincar´e symmetry strongly +constrains the possible dynamics of an open quantum system. +In this paper, we assumed an open system with a single particle. Our analysis is possible to +be extended to the case with many particles. +Considering interactions among many particles, +we can understand more general effective theories in terms of the Poincar´e symmetry. For the +particles interacting via gravity, the models which induce intrinsic gravitational decoherence have +13 + +been proposed [15–19]. These models are written in the theory of open quantum systems. In the +weak field regime of gravity, the Poincar´e symmetry may provide a guidance for establishing the +theory of an open quantum system with gravitating particles. +This paper has a potential to develop the theory of open quantum systems. To describe the +reduced dynamics of an open quantum system, a quantum master equation is often adopted. It +has been discussed how the quantum Markov dynamics given by the equation is consistent with +a relativistic theory [27, 28]. Applying the present approach, it will be possible to discuss the +quantum Markov dynamics with the Poincar´e symmetry. +It is hoped that this paper paves the way to understand a relativistic theory of open quantum +systems and to study the interplay between quantum theory and gravity. +Acknowledgments +We thank Y. Kuramochi for useful discussions and comments related to this paper. A.M. was +supported by 2022 Research Start Program 202203. +Appendix A: Derivation of Eqs.(54),(55),(56),(57),(58) and (59) +We present the transformation rules of Aℓ,ξ, Bℓ,ξ and Cℓ,ξ given in Eqs.(54),(55),(56),(57),(58) +and (59). Using the assumed form of the Kraus operators ˆEℓ,ξ defined by (53), we can compute +the right hand side of Eq.(47) as +ˆT †(a) ˆEℓ,ξ ˆT(a) = Aℓ,ξˆI + +� +d3p +� +σ +Bℓ,ξ(p, σ)e−ipµaµˆa(p, σ) ++ +� +d3p′d3p +� +σ′,σ +Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµˆa†(p′, σ′)ˆa(p, σ). +From Eq.(47), we have +Aℓ,ξ = e−iℓµaµAℓ,ξ, +(A1) +Bℓ,ξ(p, σ)e−ipµaµ = Bℓ,ξ(p, σ)e−iℓµaµ +(A2) +Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cℓ,ξ(p′, σ′, p, σ)e−iℓµaµ. +(A3) +14 + +The right hand side of Eq.(48) is evaluated as +ˆV †(W) ˆEℓ,ξ ˆV (W) += Aℓ,ξˆI + +� +d3p +� +σ +Bℓ,ξ(p, σ) +� +EpW −1 +Ep +� +σ′ +D∗ +σ′σ(Q(W −1, p))ˆa(pW −1, σ′) ++ +� +d3p′d3p +� +σ′,σ +Cℓ,ξ(p′, σ′, p, σ) +× +� +Ep′ +W −1 +Ep′ +� +EpW −1 +Ep +� +¯σ,¯σ′ +D¯σ′σ′(Q(W −1, p′))D∗ +¯σσ(Q(W −1, p))ˆa†(p′ +W −1, ¯σ′)ˆa(pW −1, ¯σ) += Aℓ,ξˆI + +� +d3p +� +σ +Bℓ,ξ(pW, σ) +� +EpW +Ep +� +σ′ +D∗ +σ′σ(Q(W −1, Wp))ˆa(p, σ′) ++ +� +d3p′d3p +� +σ′,σ +Cℓ,ξ(p′ +W, σ′, pW , σ) +× +� +Ep′ +W +Ep′ +� +EpW +Ep +� +¯σ,¯σ′ +D¯σ′σ′(Q(W −1, Wp′))D∗ +¯σσ(Q(W −1, Wp))ˆa†(p′, ¯σ′)ˆa(p, ¯σ), +where note that the Lorentz invariant measure is d3p/Ep and hence f(p)d3p = Epf(p)d3p/Ep = +EpΛf(pΛ)d3p/Ep. From Eq.(48), we have +Aℓ,ξ = +� +ξ′ +D∗ +ξ′ξ(W −1)Aℓ,ξ′, +(A4) +� +EpW +Ep +� +σ +Bℓ,ξ(pW, σ)D∗ +σ′σ(Q) = +� +ξ′ +D∗ +ξ′ξ(W −1)Bℓ,ξ′(p, σ′) +(A5) +� +Ep′ +W EpW +Ep′Ep +� +σ′,σ +Cℓ,ξ(p′ +W , σ′, pW, σ)D¯σ′σ′(Q′)D∗ +¯σσ(Q) = +� +ξ′ +D∗ +ξ′ξ(W −1)Cℓ,ξ′(p′, ¯σ′, p, ¯σ), +(A6) +where Q = Q(W −1, Wp) and Q′ = Q(W −1, Wp′). +Appendix B: Analysis of a massive particle +We assume that the spectrum of ˆP µ on any state |Ψ⟩ in the Hilbert space of one-particle states, +H1, satisfies +ˆP µ ˆPµ|Ψ⟩ = −m2|Ψ⟩, +⟨Ψ| ˆP 0|Ψ⟩ > 0. +(B1) +The above equations are equivalent to the fact that the Hamiltonian ˆH = ˆP 0 has the form ˆH = +� +ˆPk ˆP k + m2, which implies that |Ψ⟩ is the state of a massive particle. In this appendix, we derive +the form of the dynamical map Et,s for a massive particle. +15 + +Case (a) ℓµ = [M, 0, 0, 0], M > 0 or (b) ℓµ = [−M, 0, 0, 0], M > 0 : We focus on the spectrum +ℓµ = [±M, 0, 0, 0], M > 0. From Eq.(A1) for all aµ = [a, 0, 0, 0], we have +Aℓ,ξ = e±iMaAℓ,ξ +∴ +Aℓ,ξ = 0. +Eq.(A2) for all aµ = [0, a] leads to +Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ) +∴ +Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p). +From Eq.(A2) for all aµ = [a, 0, 0, 0], we get +Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iMa, +and combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we obtain +Bℓ,ξ(σ)eima = Bℓ,ξ(σ)e±iMa. +Since the mass m is positive, to get a nontrivial result, we should choose +M with M = m. Using +Eq.(A5) for Q = R ∈ SO(3) and adopting the result Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we find +� +σ +Bℓ,ξ(σ)D∗ +σ′σ(R−1) = +� +ξ′ +D∗ +ξ′ξ(R−1)Bℓ,ξ′(σ′), +where note that Q = Q(W −1, Wp) = Q(R−1, Rℓ) = S−1 +ℓ R−1SRℓ = R−1 for ℓµ = [m, 0, 0, 0]. Since +the representations Dσ′σ and Dξ′ξ are irreducible and unitary, by Schur’s lemma we have +Bℓ,ξ(σ) = Bℓ uξσ, +where uξσ is a unitary matrix. From Eq.(A3) for all aµ = [0, a], we deduce +Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ) +∴ +Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p). +Eq.(A3) for all aµ = [a, 0, 0, 0] leads to +Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iMa, +and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p) into the above equation, we have +Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(p, σ′, σ)e±iMa +∴ +Cℓ,ξ(p, σ′, σ) = 0. +The above results imply that the Kraus operator ˆEℓ,ξ with ℓµ = [m, 0, 0, 0] has the following form, +ˆEℓ,ξ = Bℓ +� +σ +uξσˆa(0, σ). +16 + +Eq.(46) tells us that +ˆEq,ξ = N ∗ +q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = N ∗ +q Bℓ +� +Eq +m +� +σ +uξσˆa(q, σ), +where Eq = (Sq ℓ)0 and qi = (Sq ℓ)i. To determine the normalization Nq, the inner product v† +q,ξvq,ξ +is assumed to be +v† +q′,s′vq,ξ = δ3(q′ − q)δξ′ξ, +which leads to Nq = +� +m/Eq up to a phase factor. For this normalization, the following complete- +ness condition is given as +� +d3q +� +s +vq,ξv† +q,ξ = I. +Under the completeness condition, we derive a part of the dynamical map Et,s as +Et,s[ρ(s)] ⊃ |Bℓ|2 +� +d3q +� +σ +ˆa(q, σ)ρ(s)ˆa†(q, σ), +(B2) +where we used the fact that uξσ is the unitary matrix. +Case (c) ℓµ = [κ, 0, 0, κ], κ > 0 or (d) ℓµ = [−κ, 0, 0, κ], κ > 0 : We consider the spectrum +ℓµ = [±κ, 0, 0, κ], κ > 0. From Eq.(A1) for all aµ = [a, 0, 0, 0], we have +Aℓ,ξ = e±iκaAℓ,ξ +∴ +Aℓ,ξ = 0. +From Eq.(A2) for all aµ = [0, a], we get +Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a +∴ +Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), +where ℓ = [0, 0, κ]T. Eq.(A2) for all aµ = [a, 0, 0, 0] leads to +Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iκa, +and from the equation Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we obtain +Bℓ,ξ(σ)ei +√ +κ2+m2a = Bℓ,ξ(σ)e±iκa +∴ +Bℓ,ξ(σ) = 0, +where Eℓ = +√ +ℓ2 + m2 = +√ +κ2 + m2. Eq.(A3) for all aµ = [0, a] gives +Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a +∴ +Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ). +Using Eq.(A3) for all aµ = [a, 0, 0, 0], we get +Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iκa, +17 + +and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have +Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)e±iκa. +Noticing the fact that Ep−ℓ − Ep ± κ ̸= 0, we get the result Cℓ,ξ(p, σ′, σ) = 0. Combined with the +above analysis, the Kraus operator ˆEℓ,ξ vanishes: +ˆEℓ,ξ = 0 +∴ +ˆEq,ξ = Nq ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0. +(B3) +Case (e) ℓµ = [0, 0, 0, N], N 2 > 0 : We focus on the spectrum ℓµ = [0, 0, 0, N], N 2 > 0. From +Eq.(A1) for all aµ = [0, a], we have +Aℓ,ξ = e−iℓ·aAℓ,ξ +∴ +Aℓ,ξ = 0. +Eq.(A2) for all aµ = [a, 0, 0, 0] leads to +Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ) +∴ +Bℓ,ξ(p, σ) = 0, +where note that Eq = +� +q2 + m2 ̸= 0. From Eq.(A3) for all aµ = [0, a], we deduce +Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a +∴ +Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ), +where ℓ = [0, 0, N]T. From Eq.(A3) for all aµ = [a, 0, 0, 0], we get +Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ), +and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have +Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ) +∴ +Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(σ′, σ)δ3(p − ℓ/2). +Combined with the above analysis, the function Cℓ,ξ(p′, σ′, p, σ) is +Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(σ′, σ)δ3(p′ + ℓ/2)δ3(p − ℓ/2), +and the Kraus operator ˆEℓ,ξ is written as +ˆEℓ,ξ = +� +σ′,σ +Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ). +By the completeness condition of the Kraus operators, Eq.(31), the above Kraus operator ˆEℓ,ξ +should satisfy ˆE† +ℓ,ξ ˆEℓ,ξ ≤ ˆI. Concretely, ˆE† +ℓ,ξ ˆEℓ,ξ is evaluated as +ˆE† +ℓ,ξ ˆEℓ,ξ = +� +¯σ′,¯σ +C∗ +ℓ,ξ(¯σ′, ¯σ)ˆa†(ℓ/2, ¯σ)ˆa(−ℓ/2, ¯σ′) +� +σ′,σ +Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ) += δ3(0) +� +σ′ +� � +¯σ +Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) +�† � +σ +Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ), +18 + +where the term given by the linear combination of ˆa†ˆa†ˆaˆa vanishes on H0 +� H1. +To satisfy +ˆE† +ℓ,ξ ˆEℓ,ξ ≤ ˆI, we find that +� +σ′ +� � +¯σ +Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) +�† � +σ +Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ) = 0 +∴ +Cℓ,ξ(σ′, σ) = 0 +The consequence of Cℓ,ξ(σ′, σ) = 0 is that the Kraus operator ˆEℓ,ξ vanishes as ⟨Φ| ˆEℓ,ξ|Ψ⟩ = 0 for +all |Ψ⟩, |Φ⟩ ∈ H0 +� H1, and hence +ˆEq,ξ = N ∗ +q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0, +(B4) +on the Hilbert space H0 +� H1. +Case (f) ℓµ = [0, 0, 0, 0] : We consider the case where ℓµ = [0, 0, 0, 0]. In the following, we drop +the label ℓ. Eq.(A1) is identical for all aµ. Since the little group associated with ℓµ is SO(3, 1), +Eq.(A4) for W = Λ ∈ SO(3, 1) is given as +Aξ = +� +ξ′ +D∗ +ξ′ξ(Λ−1)Aξ′. +(B5) +Eq.(A2) for all aµ = [a, 0, 0, 0] gives the condition +Bξ(p, σ)eiEpa = Bξ(p, σ) +∴ +Bξ(p, σ) = 0, +where note that Eq = +� +q2 + m2 ̸= 0. From Eq.(A3) for all aµ, we obtain +Cξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cξ(p′, σ′, p, σ) +∴ +Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p). +Eq. (A6) for W = Λ ∈ SO(3, 1) is written as +� +Ep′ +ΛEpΛ +Ep′Ep +� +σ′,σ +Cξ(p′ +Λ, σ′, pΛ, σ)D¯σ′σ′(Q′)D∗ +¯σσ(Q) = +� +ξ′ +D∗ +ξ′ξ(Λ−1)Cξ′(p′, ¯σ′, p, ¯σ), +where Q += +Q(Λ−1, Λp) and Q′ += +Q(Λ−1, Λp′). +From the equation Cξ(p′, σ′, p, σ) += +Cξ(p, σ′, σ)δ3(p′ − p) and noticing the fact that the invariant delta function is Epδ3(p − p′), we +get the condition +� +σ′,σ +Cξ(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ +¯σσ(Q) = +� +ξ′ +D∗ +ξ′ξ(Λ−1)Cξ′(p, ¯σ′, ¯σ), +(B6) +where Q′ = Q(Λ−1, Λp′) turns out to be Q = Q(Λ−1, Λp) by the presence of the delta function +δ3(p − p′). It is known that the dimension of irreducible unitary representations Dξ′ξ of SO(3,1) +19 + +is one or infinite [29]. For the one-dimensional representation, dropping the label ξ, we find that +Eq.(B5) trivially holds and that Eq.(B6) is reduced to +� +σ′,σ +C(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ +¯σσ(Q) = C(p, ¯σ′, ¯σ). +For p = 0 and Λ = R ∈ SO(3), we get +� +σ′,σ +C(0, σ′, σ)D¯σ′σ′(R−1)D∗ +¯σσ(R−1) = C(0, ¯σ′, ¯σ) +∴ +C(0, σ′, σ) = Cδσ′σ, +where this holds by the Schur’s lemma. Choosing p = 0 and Λ = Sp with (Sp)µνkν = pµ for +kµ = [m, 0, 0, 0], we have +C(p, σ′, σ) = C(0, σ′, σ), +where we used Q = Q(S−1 +p , p) = S−1 +k S−1 +p Sp = I and Dσ′σ(I) = δσ′σ. Hence C(p, σ′, σ) = Cδσ′σ. +For the infinite dimensional representation, Eq.(B5) leads to Aℓ,ξ = 0, and Eq.(B6) for p = 0 and +Λ = R ∈ SO(3) gives +� +σ′,σ +Cξ(0, σ′, σ)D¯σ′σ′(R−1)D∗ +¯σσ(R−1) = +� +ξ′ +D∗ +ξ′ξ(R−1)Cξ′(0, ¯σ′, ¯σ). +Assuming that the massive particle has a finite spin and using the Schur’s lemma, we get +Cξ′(0, ¯σ′, ¯σ) = 0. Eq.(B6) for p = 0 and Λ = Sp with (Sp)µνkν = pµ for kµ = [m, 0, 0, 0] pro- +vides +Cξ(p, σ′, σ) = +� +ξ′ +D∗ +ξ′ξ(S−1 +p )Cξ′(0, σ′, σ) = 0. +The above analysis on ℓµ = [0, 0, 0, 0] tells us the following Kraus operator +ˆE = AˆI + C ˆN, +where ˆN is the number operator defined in (61). A part of the dynamical map Et,s is given as +Et,s[ρ(s)] ⊃ +� +AˆI + C ˆN +� +ρ(s) +� +AˆI + C ˆN +�† +. +(B7) +The above results given in Eqs.(B2), (B3), (B4) and (B7) provide the following form of Et,s: +Et,s[ρ(s)] = |Bℓ|2 +� +d3q +� +σ +ˆa(q, σ)ρ(s)ˆa†(q, σ) + +� +AˆI + C ˆN +� +ρ(s) +� +AˆI + C ˆN +�† +. +(B8) +Recovering other degeneracies labeled by j differently from q and ξ, introducing � +j and redefining +the parameters as |A|2 = α(j) +t,s , C/A = γ(j) +t,s and |Bℓ|2 = β(j) +t,s , we get the form of the dynamical map +Et,s given in (60). +20 + +Appendix C: Analysis on a massless particle +We assume that the spectrum of ˆP µ on any state |Ψ⟩ in the Hilbert space of one-particle states, +H1, satisfies +ˆP µ ˆPµ|Ψ⟩ = 0, +⟨Ψ| ˆP 0|Ψ⟩ > 0. +(C1) +The above equations leads to the fact that the Hamiltonian ˆH = ˆP 0 has the form ˆH = +� +ˆPk ˆP k, +which means that |Ψ⟩ is the state of a massless particle. In this appendix, we derive the form of +the dynamical map Et,s for a massless particle with nonzero momentum. +Case (a) ℓµ = [M, 0, 0, 0], M > 0 or (b) ℓµ = [−M, 0, 0, 0], M > 0 : We focus on the spectrum +ℓµ = [±M, 0, 0, 0], M > 0. Eq.(A1) for all aµ = [a, 0, 0, 0] gives +Aℓ,ξ = e±iMaAℓ,ξ +∴ +Aℓ,ξ = 0. +Eq.(A2) for all aµ = [0, a] leads to +Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ) +∴ +Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p). +From Eq.(A2) for all aµ = [a, 0, 0, 0], we get +Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iMa, +and combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we obtain +Bℓ,ξ(σ) = Bℓ,ξ(σ)e±iMa +∴ +Bℓ,ξ(σ) = 0. +Using Eq.(A3) for all aµ = [0, a], we deduce +Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ) +∴ +Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p). +Eq.(A3) for all aµ = [a, 0, 0, 0] leads to +Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iMa, +and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p) into the above equation, we have +Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(p, σ′, σ)e±iMa +∴ +Cℓ,ξ(p, σ′, σ) = 0. +The above results imply that the Kraus operator ˆEℓ,ξ vanishes and has the following form, +ˆEq,ξ = N ∗ +q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0. +(C2) +21 + +Case (c) ℓµ = [κ, 0, 0, κ], κ > 0 or (d) ℓµ = [−κ, 0, 0, κ], κ > 0 : We consider the spectrum +ℓµ = [±κ, 0, 0, κ], κ > 0. From Eq.(A1) for all aµ = [a, 0, 0, 0], we have +Aℓ,ξ = e±iκaAℓ,ξ +∴ +Aℓ,ξ = 0. +Eq.(A2) for all aµ = [0, a] gives +Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a +∴ +Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), +where ℓ = [0, 0, κ]T. Eq.(A2) for all aµ = [a, 0, 0, 0] leads to +Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iκa, +and from the equation Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we have +Bℓ,ξ(σ)eiκa = Bℓ,ξ(σ)e±iκa, +where Eℓ = +√ +ℓ2 = κ. +To get a nontrivial result, we should choose +κ. +Using Eq.(A5) for +Q = L ∈ ISO(2) and adopting the result Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we find +� +σ +Bℓ,ξ(σ)D∗ +σ′σ(L−1) = +� +ξ′ +D∗ +ξ′ξ(L−1)Bℓ,ξ′(σ′), +where note that Q = Q(W −1, Wp) = Q(L−1, Lℓ) = S−1 +ℓ L−1SLℓ = L−1 for ℓµ = [κ, 0, 0, κ]. Since +the representations Dσ′σ and Dξ′ξ are irreducible and unitary, by Schur’s lemma we get +Bℓ,ξ(σ) = Bℓ uξσ, +where uξσ is a unitary matrix. Using Eq.(A3) for all aµ = [0, a], we deduce +Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a +∴ +Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ). +Eq.(A3) for all aµ = [a, 0, 0, 0] leads to +Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iκa, +and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have +Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)e±iκa. +The condition of Ep−ℓ − Ep + κ = 0 is written by p⊥ = [p1, p2] = 0 and p3 ≥ κ, and the condition +of Ep−ℓ − Ep − κ = 0 is given by p⊥ = 0 and p3 ≤ 0. Hence, the form of Cℓ,ξ(p, σ′, σ) is +Cℓ,ξ(p, σ′, σ) = +� +C+ +ℓ,ξ(p3, σ′, σ)θ(p3 − κ) + C− +ℓ,ξ(p3, σ′, σ)θ(−p3) +� +δ2(p⊥) +22 + +Combined with the above analysis, the Kraus operator ˆE+ +ℓ,ξ for +κ is +ˆE+ +ℓ,ξ = Bℓ +� +σ +uξσˆa(ℓ, σ) + +� +dp3 +� +σ,σ′ +C+ +ℓ,ξ(p3, σ′, σ)θ(p3 − κ)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ), +and the Kraus operator ˆE− +ℓ,ξ for −κ is +ˆE− +ℓ,ξ = +� +dp3 +� +σ,σ′ +C− +ℓ,ξ(p3, σ′, σ)θ(−p3)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ). +By the completeness condition of the Kraus operators, Eq.(31), the above Kraus operator ˆE± +ℓ,ξ +should satisfy ˆE±† +ℓ,ξ ˆE± +ℓ,ξ ≤ ˆI. Concretely, ˆE+† +ℓ,ξ ˆE+ +ℓ,ξ is evaluated as +ˆE+† +ℓ,ξ ˆE+ +ℓ,ξ = +� +Bℓ +� +σ +uξσˆa(ℓ, σ) + +� +dp3 +� +σ,σ′ +C+ +ℓ,ξ(p3, σ′, σ)θ(p3 − κ)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ) +�† +× +� +Bℓ +� +σ +uξσˆa(ℓ, σ) + +� +dp3 +� +σ,σ′ +C+ +ℓ,ξ(p3, σ′, σ)θ(p3 − κ)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ) +� += |Bℓ|2 � +σ +u∗ +sσˆa†(ℓ, σ) +� +σ′ +usσ′ˆa(ℓ, σ′) ++ δ2(0) +� +dp3 +� +σ′ +� +σ,¯σ +C+∗ +ℓ,ξ (p3, σ′, σ)C+ +ℓ,ξ(p3, σ′, ¯σ)θ(p3 − κ)ˆa†(0, p3, σ)ˆa(0, p3, ¯σ), +where the term given by the linear combination of ˆa†ˆaˆa, ˆa†ˆa†ˆa, and ˆa†ˆa†ˆaˆa vanishes on H0 +� H1. +To satisfy ˆE+† +ℓ,ξ ˆE+ +ℓ,ξ ≤ ˆI, we find +� +dp3 +� +σ′ +� +σ,¯σ +C+∗ +ℓ,ξ (p3, σ′, σ)C+ +ℓ,ξ(p3, σ′, ¯σ)θ(p3−κ)ˆa†(0, p3, σ)ˆa(0, p3, ¯σ) = 0 +∴ +C+ +ℓ,ξ(p3, σ′, ¯σ) = 0 +In the same manner, we have +ˆE−† +ℓ,ξ ˆE− +ℓ,ξ = δ2(0) +� +dp3 +� +σ′ +� +σ,¯σ +C−∗ +ℓ,ξ (p3, σ′, σ)C− +ℓ,ξ(p3, σ′, ¯σ)θ(−p3)ˆa†(0, p3, σ)ˆa(0, p3, ¯σ), +and to satisfy ˆE−† +ℓ,ξ ˆE− +ℓ,ξ ≤ ˆI, we obtain +� +dp3 +� +σ′ +� +σ,¯σ +C−∗ +ℓ,ξ (p3, σ′, σ)C− +ℓ,ξ(p3, σ′, ¯σ)θ(−p3)ˆa†(0, p3, σ)ˆa(0, p3, ¯σ) = 0 +∴ +C− +ℓ,ξ(p3, σ′, ¯σ) = 0. +These analyses give the following form of the Kraus operators, +ˆE+ +ℓ,ξ = Bℓ +� +σ +uξσˆa(ℓ, σ), +ˆE− +ℓ,ξ = 0, +on the Hilbert space H0 +� H1. By Eq. (46), we get +ˆE+ +q,ξ = N ∗ +q ˆV (Sq) ˆE+ +ℓ,ξ ˆV †(Sq) = N ∗ +q Bℓ +� +Eq +κ +� +σ +uξσˆa(q, σ), +ˆE− +q,ξ = N ∗ +q ˆV (Sq) ˆE− +ℓ,ξ ˆV †(Sq) = 0. +23 + +Setting that the inner product v† +q,ξvq,ξ is +v† +q′,s′vq,ξ = δ3(q′ − q)δξ′ξ, +the normalization Nq is given as Nq = +� +κ/Eq up to a phase factor. For this normalization, we +get the following completeness condition as +� +d3q +� +s +vq,ξv† +q,ξ = I. +Taking account for the completeness, we can derive a part of the dynamical map Et,sas +Et,s[ρ(s)] ⊃ |Bℓ|2 +� +d3q +� +σ +ˆa(q, σ)ρ(s)ˆa†(q, σ), +(C3) +where we used the fact that uξσ is the unitary matrix. +Case (e) ℓµ = [0, 0, 0, N], N 2 > 0 : We focus on the spectrum ℓµ = [0, 0, 0, N], N 2 > 0. From +Eq.(A1) for all aµ = [0, a], we have +Aℓ,ξ = e−iℓ·aAℓ,ξ +∴ +Aℓ,ξ = 0. +Eq.(A2) for all aµ = [0, a] leads to +Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a +∴ +Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ). +Eq.(55) for all aµ = [a, 0, 0, 0] gives +Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ), +and then combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we get +Bℓ,ξ(σ)eiκa = Bℓ,ξ(σ) +∴ +Bℓ,ξ(σ) = 0 +where we used Eℓ = +√ +ℓ2 = κ > 0. Adopting Eq.(A3) for all aµ = [0, a], we deduce +Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a +∴ +Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ), +where ℓ = [0, 0, N]T. From Eq.(A3) for all aµ = [a, 0, 0, 0], we get +Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ), +and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have +Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ) +∴ +Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(σ′, σ)δ3(p − ℓ/2). +24 + +Combined with the above analysis, the function Cℓ,ξ(p′, σ′, p, σ) is +Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(σ′, σ)δ3(p′ + ℓ/2)δ3(p − ℓ/2), +and the Kraus operator ˆEℓ,ξ is written as +ˆEℓ,ξ = +� +σ′,σ +Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ). +By the completeness condition of the Kraus operators, Eq.(31), the above Kraus operator ˆEℓ,ξ +should satisfy ˆE† +ℓ,ξ ˆEℓ,ξ ≤ ˆI. Explicitly, ˆE† +ℓ,ξ ˆEℓ,ξ is evaluated as +ˆE† +ℓ,ξ ˆEℓ,ξ = +� +¯σ′,¯σ +C∗ +ℓ,ξ(¯σ′, ¯σ)ˆa†(ℓ/2, ¯σ)ˆa(−ℓ/2, ¯σ′) +� +σ′,σ +Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ) += δ3(0) +� +σ′ +� � +¯σ +Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) +�† � +σ +Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ), +where the term associated with the linear combination of ˆa†ˆa†ˆaˆa vanishes on H0 +� H1. To satisfy +ˆE† +ℓ,ξ ˆEℓ,ξ ≤ ˆI, we find that +� +σ′ +� � +¯σ +Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) +�† � +σ +Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ) = 0 +∴ +Cℓ,ξ(σ′, σ) = 0 +Hence, the Kraus operator ˆEℓ,ξ vanishes, and we have that +ˆEq,ξ = N ∗ +q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0, +(C4) +on the Hilbert space H0 +� H1. +Case (f) ℓµ = [0, 0, 0, 0] : We focus on the spectrum ℓµ = [0, 0, 0, 0]. In the following, we do +not write the label ℓ. Eq.(A1) is identical for all aµ. Since the little group associated with ℓµ is +SO(3, 1), Eq.(A4) for W = Λ ∈ SO(3, 1) is given as +Aξ = +� +ξ′ +D∗ +ξ′ξ(Λ−1)Aξ′. +(C5) +Eq.(A2) for all aµ = [0, a] gives the condition +Bξ(p, σ)e−ip·a = Bξ(p, σ) +∴ +Bξ(p, σ) = Bξ(σ)δ3(p). +This equation makes Eq.(A2) for all aµ = [a, 0, 0, 0] and Eq. (A5) for W = Λ ∈ SO(3, 1) trivial. +This form Bξ(p, σ) = Bξ(σ)δ3(p) leads to ˆEℓ,ξ ⊃ � +σ Bξ(σ)ˆa(0, σ). However, this operator vanishes +on the Hilbert space of massless particles since we assumed that there are no states with zero +momentum. Eq.(A3) for all aµ gives us the condition +Cξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cξ(p′, σ′, p, σ) +∴ +Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p). +25 + +Eq. (A6) for W = Λ ∈ SO(3, 1) is written as +� +Ep′ +ΛEpΛ +Ep′Ep +� +σ′,σ +Cξ(p′ +Λ, σ′, pΛ, σ)D¯σ′σ′(Q′)D∗ +¯σσ(Q) = +� +ξ′ +D∗ +ξ′ξ(Λ−1)Cξ′(p′, ¯σ′, p, ¯σ), +where Q += +Q(Λ−1, Λp) and Q′ += +Q(Λ−1, Λp′). +From the equation Cξ(p′, σ′, p, σ) += +Cξ(p, σ′, σ)δ3(p′ − p) and noticing the fact that the invariant delta function is Epδ3(p − p′), we +get the condition +� +σ′,σ +Cξ(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ +¯σσ(Q) = +� +ξ′ +D∗ +ξ′ξ(Λ−1)Cξ′(p, ¯σ′, ¯σ), +(C6) +where note that the delta function δ3(p − p′) leads to Q′ = Q(Λ−1, Λp′) = Q(Λ−1, Λp) = Q. The +(proper orthochronous) Lorentz group SO(3, 1) has one and infinite dimensional unitary irreducible +representations [29]. Choosing the one-dimensional representation of Dξ′,ξ and dropping the label +ξ, we find that Eq.(C5) trivially holds and that Eq.(C6) is reduced to +� +σ′,σ +C(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ +¯σσ(Q) = C(p, ¯σ′, ¯σ). +For p = ℓ = [0, 0, κ] and Λ = L ∈ ISO(2), we get +� +σ′,σ +C(ℓ, σ′, σ)D¯σ′σ′(L−1)D∗ +¯σσ(L−1) = C(ℓ, ¯σ′, ¯σ) +∴ +C(ℓ, σ′, σ) = Cδσ′σ, +where we used the Schur’s lemma. Choosing p = ℓ and Λ = Sp with (Sp)µνℓν = pµ for ℓµ = +[κ, 0, 0, κ], we have +C(p, σ′, σ) = C(ℓ, σ′, σ), +where we used Q = Q(S−1 +p , p) = S−1 +k S−1 +p Sp = I and Dσ′σ(I) = δσ′σ. Hence C(p, σ′, σ) = Cδσ′σ. +If we adopt the infinite dimensional representation of Dξ′ξ, Eq.(C5) leads to Aℓ,ξ = 0 and Eq.(C6) +for p = ℓ and Λ = L ∈ ISO(2) gives +� +σ′,σ +Cξ(ℓ, σ′, σ)D¯σ′σ′(L−1)D∗ +¯σσ(L−1) = +� +ξ′ +D∗ +ξ′ξ(L−1)Cξ′(ℓ, ¯σ′, ¯σ). +Assuming that the massless particle has a finite spin and using the Schur’s lemma, we get +Cξ′(ℓ, ¯σ′, ¯σ) = 0. Eq.(C6) for p = ℓ and Λ = Sp with (Sp)µνℓν = pµ for ℓµ = [κ, 0, 0, κ] pro- +vides +Cξ(p, σ′, σ) = +� +ξ′ +D∗ +ξ′ξ(S−1 +p )Cξ′(ℓ, σ′, σ) = 0. +26 + +The above analysis tells us that the Kraus operator has the following form +ˆE = AˆI + C ˆN, +where ˆN is the number operator defined in (61). +A part of the dynamical map Et,s with the +Poincar´e symmetry is given as +Et,s[ρ(s)] ⊃ +� +AˆI + C ˆN +� +ρ(s) +� +AˆI + C ˆN +�† +. +(C7) +Gathering the above results (C2), (C3), (C4) and (C7), we have the following form of Et,s: +Et,s[ρ(s)] = |Bℓ|2 +� +d3q +� +σ +ˆa(q, σ)ρ(s)ˆa†(q, σ) + +� +AˆI + C ˆN +� +ρ(s) +� +AˆI + C ˆN +�† +. +(C8) +In the same manner performed around (B8), we obtain the form of the dynamical map Et,s given +in (60). +[1] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, “Colloquium: Non-Markovian Dynamics in Open +Quantum Systems”, Rev. Mod. Phys. 88, 021002 (2016). +[2] H.-P. Breuer and F. 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Math. 48, 568 (1947). +28 + diff --git a/3dAzT4oBgHgl3EQffPzi/content/tmp_files/load_file.txt b/3dAzT4oBgHgl3EQffPzi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d6e69c228a0ca451c2ee75f0393b8dde9c9966e --- /dev/null +++ b/3dAzT4oBgHgl3EQffPzi/content/tmp_files/load_file.txt @@ -0,0 +1,700 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf,len=699 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='01451v1 [quant-ph] 4 Jan 2023 Reduced dynamics with Poincar´e symmetry in open quantum system Akira Matsumura∗ Department of Physics, Kyushu University, Fukuoka, 819-0395, Japan Abstract We consider how the reduced dynamics of an open quantum system coupled to an environment admits the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The reduced dynamics is described by a dynamical map, which is given by tracing out the environment from the total dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Introducing the notion of covariant map, we investigate the dynamical map which is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Based on the representation theory of the Poincar´e group, we develop a systematic way to give the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using this way, we derive the dynamical map for a massive particle with a finite spin and a massless particle with a finite spin and a nonzero momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We show that the derived map gives the unitary evolution of a particle when its energy is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We also find that the dynamical map for a particle does not have the Poincar´e symmetry when the superposition state of the particle decoheres into a mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ∗Electronic address: matsumura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='akira@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='kyushu-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='jp 1 Contents I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Introduction 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Quantum dynamical map and its symmetry 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Dynamical map with Poincar´e symmetry 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A model of the dynamical map for a single particle 10 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Conclusion 13 Acknowledgments 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (54),(55),(56),(57),(58) and (59) 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Analysis of a massive particle 15 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Analysis on a massless particle 21 References 27 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' INTRODUCTION It is difficult to isolate a quantum system perfectly, which is affected by the inevitable influence of a surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Such a quantum system is called an open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since we encounter open quantum systems in a wide range of fields such as quantum information science [1, 2], condensed matter physics [3, 4] and high energy physics [5], it is important to understand their dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In general, the dynamics of an open quantum system, the so-called reduced dynamics, is very complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This is because the environment may have infinitely many degrees of freedom and they are uncontrollable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' One needs the effective theory with relevant degrees of freedom to describe the reduced dynamics of an open quantum system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' As is well-known, symmetry gives a powerful tool for capturing relevant degrees of freedom in the dynamics of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For example, let us focus on the symmetry in the Minkowski spacetime, which is called the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Imposing the Poincar´e symmetry on a quantum theory, one finds that quantum dynamics in the theory is described by the fundamental degrees of freedom such as a massive particle and a massless particle [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The approach based on symmetries provides 2 a way to get the effective theory of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this paper, we discuss the consequences of the Poincar´e symmetry on the reduced dynamics of an open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This may give the understanding of the relativistic theories of open quantum systems (for example, [7–14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The present paper is also motivated by the theory of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the unification of quantum mechanics and gravity has not been completed yet, we do not exactly know how gravity is incorporated in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This situation has prompted to propose many models on the gravity of quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the previous work [15], the model with a classical gravitational interaction between quantum systems was proposed, which is called the Kafri-Taylor-Milburn model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In addition, the Diosi-Penrose model [16–18] and the Tilloy-Diosi model [19] were advocated, for which the gravity of a quantum system intrinsically induces decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' They are formulated by the theory of open quantum systems in a non- relativistic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' One may concern how the models are consistent with a relativistic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Our analysis on reduced dynamics with the Poincar´e symmetry would help to obtain a relativistic extension of the above proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For our analysis, we assume that the reduced dynamics of an open quantum system is described by a dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The dynamical map is obtained by tracing out the environment from the total unitary evolution with an initial product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It is known that the dynamical map is represented by using the Kraus operators [2, 20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The notion of covariant map is adopted for incorporating a symmetry into a dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We derive the condition of a dynamical map with the Poincar´e symmetry in terms of the Kraus operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' With the help of the representation theory of the Poincar´e group, we obtain a systematic way to deduce those Kraus operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Applying the way, we exemplify the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To get the concrete Kraus operators, we focus on the dynamics of a single particle, which is possible to decay into the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Assuming that the particle is a massive particle with a finite spin or a massless particle with a finite spin and a nonzero momentum, we get a model of the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the model, we find the following consequences: (i) if the particle is stable or the energy of particle is conserved, the obtained map turns out to be the unitary map given by the Hamiltonian of particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (ii) If the superposition state of a particle decoheres into a mixed state, the dynamical map for the particle does not have the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' These consequences imply that the Poincar´e symmetry can strongly constraint the reduced dynamics of an open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='II, we discuss a dynamical map describing the reduced dynamics of an open quantum system and consider how symmetries are introduced in the 3 dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='III, we derive the condition that the dynamical map is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='IV, focusing on the dynamics of a single particle, we present a model of the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We then investigate the properties of the dynamical map in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='V is devoted as the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We use the unit ℏ = c = 1 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' QUANTUM DYNAMICAL MAP AND ITS SYMMETRY In this section, we consider the reduced dynamics of an open quantum system and discuss the symmetry of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The reduced dynamics is given as the time evolution of the density operator of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The time evolution from a time τ = s to τ = t is assumed to be given by ρ(t) = Φt,s[ρ(s)] = TrE[ ˆU(t, s)ρ(s) ⊗ ρE ˆU †(t, s)], (1) where ρ(τ) is the system density operator, ρE is the density operator of an environment and ˆU(t, s) is the unitary evolution operator of the total system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this paper, the map Φt,s is called a dynamical map, which has the property called completely positive and trace-preserving (CPTP) [2, 20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The dynamical map Φt,s is rewritten in the operator-sum representation, Φt,s[ρ(s)] = � λ ˆF t,s λ ρ(s) ˆF t,s † λ , (2) where ˆF t,s λ called the Kraus operators satisfy the completeness condition, � λ ˆF t,s † λ ˆF t,s λ = ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (3) In this notation, λ takes discrete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' When the label λ is continuous, we should replace the summation � λ with the integration � dµ(λ) with an appropriate measure µ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It is known that two dynamical maps Φ and Φ′ with Φ[ρ] = � λ ˆFλ ρ ˆF † λ, Φ′[ρ] = � λ ˆF ′ λ ρ ˆF ′† λ , (4) are equivalent to each other (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Φ[ρ] = Φ′[ρ] for any density operator ρ) if and only if there is a unitary matrix Uλλ′ satisfying � λ Uλ1λU∗ λ2λ = δλ1λ2 = � λ Uλλ1U∗ λλ2 and ˆF ′ λ = � λ′ Uλλ′ ˆFλ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (5) This is the uniqueness of a dynamical map [2, 20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We introduce the notion of covariant map [22–24] to impose symmetry on dynamical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A dynamical map Φt,s is covariant under a group G if Φt,s[ ˆUs(g) ρ(s) ˆU † s (g)] = ˆUt(g)Φt,s[ρ(s)] ˆU † t (g), (6) 4 where ˆUs(g) and ˆUt(g) with g ∈ G are the unitary representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this paper, the dynamical map Φt,s satisfying (6) is called symmetric under the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the next section, we will discuss the dynamical map which is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' DYNAMICAL MAP WITH POINCAR´E SYMMETRY In this section, we consider a quantum theory with the Poincar´e symmetry and discuss the general conditions on a dynamical map with the Poincare symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The generators of the unitary representation of the Poincar´e group in the Schr¨odinger picture [6] are given by ˆPµ = � d3x ˆT 0 µ, ˆJµν = � d3x ˆ Mµν0, (7) where ˆTµν is the energy-momentum tensor satisfying ∂µ ˆT µ ν = 0, ˆTµν = ˆTνµ (8) and ˆ Mµνρ with ˆ Mµνρ = xµ ˆT ρ ν − xν ˆT ρ µ (9) is the Noether current associated with the Lorentz transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (8), we can show that ∂ρ ˆ Mµνρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Focusing on each component of the generators, we have ˆH = ˆP 0 = � d3x ˆT 00, ˆP i = � d3x ˆT 0i, (10) ˆJk = 1 2ǫijk ˆJij = � d3x ǫijkxi ˆT 0 j , ˆKk = � d3(xk ˆT 00 − t ˆT 0k), (11) where note that the boost generator ˆKk explicitly depends on a time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' These operators satisfy the commutation relations, [ ˆPi, ˆPj] = 0, (12) [ ˆPi, ˆH] = 0, (13) [ ˆJi, ˆH] = 0, (14) [ ˆJi, ˆJj] = iǫijk ˆJk, (15) [ ˆJi, ˆPj] = iǫijk ˆP k, (16) [ ˆJi, ˆKj] = iǫijk ˆKk, (17) [ ˆKi, ˆPj] = iδij ˆH, (18) [ ˆKi, ˆH] = i ˆPi, (19) [ ˆKi, ˆKj] = −iǫijk ˆJk, (20) 5 which correspond to the Poincar´e algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We consider a dynamical map Φt,s from ρ(s) to ρ(t) = Φt,s[ρ(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The Poincar´e symmetry of the dynamical map requires that ˆUt(Λ, a)Φt,s[ρ(s)] ˆU † t (Λ, a) = Φt,s[ ˆUs(Λ, a)ρ(s) ˆU † s (Λ, a)], (21) where the unitary operator ˆUt(Λ, a) depends on the proper (detΛ = 1) orthochronous (Λ00 ≥ 1) Lorentz transformation matrix Λµν and the real parameters aµ for the spacetime translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The unitary operator ˆUt(Λ, a) generated by ˆH, ˆPi, ˆJi and ˆKi has the group multiplication rule ˆUt(Λ′, a′) ˆUt(Λ, a) = ˆUt(Λ′Λ, a′ + Λ′a), (22) where we used the fact that we can always adopt the non-projective unitary representation of the Poincar´e group [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The explicit time dependence of ˆUt comes from the boost generator ˆKi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using the operator-sum representation, we have ˆUt(Λ, a) � λ ˆF t,s λ ρ(s) ˆF t,s† λ ˆU † t (Λ, a) = � λ ˆF t,s λ ˆUs(Λ, a)ρ(s) ˆU † s (Λ, a) ˆF t,s† λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From the uniqueness of the Kraus operators ˆF t,s λ (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (5)), we obtain ˆU † t (Λ, a) ˆF t,s λ ˆUs(Λ, a) = � λ′ Uλλ′(Λ, a) ˆF t,s λ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (23) We can always choose ˆF t,s λ so that { ˆF t,s λ }λ is the set of linearly independent operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This linear independence and the group multiplication rule of ˆUt(Λ, a) given in (22) lead to the fact that the unitary matrix Uλλ′(Λ, a) satisfies the group multiplication rule � λ′ Uλλ′(Λ′, a′)Uλ′λ′′(Λ, a) = Uλλ′′(Λ′Λ, Λa + a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (24) Hence, the unitary matrix Uλλ′(Λ, a) is a representation of the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Before discussing the condition of symmetry, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (23), we present the useful relation ˆKi = e−i ˆ Ht ˆKi 0 ei ˆ Ht, (25) where ˆKi 0 = � d3x xi ˆT 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (26) According to the Poincar´e algebra, we have ˆUt(Λ, a) = e−i ˆ Ht ˆU0(Λ, a)ei ˆ Ht, (27) 6 where ˆU0(Λ, a) is the unitary representation of the Poincar´e group with the genrators ˆH, ˆP i, ˆKi 0 and ˆJi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the scattering theory, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (27) is consistent with the Poincar´e invariance of the S-operator ˆS(∞, −∞), where ˆS(tf, ti) = ei ˆ H0tfe−i ˆ H(tf−ti)e−i ˆ H0ti and ˆH = ˆH0 + ˆV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This is because ˆU I† tf (Λ, a) ˆS(tf, ti) ˆU I ti(Λ, a) = ei ˆ H0tf ˆU † tf(Λ, a)e−i ˆ H(tf−ti) ˆUti(Λ, a)e−i ˆ H0ti = ei ˆ H0tfe−i ˆ Htf ˆU † 0(Λ, a) ˆU0(Λ, a)ei ˆ Htie−i ˆ H0ti = ˆS(tf, ti), (28) where ˆU I t(Λ, a) = ei ˆ H0t ˆUt(Λ, a)e−i ˆ H0t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (27) also implies that the unitary evolution generated by ˆH is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Indeed, we can show that the unitary map Ut,s[ρ(s)] = e−i ˆ H(t−s) ρ(s) ei ˆ H(t−s) (29) satisfies the condition of symmetry (21) as Ut,s[ ˆUs(Λ, a)ρ(s) ˆU † s (Λ, a)] = e−i ˆ H(t−s) ˆUs(Λ, a) ρ(s) ˆU † s (Λ, a)ei ˆ H(t−s) = e−i ˆ H(t−s) ˆUs(Λ, a)ei ˆ H(t−s) Ut,s[ρ(s)] e−i ˆ H(t−s) ˆU † s(Λ, a)ei ˆ H(t−s) = ˆUt(Λ, a) Ut,s[ρ(s)] ˆU † t (Λ, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (27) helps us to simplify the condition of symmetry, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (23), on the Kraus operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Defining the Kraus operators ˆEt,s λ as ˆEt,s λ = ei ˆ Ht ˆF t,s λ e−i ˆ Hs (30) which have the completeness condition, � λ ˆEt,s† λ ˆEt,s λ = ˆI, (31) we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (23) as ˆU † 0(Λ, a) ˆE ˆU0(Λ, a) = U(Λ, a) ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) Here, we introduced the vector ˆE with the λ component given by ˆEt,s λ and the matrix U(Λ, a) with the (λ, λ′) component given by Uλλ′(Λ, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We define the dynamical map Et,s as Et,s[ρ] = � λ ˆEt,s λ ρ ˆEt,s† λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (33) The condition (32) gives the fact that the map Et,s is symmetric under the Poincar`e group in the sense that ˆU0(Λ, a)Et,s[ρ] ˆU † 0(Λ, a) = Et,s[ ˆU0(Λ, a) ρ ˆU † 0(Λ, a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (34) 7 The dynamical map Φt,s is written by the unitary map Ut,s and the dynamical map Et,s as Φt,s[ρ] = � λ ˆF t,s λ ρ ˆF t,s† λ = e−i ˆ Ht � λ ˆEt,s λ ei ˆ Hsρe−i ˆ Hs ˆEt,s† λ ei ˆ Ht = e−i ˆ HtEt,s[ei ˆ Hsρe−i ˆ Hs]ei ˆ Ht = e−i ˆ H(t−s)Et,s[ρ]ei ˆ H(t−s) = Ut,s ◦ Et,s[ρ], (35) where in the fourth equality we used the symmetric condition (34) noticing that ei ˆ Hs is the unitary transformation of the time translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Our task is to determine ˆE satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) (or Et,s satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='(34)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) is decomposed into equations for each irreducible representation subspace, the irreducible unitary representations of the Poincar´e group is useful for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Let us present how to classify the unitary representations of the Poincar´e group [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We consider the standard momentum ℓµ and the Lorentz transformation matrix (Sq)µν with qµ = (Sq)µνℓν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (36) The unitary matrix U(Λ, a) is written as U(Λ, a) = U(I, a)U(Λ, 0) = T (a)V(Λ), (37) where I is the identity matrix, U(I, a) = T (a) = e−iPµaµ and U(Λ, 0) = V(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We define the vector vq,ξ as vq,ξ = NqV(Sq)vℓ,ξ, (38) where Pµvℓ,ξ = ℓµvℓ,ξ, Nq is the normalization and the label ξ describes the degrees of freedom other than them determined by ℓµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We obtain the following transformation rules for the vector vq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ: T (a)vq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = Nqe−iP µaµV(Sq)vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(Sq)e−i(Sq)µνP νaµvℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(Sq)e−i(Sq)µνℓνaµvℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(Sq)e−iqµaµvℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = e−iqµaµvq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (39) 8 and V(Λ)vq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(Λ)V(Sq)vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(ΛSq)vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(SΛq)V(S−1 Λq ΛSq)vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = NqV(SΛq) � ξ′ Dξ′ξ(Q(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' q))vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='s′ = Nq NΛq � ξ′ Dξ′ξ(Q(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' q))vΛq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (40) where Q(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' q) = S−1 Λq ΛSq is an element of the little group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' which satisfies Qµνℓν = ℓµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' and Dξ′ξ(Q) is the unitary representation of the little group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The irreducible unitary representations of the Poincar´e group are classified by the standard momentum ℓµ and the irreducible unitary represen- tations of the little group which does not change ℓµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' standard momentum ℓµ little group composed of Qµν with Qµνℓν = ℓµ (a) ℓµ = [M, 0, 0, 0], M > 0 SO(3) (b) ℓµ = [−M, 0, 0, 0], M > 0 SO(3) (c) ℓµ = [κ, 0, 0, κ], κ > 0 ISO(2) (d) ℓµ = [−κ, 0, 0, κ], κ > 0 ISO(2) (e) ℓµ = [0, 0, 0, N], N 2 > 0 SO(2,1) (f) ℓµ = [0, 0, 0, 0] SO(3,1) TABLE I: Classification of the standard momentum ℓµ and the little group associated with ℓµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For simplicity, ξ is regarded as the label of basis vectors of the irreducible representation sub- spaces of the little group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Other degeneracies not represented by q and ξ will be reintroduced in the form of the dynamical map Φt,s, which we will see in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We investigate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) restricted on each irreducible representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For convenience, we separately focus on the Lorentz transformation and the spacetime translation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='(32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The unitary operator ˆU0(Λ, a) is written as ˆU0(Λ, a) = ˆU0(I, a) ˆU0(Λ, 0) = ˆT(a) ˆV (Λ), (41) where ˆU0(I, a) = ˆT(a) = e−i ˆPµaµ with ˆP µ = [ ˆH, ˆP 1, ˆP 2, ˆP 3] and ˆU0(Λ, 0) = ˆV (Λ) with the genera- tors ˆJi and ˆKi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) for Λ = I, we have ˆT †(a) ˆE ˆT(a) = T (a) ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (42) 9 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (32) for aµ = 0 gives ˆV †(Λ) ˆE ˆV (Λ) = V(Λ) ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (43) Introducing ˆEq,ξ = v† q,ξ ˆE, we obtain the following equations from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (42) and (43): ˆT †(a) ˆEq,ξ ˆT(a) = e−iqµaµ ˆEq,ξ (44) and ˆV †(Λ) ˆEq,ξ ˆV (Λ) = N ∗ q N ∗ Λ−1q � ξ′ D∗ ξ′ξ(Q(Λ−1, q)) ˆEΛ−1q,ξ′, (45) where we used Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (39) and (40), and Q(Λ, q) = S−1 Λq ΛSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The label ξ can take discrete or contin- uous values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the continous case, the summation � ξ is replaced with the integration � dµ(ξ) with a measure µ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Focusing on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (45) for Λ = Sq, we get ˆV †(Sq) ˆEq,ξ ˆV (Sq) = N ∗ q ˆEℓ,ξ, (46) where note that Nℓ = 1 and Q(S−1 q , q) = S−1 S−1 q qS−1 q Sq = S−1 ℓ = I hold by the definition of vq,ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (46) tells us that the Kraus operators ˆEq,ξ is determined from the Kraus operators ˆEℓ,ξ with the standard momentum ℓµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' All we have to do is to give the form of the Kraus operators ˆEℓ,ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To this end, we present the following equations given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (44) for qµ = ℓµ and by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (45) for qµ = ℓµ and Λ = W with W µνℓν = ℓµ, respectively: ˆT †(a) ˆEℓ,ξ ˆT(a) = e−iℓµaµ ˆEℓ,ξ, (47) ˆV †(W) ˆEℓ,ξ ˆV (W) = � ξ′ D∗ ξ′ξ(W −1) ˆEℓ,ξ′, (48) where Q(Λ−1, q) = Q(W −1, ℓ) = S−1 W −1ℓW −1Sℓ = W −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the next section, we construct a model of the dynamical map with the Poincar´e symmetry to describe the reduced dynamics of a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A MODEL OF THE DYNAMICAL MAP FOR A SINGLE PARTICLE In this section, based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (47) and (48), we give a model of the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To simplify the analysis, we consider the Hilbert space H0 ⊗ H1, where H0 is the one-dimensional Hilbert space with a vacuum state |0⟩ and H1 is the irreducible subspace with one-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Any state vector |Ψ⟩ in H1 ( |Ψ⟩ ∈ H1 ) is given by |Ψ⟩ = � d3q � σ Ψ(p, σ) ˆa†(p, σ)|0⟩, (49) 10 where |0⟩ is the vacuum state satisfying ˆa(p, σ)|0⟩ = 0, Ψ(p, σ) with the momentum p and the spin σ is the wave function, ˆa(p, σ) and ˆa†(p, σ) are the annihilation and creation operators satisfying [ˆa(p, σ), ˆa(p′, σ′)]± = 0 = [ˆa†(p, σ), ˆa†(p′, σ′)]±, [ˆa(p, σ), ˆa†(p′, σ′)]± = δ3(p − p′)δσσ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (50) In the above notation, [ ˆA, ˆB]± is defined as [ ˆA, ˆB]± = ˆA ˆB ± ˆB ˆA, in which the signs − and + apply bosons and fermions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [6, 26], the transformation rules of ˆa†(p, σ) are given by ˆT(a)ˆa†(p, σ) ˆT †(a) = e−ipµaµˆa†(p, σ), (51) ˆV (Λ)ˆa†(p, σ) ˆV †(Λ) = � EpΛ Ep � σ′ Dσ′σ(Q(Λ, p))ˆa†(pΛ, σ′), (52) where Ep = p0, EpΛ = (Λp)0 and pΛ is the vector with the component (pΛ)i = (Λp)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The matrix Q(Λ, p) = S−1 Λp ΛSp is the element of the little group which satisfies Q(Λ, p)µνkν = kµ, where kµ is the standard momentum for a massive particle (kµ = [m, 0, 0, 0], m > 0) or a massless particle (kµ = [k, 0, 0, k], k > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The momentum pµ and the standard momentum kµ are connected with (Sp)µνkν = pµ, and Dσ′σ(Q(Λ, p)) is the irreducible unitary representation of the little group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We consider the Kraus operators ˆEℓ,ξ acting on the Hilbert space H0 � H1, that is, ˆEℓ,ξ : H0 � H1 → H0 � H1, which have the following form ˆEℓ,ξ = Aℓ,ξˆI + � d3p � σ Bℓ,ξ(p, σ)ˆa(p, σ) + � d3p′d3p � σ′,σ Cℓ,ξ(p′, σ′, p, σ)ˆa†(p′, σ′)ˆa(p, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (53) The dynamical map given by these operators describes the reduced dynamics of a single particle, which can possibly decay into the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Substituting the above operators into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (47) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (48), we obtain the equations Aℓ,ξ = e−iℓµaµAℓ,ξ, (54) Bℓ,ξ(p, σ)e−ipµaµ = Bℓ,ξ(p, σ)e−iℓµaµ, (55) Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cℓ,ξ(p′, σ′, p, σ)e−iℓµaµ, (56) and Aℓ,ξ = � ξ′ D∗ ξ′ξ(W −1)Aℓ,ξ′, (57) � EpW Ep � σ Bℓ,ξ(pW , σ)D∗ σ′σ(Q) = � ξ′ D∗ ξ′ξ(W −1)Bℓ,ξ′(p, σ′), (58) � Ep′ W EpW Ep′Ep � σ′,σ Cℓ,ξ(p′ W, σ′, pW , σ)D¯σ′σ′(Q′)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(W −1)Cℓ,ξ′(p′, ¯σ′, p, ¯σ), (59) 11 where Q = Q(W −1, Wp) and Q′ = Q(W −1, Wp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The derivation of these equations is devoted in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We can analyze the explicit form of Aℓ,ξ, Bℓ,ξ(p, σ) and Cℓ,ξ(p′, σ′, p, σ) for a massive particle and a massless particle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the analysis, we assume that the massive particle has a finite spin and the massless particle has a finite spin and a nonzero momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Through the long computations presented in Appendices B and C, we get the following dynamical map with the Poincar´e symmetry, Φt,s[ρ(s)] = Ut,s◦Et,s[ρ(s)], Et,s[ρ] = � j � β(j) t,s � d3p � σ ˆa(p, σ)ρˆa†(p, σ)+α(j) t,s �ˆI+γ(j) t,s ˆN � ρ �ˆI+γ(j)∗ t,s ˆN �� , (60) where α(j) t,s , β(j) t,s and γ(j) t,s are the parameters depending on time, ˆN is the number operator defined as ˆN = � d3p � σ ˆa†(p, σ)ˆa(p, σ), (61) and Ut,s is the unitary map given in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the form of the dynamical map Φt,s, we recovered the labels j which represent the degeneracies other than the labels q and ξ appearing in the Kraus operators ˆEq,ξ defined around (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The parameters α(j) t,s, β(j) t,s and γ(j) t,s in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (60) satisfy 0 ≤ α(j) t,s ≤ 1, � j α(j) t,s = 1, 0 ≤ β(j) t,s , 0 ≤ � j β(j) t,s ≤ 1 � j � β(j) t,s + α(j) t,s � γ(j) t,s + γ(j)∗ t,s + |γ(j) t,s |2�� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (62) These conditions come from the completeness condition of the Kraus operators (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the com- putation of the completeness condition, note that the number operator ˆN satisfies ˆN 2 = ˆN on the Hilbert space H0 ⊗ H1, since we assume that the dynamical map describes the reduced dynamics of a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From the transformation rules of the creation and the annihilation operators, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (51) and (52), it is easy to check that the map Et,s satisfies the condition of symmetry given in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the unitary map Ut,s is symmetric under the Poincar´e group, which is checked around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (29), we can confirm that Φt,s is also symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Let us consider the case where there is no decay under the dynamical map Φt,s and focus on the dynamics of one-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this case, the parameter � j β(j) t,s vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the density operator ρ given by one-particle states satisfies ˆNρ = ρ = ρ ˆN, we have Φt,s[ρ(s)] = Ut,s ◦ Et,s[ρ(s)] = � j α(j) t,s |1 + γ(j) t,s |2Ut,s[ρ(s)] = Ut,s[ρ(s)], (63) 12 where we used the condition (62) with � j β(j) t,s = 0 in the third equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This means that the dynamical map with the Poincar´e symmetry for a non-decaying particle is the unitary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The result corresponds to a non-perturbative extension of the analysis in [25], which gives an implication on the particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For example, if the superposition state of a particle decoheres under a non-unitary evolution, the Poincar´e symmetry breaks in the particle dynamics described by a dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' We discuss the energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The expectation value of ˆHn at a time t, where n is a natural number, is computed as Tr[ ˆHnρ(t)] = � j � β(j) t,s Tr[ ˆHn � d3p � σ ˆa(p, σ)ρ(s)ˆa†(p, σ)] + α(j) t,s Tr[ ˆHn(ˆI + γ(j) t,s ˆN)ρ(s)(ˆI + γ(j)∗ t,s ˆN)] � = (1 − � j β(j) t,s )Tr[ ˆHnρ(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (64) In the reduced dynamics by the dynamical map Φt,s, the energy of a single particle is not conserved unless � j β(j) t,s is a constant, even when the map is symmetric under the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Such a deviation between symmetry and conservation law was discussed in, for example, Refs [23] and [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' If the parameter � j β(j) t,s is a constant, then � j β(j) t,s = � j β(j) s,s = 0 and the dynamical map Φt,s is reduced to the unitary map Ut,s as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' CONCLUSION We discussed what a dynamical map describing the reduced dynamics of an open quantum system is realized under the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The unitary representation theory of the Poincar´e group refines the condition for the dynamical map with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For a massive particle and a massless particle, we derived a concrete model of the dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the model, the particle can decay into the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' If there is no decay process, the dynamical map describes the unitary evolution generated by the Hamiltonian of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This means that the non-decaying single particle does not decohere as long as the dynamical map for the particle has the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this way, it was exemplified that the Poincar´e symmetry strongly constrains the possible dynamics of an open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this paper, we assumed an open system with a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Our analysis is possible to be extended to the case with many particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Considering interactions among many particles, we can understand more general effective theories in terms of the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the particles interacting via gravity, the models which induce intrinsic gravitational decoherence have 13 been proposed [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' These models are written in the theory of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the weak field regime of gravity, the Poincar´e symmetry may provide a guidance for establishing the theory of an open quantum system with gravitating particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This paper has a potential to develop the theory of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To describe the reduced dynamics of an open quantum system, a quantum master equation is often adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It has been discussed how the quantum Markov dynamics given by the equation is consistent with a relativistic theory [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Applying the present approach, it will be possible to discuss the quantum Markov dynamics with the Poincar´e symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It is hoped that this paper paves the way to understand a relativistic theory of open quantum systems and to study the interplay between quantum theory and gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Acknowledgments We thank Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Kuramochi for useful discussions and comments related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' was supported by 2022 Research Start Program 202203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Appendix A: Derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (54),(55),(56),(57),(58) and (59) We present the transformation rules of Aℓ,ξ, Bℓ,ξ and Cℓ,ξ given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (54),(55),(56),(57),(58) and (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using the assumed form of the Kraus operators ˆEℓ,ξ defined by (53), we can compute the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (47) as ˆT †(a) ˆEℓ,ξ ˆT(a) = Aℓ,ξˆI + � d3p � σ Bℓ,ξ(p, σ)e−ipµaµˆa(p, σ) + � d3p′d3p � σ′,σ Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµˆa†(p′, σ′)ˆa(p, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (47), we have Aℓ,ξ = e−iℓµaµAℓ,ξ, (A1) Bℓ,ξ(p, σ)e−ipµaµ = Bℓ,ξ(p, σ)e−iℓµaµ (A2) Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cℓ,ξ(p′, σ′, p, σ)e−iℓµaµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) 14 The right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (48) is evaluated as ˆV †(W) ˆEℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆV (W) = Aℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξˆI + � d3p � σ Bℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) � EpW −1 Ep � σ′ D∗ σ′σ(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p))ˆa(pW −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′) + � d3p′d3p � σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='σ Cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) × � Ep′ W −1 Ep′ � EpW −1 Ep � ¯σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ′ D¯σ′σ′(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p′))D∗ ¯σσ(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p))ˆa†(p′ W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ′)ˆa(pW −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = Aℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξˆI + � d3p � σ Bℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(pW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) � EpW Ep � σ′ D∗ σ′σ(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Wp))ˆa(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′) + � d3p′d3p � σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='σ Cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p′ W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' pW ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) × � Ep′ W Ep′ � EpW Ep � ¯σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ′ D¯σ′σ′(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Wp′))D∗ ¯σσ(Q(W −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Wp))ˆa†(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ′)ˆa(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' where note that the Lorentz invariant measure is d3p/Ep and hence f(p)d3p = Epf(p)d3p/Ep = EpΛf(pΛ)d3p/Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (48), we have Aℓ,ξ = � ξ′ D∗ ξ′ξ(W −1)Aℓ,ξ′, (A4) � EpW Ep � σ Bℓ,ξ(pW, σ)D∗ σ′σ(Q) = � ξ′ D∗ ξ′ξ(W −1)Bℓ,ξ′(p, σ′) (A5) � Ep′ W EpW Ep′Ep � σ′,σ Cℓ,ξ(p′ W , σ′, pW, σ)D¯σ′σ′(Q′)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(W −1)Cℓ,ξ′(p′, ¯σ′, p, ¯σ), (A6) where Q = Q(W −1, Wp) and Q′ = Q(W −1, Wp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Appendix B: Analysis of a massive particle We assume that the spectrum of ˆP µ on any state |Ψ⟩ in the Hilbert space of one-particle states, H1, satisfies ˆP µ ˆPµ|Ψ⟩ = −m2|Ψ⟩, ⟨Ψ| ˆP 0|Ψ⟩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B1) The above equations are equivalent to the fact that the Hamiltonian ˆH = ˆP 0 has the form ˆH = � ˆPk ˆP k + m2, which implies that |Ψ⟩ is the state of a massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this appendix, we derive the form of the dynamical map Et,s for a massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 15 Case (a) ℓµ = [M, 0, 0, 0], M > 0 or (b) ℓµ = [−M, 0, 0, 0], M > 0 : We focus on the spectrum ℓµ = [±M, 0, 0, 0], M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [a, 0, 0, 0], we have Aℓ,ξ = e±iMaAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] leads to Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ) ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0], we get Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iMa, and combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we obtain Bℓ,ξ(σ)eima = Bℓ,ξ(σ)e±iMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the mass m is positive, to get a nontrivial result, we should choose +M with M = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A5) for Q = R ∈ SO(3) and adopting the result Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we find � σ Bℓ,ξ(σ)D∗ σ′σ(R−1) = � ξ′ D∗ ξ′ξ(R−1)Bℓ,ξ′(σ′), where note that Q = Q(W −1, Wp) = Q(R−1, Rℓ) = S−1 ℓ R−1SRℓ = R−1 for ℓµ = [m, 0, 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the representations Dσ′σ and Dξ′ξ are irreducible and unitary, by Schur’s lemma we have Bℓ,ξ(σ) = Bℓ uξσ, where uξσ is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ) ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0] leads to Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iMa, and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p) into the above equation, we have Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(p, σ′, σ)e±iMa ∴ Cℓ,ξ(p, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The above results imply that the Kraus operator ˆEℓ,ξ with ℓµ = [m, 0, 0, 0] has the following form, ˆEℓ,ξ = Bℓ � σ uξσˆa(0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 16 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (46) tells us that ˆEq,ξ = N ∗ q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = N ∗ q Bℓ � Eq m � σ uξσˆa(q, σ), where Eq = (Sq ℓ)0 and qi = (Sq ℓ)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To determine the normalization Nq, the inner product v† q,ξvq,ξ is assumed to be v† q′,s′vq,ξ = δ3(q′ − q)δξ′ξ, which leads to Nq = � m/Eq up to a phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For this normalization, the following complete- ness condition is given as � d3q � s vq,ξv† q,ξ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Under the completeness condition, we derive a part of the dynamical map Et,s as Et,s[ρ(s)] ⊃ |Bℓ|2 � d3q � σ ˆa(q, σ)ρ(s)ˆa†(q, σ), (B2) where we used the fact that uξσ is the unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (c) ℓµ = [κ, 0, 0, κ], κ > 0 or (d) ℓµ = [−κ, 0, 0, κ], κ > 0 : We consider the spectrum ℓµ = [±κ, 0, 0, κ], κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [a, 0, 0, 0], we have Aℓ,ξ = e±iκaAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a], we get Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), where ℓ = [0, 0, κ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] leads to Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iκa, and from the equation Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we obtain Bℓ,ξ(σ)ei √ κ2+m2a = Bℓ,ξ(σ)e±iκa ∴ Bℓ,ξ(σ) = 0, where Eℓ = √ ℓ2 + m2 = √ κ2 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a] gives Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0], we get Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iκa, 17 and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)e±iκa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Noticing the fact that Ep−ℓ − Ep ± κ ̸= 0, we get the result Cℓ,ξ(p, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Combined with the above analysis, the Kraus operator ˆEℓ,ξ vanishes: ˆEℓ,ξ = 0 ∴ ˆEq,ξ = Nq ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B3) Case (e) ℓµ = [0, 0, 0, N], N 2 > 0 : We focus on the spectrum ℓµ = [0, 0, 0, N], N 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [0, a], we have Aℓ,ξ = e−iℓ·aAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] leads to Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ) ∴ Bℓ,ξ(p, σ) = 0, where note that Eq = � q2 + m2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ), where ℓ = [0, 0, N]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0], we get Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ), and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ) ∴ Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(σ′, σ)δ3(p − ℓ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Combined with the above analysis, the function Cℓ,ξ(p′, σ′, p, σ) is Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(σ′, σ)δ3(p′ + ℓ/2)δ3(p − ℓ/2), and the Kraus operator ˆEℓ,ξ is written as ˆEℓ,ξ = � σ′,σ Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' By the completeness condition of the Kraus operators, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (31), the above Kraus operator ˆEℓ,ξ should satisfy ˆE† ℓ,ξ ˆEℓ,ξ ≤ ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Concretely, ˆE† ℓ,ξ ˆEℓ,ξ is evaluated as ˆE† ℓ,ξ ˆEℓ,ξ = � ¯σ′,¯σ C∗ ℓ,ξ(¯σ′, ¯σ)ˆa†(ℓ/2, ¯σ)ˆa(−ℓ/2, ¯σ′) � σ′,σ Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ) = δ3(0) � σ′ � � ¯σ Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) �† � σ Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ), 18 where the term given by the linear combination of ˆa†ˆa†ˆaˆa vanishes on H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To satisfy ˆE† ℓ,ξ ˆEℓ,ξ ≤ ˆI, we find that � σ′ � � ¯σ Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) �† � σ Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ) = 0 ∴ Cℓ,ξ(σ′, σ) = 0 The consequence of Cℓ,ξ(σ′, σ) = 0 is that the Kraus operator ˆEℓ,ξ vanishes as ⟨Φ| ˆEℓ,ξ|Ψ⟩ = 0 for all |Ψ⟩, |Φ⟩ ∈ H0 � H1, and hence ˆEq,ξ = N ∗ q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0, (B4) on the Hilbert space H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (f) ℓµ = [0, 0, 0, 0] : We consider the case where ℓµ = [0, 0, 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the following, we drop the label ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) is identical for all aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the little group associated with ℓµ is SO(3, 1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A4) for W = Λ ∈ SO(3, 1) is given as Aξ = � ξ′ D∗ ξ′ξ(Λ−1)Aξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B5) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] gives the condition Bξ(p, σ)eiEpa = Bξ(p, σ) ∴ Bξ(p, σ) = 0, where note that Eq = � q2 + m2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ, we obtain Cξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cξ(p′, σ′, p, σ) ∴ Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A6) for W = Λ ∈ SO(3, 1) is written as � Ep′ ΛEpΛ Ep′Ep � σ′,σ Cξ(p′ Λ, σ′, pΛ, σ)D¯σ′σ′(Q′)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(Λ−1)Cξ′(p′, ¯σ′, p, ¯σ), where Q = Q(Λ−1, Λp) and Q′ = Q(Λ−1, Λp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From the equation Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p) and noticing the fact that the invariant delta function is Epδ3(p − p′), we get the condition � σ′,σ Cξ(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(Λ−1)Cξ′(p, ¯σ′, ¯σ), (B6) where Q′ = Q(Λ−1, Λp′) turns out to be Q = Q(Λ−1, Λp) by the presence of the delta function δ3(p − p′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' It is known that the dimension of irreducible unitary representations Dξ′ξ of SO(3,1) 19 is one or infinite [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the one-dimensional representation, dropping the label ξ, we find that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B5) trivially holds and that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B6) is reduced to � σ′,σ C(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ ¯σσ(Q) = C(p, ¯σ′, ¯σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For p = 0 and Λ = R ∈ SO(3), we get � σ′,σ C(0, σ′, σ)D¯σ′σ′(R−1)D∗ ¯σσ(R−1) = C(0, ¯σ′, ¯σ) ∴ C(0, σ′, σ) = Cδσ′σ, where this holds by the Schur’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Choosing p = 0 and Λ = Sp with (Sp)µνkν = pµ for kµ = [m, 0, 0, 0], we have C(p, σ′, σ) = C(0, σ′, σ), where we used Q = Q(S−1 p , p) = S−1 k S−1 p Sp = I and Dσ′σ(I) = δσ′σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Hence C(p, σ′, σ) = Cδσ′σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For the infinite dimensional representation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B5) leads to Aℓ,ξ = 0, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B6) for p = 0 and Λ = R ∈ SO(3) gives � σ′,σ Cξ(0, σ′, σ)D¯σ′σ′(R−1)D∗ ¯σσ(R−1) = � ξ′ D∗ ξ′ξ(R−1)Cξ′(0, ¯σ′, ¯σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Assuming that the massive particle has a finite spin and using the Schur’s lemma, we get Cξ′(0, ¯σ′, ¯σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B6) for p = 0 and Λ = Sp with (Sp)µνkν = pµ for kµ = [m, 0, 0, 0] pro- vides Cξ(p, σ′, σ) = � ξ′ D∗ ξ′ξ(S−1 p )Cξ′(0, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The above analysis on ℓµ = [0, 0, 0, 0] tells us the following Kraus operator ˆE = AˆI + C ˆN, where ˆN is the number operator defined in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A part of the dynamical map Et,s is given as Et,s[ρ(s)] ⊃ � AˆI + C ˆN � ρ(s) � AˆI + C ˆN �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B7) The above results given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B2), (B3), (B4) and (B7) provide the following form of Et,s: Et,s[ρ(s)] = |Bℓ|2 � d3q � σ ˆa(q, σ)ρ(s)ˆa†(q, σ) + � AˆI + C ˆN � ρ(s) � AˆI + C ˆN �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (B8) Recovering other degeneracies labeled by j differently from q and ξ, introducing � j and redefining the parameters as |A|2 = α(j) t,s , C/A = γ(j) t,s and |Bℓ|2 = β(j) t,s , we get the form of the dynamical map Et,s given in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 20 Appendix C: Analysis on a massless particle We assume that the spectrum of ˆP µ on any state |Ψ⟩ in the Hilbert space of one-particle states, H1, satisfies ˆP µ ˆPµ|Ψ⟩ = 0, ⟨Ψ| ˆP 0|Ψ⟩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C1) The above equations leads to the fact that the Hamiltonian ˆH = ˆP 0 has the form ˆH = � ˆPk ˆP k, which means that |Ψ⟩ is the state of a massless particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In this appendix, we derive the form of the dynamical map Et,s for a massless particle with nonzero momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (a) ℓµ = [M, 0, 0, 0], M > 0 or (b) ℓµ = [−M, 0, 0, 0], M > 0 : We focus on the spectrum ℓµ = [±M, 0, 0, 0], M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [a, 0, 0, 0] gives Aℓ,ξ = e±iMaAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] leads to Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ) ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0], we get Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iMa, and combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we obtain Bℓ,ξ(σ) = Bℓ,ξ(σ)e±iMa ∴ Bℓ,ξ(σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ) ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0] leads to Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iMa, and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p) into the above equation, we have Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(p, σ′, σ)e±iMa ∴ Cℓ,ξ(p, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The above results imply that the Kraus operator ˆEℓ,ξ vanishes and has the following form, ˆEq,ξ = N ∗ q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C2) 21 Case (c) ℓµ = [κ, 0, 0, κ], κ > 0 or (d) ℓµ = [−κ, 0, 0, κ], κ > 0 : We consider the spectrum ℓµ = [±κ, 0, 0, κ], κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [a, 0, 0, 0], we have Aℓ,ξ = e±iκaAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] gives Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), where ℓ = [0, 0, κ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] leads to Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iκa, and from the equation Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we have Bℓ,ξ(σ)eiκa = Bℓ,ξ(σ)e±iκa, where Eℓ = √ ℓ2 = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To get a nontrivial result, we should choose +κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A5) for Q = L ∈ ISO(2) and adopting the result Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we find � σ Bℓ,ξ(σ)D∗ σ′σ(L−1) = � ξ′ D∗ ξ′ξ(L−1)Bℓ,ξ′(σ′), where note that Q = Q(W −1, Wp) = Q(L−1, Lℓ) = S−1 ℓ L−1SLℓ = L−1 for ℓµ = [κ, 0, 0, κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the representations Dσ′σ and Dξ′ξ are irreducible and unitary, by Schur’s lemma we get Bℓ,ξ(σ) = Bℓ uξσ, where uξσ is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0] leads to Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iκa, and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)e±iκa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The condition of Ep−ℓ − Ep + κ = 0 is written by p⊥ = [p1, p2] = 0 and p3 ≥ κ, and the condition of Ep−ℓ − Ep − κ = 0 is given by p⊥ = 0 and p3 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Hence, the form of Cℓ,ξ(p, σ′, σ) is Cℓ,ξ(p, σ′, σ) = � C+ ℓ,ξ(p3, σ′, σ)θ(p3 − κ) + C− ℓ,ξ(p3, σ′, σ)θ(−p3) � δ2(p⊥) 22 Combined with the above analysis, the Kraus operator ˆE+ ℓ,ξ for +κ is ˆE+ ℓ,ξ = Bℓ � σ uξσˆa(ℓ, σ) + � dp3 � σ,σ′ C+ ℓ,ξ(p3, σ′, σ)θ(p3 − κ)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ), and the Kraus operator ˆE− ℓ,ξ for −κ is ˆE− ℓ,ξ = � dp3 � σ,σ′ C− ℓ,ξ(p3, σ′, σ)θ(−p3)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' By the completeness condition of the Kraus operators, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (31), the above Kraus operator ˆE± ℓ,ξ should satisfy ˆE±† ℓ,ξ ˆE± ℓ,ξ ≤ ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Concretely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ˆE+† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ is evaluated as ˆE+† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = � Bℓ � σ uξσˆa(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) + � dp3 � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='σ′ C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)θ(p3 − κ)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3 − κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) �† × � Bℓ � σ uξσˆa(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) + � dp3 � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='σ′ C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)θ(p3 − κ)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3 − κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) � = |Bℓ|2 � σ u∗ sσˆa†(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ) � σ′ usσ′ˆa(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′) + δ2(0) � dp3 � σ′ � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ C+∗ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ)θ(p3 − κ)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' where the term given by the linear combination of ˆa†ˆaˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ˆa†ˆa†ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' and ˆa†ˆa†ˆaˆa vanishes on H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To satisfy ˆE+† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ≤ ˆI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' we find � dp3 � σ′ � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ C+∗ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ)θ(p3−κ)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = 0 ∴ C+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = 0 In the same manner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' we have ˆE−† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ = δ2(0) � dp3 � σ′ � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ C−∗ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)C− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ)θ(−p3)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' and to satisfy ˆE−† ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ˆE− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ ≤ ˆI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' we obtain � dp3 � σ′ � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='¯σ C−∗ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)C− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ)θ(−p3)ˆa†(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ)ˆa(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = 0 ∴ C− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='ξ(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' ¯σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' These analyses give the following form of the Kraus operators, ˆE+ ℓ,ξ = Bℓ � σ uξσˆa(ℓ, σ), ˆE− ℓ,ξ = 0, on the Hilbert space H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (46), we get ˆE+ q,ξ = N ∗ q ˆV (Sq) ˆE+ ℓ,ξ ˆV †(Sq) = N ∗ q Bℓ � Eq κ � σ uξσˆa(q, σ), ˆE− q,ξ = N ∗ q ˆV (Sq) ˆE− ℓ,ξ ˆV †(Sq) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 23 Setting that the inner product v† q,ξvq,ξ is v† q′,s′vq,ξ = δ3(q′ − q)δξ′ξ, the normalization Nq is given as Nq = � κ/Eq up to a phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For this normalization, we get the following completeness condition as � d3q � s vq,ξv† q,ξ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Taking account for the completeness, we can derive a part of the dynamical map Et,sas Et,s[ρ(s)] ⊃ |Bℓ|2 � d3q � σ ˆa(q, σ)ρ(s)ˆa†(q, σ), (C3) where we used the fact that uξσ is the unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (e) ℓµ = [0, 0, 0, N], N 2 > 0 : We focus on the spectrum ℓµ = [0, 0, 0, N], N 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) for all aµ = [0, a], we have Aℓ,ξ = e−iℓ·aAℓ,ξ ∴ Aℓ,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] leads to Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a ∴ Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (55) for all aµ = [a, 0, 0, 0] gives Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ), and then combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we get Bℓ,ξ(σ)eiκa = Bℓ,ξ(σ) ∴ Bℓ,ξ(σ) = 0 where we used Eℓ = √ ℓ2 = κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Adopting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [0, a], we deduce Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a ∴ Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ), where ℓ = [0, 0, N]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ = [a, 0, 0, 0], we get Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ), and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ) ∴ Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(σ′, σ)δ3(p − ℓ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 24 Combined with the above analysis, the function Cℓ,ξ(p′, σ′, p, σ) is Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(σ′, σ)δ3(p′ + ℓ/2)δ3(p − ℓ/2), and the Kraus operator ˆEℓ,ξ is written as ˆEℓ,ξ = � σ′,σ Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' By the completeness condition of the Kraus operators, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (31), the above Kraus operator ˆEℓ,ξ should satisfy ˆE† ℓ,ξ ˆEℓ,ξ ≤ ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Explicitly, ˆE† ℓ,ξ ˆEℓ,ξ is evaluated as ˆE† ℓ,ξ ˆEℓ,ξ = � ¯σ′,¯σ C∗ ℓ,ξ(¯σ′, ¯σ)ˆa†(ℓ/2, ¯σ)ˆa(−ℓ/2, ¯σ′) � σ′,σ Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ) = δ3(0) � σ′ � � ¯σ Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) �† � σ Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ), where the term associated with the linear combination of ˆa†ˆa†ˆaˆa vanishes on H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' To satisfy ˆE† ℓ,ξ ˆEℓ,ξ ≤ ˆI, we find that � σ′ � � ¯σ Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ) �† � σ Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ) = 0 ∴ Cℓ,ξ(σ′, σ) = 0 Hence, the Kraus operator ˆEℓ,ξ vanishes, and we have that ˆEq,ξ = N ∗ q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0, (C4) on the Hilbert space H0 � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Case (f) ℓµ = [0, 0, 0, 0] : We focus on the spectrum ℓµ = [0, 0, 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' In the following, we do not write the label ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A1) is identical for all aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Since the little group associated with ℓµ is SO(3, 1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A4) for W = Λ ∈ SO(3, 1) is given as Aξ = � ξ′ D∗ ξ′ξ(Λ−1)Aξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C5) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [0, a] gives the condition Bξ(p, σ)e−ip·a = Bξ(p, σ) ∴ Bξ(p, σ) = Bξ(σ)δ3(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This equation makes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A2) for all aµ = [a, 0, 0, 0] and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A5) for W = Λ ∈ SO(3, 1) trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' This form Bξ(p, σ) = Bξ(σ)δ3(p) leads to ˆEℓ,ξ ⊃ � σ Bξ(σ)ˆa(0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' However, this operator vanishes on the Hilbert space of massless particles since we assumed that there are no states with zero momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A3) for all aµ gives us the condition Cξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cξ(p′, σ′, p, σ) ∴ Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 25 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (A6) for W = Λ ∈ SO(3, 1) is written as � Ep′ ΛEpΛ Ep′Ep � σ′,σ Cξ(p′ Λ, σ′, pΛ, σ)D¯σ′σ′(Q′)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(Λ−1)Cξ′(p′, ¯σ′, p, ¯σ), where Q = Q(Λ−1, Λp) and Q′ = Q(Λ−1, Λp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' From the equation Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p) and noticing the fact that the invariant delta function is Epδ3(p − p′), we get the condition � σ′,σ Cξ(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ ¯σσ(Q) = � ξ′ D∗ ξ′ξ(Λ−1)Cξ′(p, ¯σ′, ¯σ), (C6) where note that the delta function δ3(p − p′) leads to Q′ = Q(Λ−1, Λp′) = Q(Λ−1, Λp) = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' The (proper orthochronous) Lorentz group SO(3, 1) has one and infinite dimensional unitary irreducible representations [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Choosing the one-dimensional representation of Dξ′,ξ and dropping the label ξ, we find that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C5) trivially holds and that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C6) is reduced to � σ′,σ C(pΛ, σ′, σ)D¯σ′σ′(Q)D∗ ¯σσ(Q) = C(p, ¯σ′, ¯σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' For p = ℓ = [0, 0, κ] and Λ = L ∈ ISO(2), we get � σ′,σ C(ℓ, σ′, σ)D¯σ′σ′(L−1)D∗ ¯σσ(L−1) = C(ℓ, ¯σ′, ¯σ) ∴ C(ℓ, σ′, σ) = Cδσ′σ, where we used the Schur’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Choosing p = ℓ and Λ = Sp with (Sp)µνℓν = pµ for ℓµ = [κ, 0, 0, κ], we have C(p, σ′, σ) = C(ℓ, σ′, σ), where we used Q = Q(S−1 p , p) = S−1 k S−1 p Sp = I and Dσ′σ(I) = δσ′σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Hence C(p, σ′, σ) = Cδσ′σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' If we adopt the infinite dimensional representation of Dξ′ξ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C5) leads to Aℓ,ξ = 0 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C6) for p = ℓ and Λ = L ∈ ISO(2) gives � σ′,σ Cξ(ℓ, σ′, σ)D¯σ′σ′(L−1)D∗ ¯σσ(L−1) = � ξ′ D∗ ξ′ξ(L−1)Cξ′(ℓ, ¯σ′, ¯σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Assuming that the massless particle has a finite spin and using the Schur’s lemma, we get Cξ′(ℓ, ¯σ′, ¯σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C6) for p = ℓ and Λ = Sp with (Sp)µνℓν = pµ for ℓµ = [κ, 0, 0, κ] pro- vides Cξ(p, σ′, σ) = � ξ′ D∗ ξ′ξ(S−1 p )Cξ′(ℓ, σ′, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 26 The above analysis tells us that the Kraus operator has the following form ˆE = AˆI + C ˆN, where ˆN is the number operator defined in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' A part of the dynamical map Et,s with the Poincar´e symmetry is given as Et,s[ρ(s)] ⊃ � AˆI + C ˆN � ρ(s) � AˆI + C ˆN �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C7) Gathering the above results (C2), (C3), (C4) and (C7), we have the following form of Et,s: Et,s[ρ(s)] = |Bℓ|2 � d3q � σ ˆa(q, σ)ρ(s)ˆa†(q, σ) + � AˆI + C ˆN � ρ(s) � AˆI + C ˆN �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' (C8) In the same manner performed around (B8), we obtain the form of the dynamical map Et,s given in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Breuer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Laine, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Piilo, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Vacchini, “Colloquium: Non-Markovian Dynamics in Open Quantum Systems”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQffPzi/content/2301.01451v1.pdf'} +page_content=' 88, 021002 (2016).' metadata={'source': 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+All rights reserved +Keywords +top quark, beyond the standard model, hierarchy problem, flavor, +dark matter, new physics +Abstract +We review scenarios of physics beyond the Standard Model in which the +top quark plays a special role. Models that aim at the stabilization of +the weak scale are presented together with the specific phenomenology +of partner states that are characteristic of this type of model. +Fur- +ther, we present models of flavor in which the top quark is singled out +as a special flavor among the SM ones. The flavor and collider phe- +nomenology of these models is broadly presented. Finally, we discuss +the possibility that dark matter interacts preferably with the top quark +flavor and broadly present the dark matter phenomenology of these +scenarios, as well as collider and flavor signals. +1 +arXiv:2301.04407v1 [hep-ph] 11 Jan 2023 + +Contents +1. Introduction ................................................................................................. +2 +2. Top quark and BSM related to the Higgs boson and the origin of the weak scale ......................... +3 +2.1. Supersymmetry.......................................................................................... +3 +2.2. Composite and pNGB/Little Higgs ..................................................................... +8 +3. EFT at current and future colliders.......................................................................... +11 +4. Top quark and BSM related to Flavor Dynamics or Dark Matter (or both)................................ +12 +4.1. Top quark and BSM related to flavor .................................................................. +12 +4.2. Flavored dark matter models ........................................................................... +15 +5. Conclusions ................................................................................................... +18 +1. Introduction +The top quark is a singular object amidst the fermions of the Standard Model as it is the +heaviest among them. This entails several peculiar properties: in the domain of QCD it +stands out as it is the only quark never to be observed into a hadron, as its decay is much +faster than the hadron formation time; after the discovery of the Higgs boson, and for +all the time before in which the Higgs mechanism has dominated the landscape of model +building for the electroweak sector of the SM, the top quark stands out as the only one +with a “normal” size coupling with the Higgs boson. This latter property has made the top +quark very interesting both for the question about the origin of the structure of flavor in +the SM and for the origin of the electroweak scale itself. The special interest about top +flavor has to do with its strong preference to decay into bottom quarks, i.e not involving +other flavor families, which in the CKM picture results in Vtb = 1 up to small corrections, +and its large mass, which can possibly act as a magnifier of the effects of physics beyond +the Higgs boson as origin of flavor. For electroweak physics the top quark plays a crucial +role in that it affects the properties of the Higgs boson, and by the Higgs mechanism for +weak bosons mass generation, also in the physics of weak gauge bosons: its effect can be +seen in their masses and decay rates, which are sensitive to the strength of the top quark +gauge and Yukawa couplings and to its mass. Deviations of these properties from the SM +predictions can be signs of new physics related to the top quark. While the importance of +the top quark can be appreciated already from these general facts, the detailed role played +by the top quark can be better understood going closer to explicit new physics models, +which will pace the exposition of the greatest part of the following material. +In sections 2.1 and 2.2 we discuss models in which the top quark plays a special role for +the origin of the electroweak symmetry. The discussion is further extended in section 3 in a +more model independent direction using a flavor-conserving effective field theory of the top +quark sector, which also allow to discuss prospects for top quark physics at future colliders. +In section 4.1 we attack a different problem, that of the origin of the flavors of the SM. In +section 4.2 we extend the discussion to the possibility that SM flavor plays a part in the +stabilization of the dark matter in a way that makes the dark matter interact preferably +with the top quark flavor and discuss the phenomenology of dark matter in these scenarios. +Finally in section 5 we offer some conclusions. +Being the subjects list rather large, the discussion is necessarily kept free from some +details, which are available in the provided references. This review is conceived so that it +2 + +can also be useful for younger graduate students seeking an high-level introduction to the +subject(s) discussed. Hopefully the readers can start here their own exploration on topics +that would otherwise require to go through a large stack of literature. References are kept +to a minimum of key works as to encourage the reader to actually study these selected +works. +2. Top quark and BSM related to the Higgs boson and the origin of the weak +scale +2.1. Supersymmetry +Supersymmetry has been proposed as a space-time symmetry involving fermionic genera- +tors. Unlike in gauge symmetries, this makes possible to involve spin and momentum in the +definition of the symmetry algebra, which, up to violations of the symmetry itself, would +require interactions and masses of bosonic and fermionic particles to be tightly related. +One such relation would require the electron to be accompanied by exactly mass degener- +ate states of spin-0, pretty much the same as Lorentz symmetry of space-time built-in the +Dirac equation implies the existence of exactly mass degenerate anti-particles of the elec- +tron. The absence of any evidence in experiments for spin-0 electron-like state motivates +to consider supersymmetry as an approximate symmetry, broken at some unknown scale so +that all the supersymmetric partners of the SM states are pushed beyond the mass scale +presently probed by experiments. +The mechanism for supersymmetry breaking is a subject for model building, which is +outside of the scope of this review. For our purpose it is key to recall that the supersym- +metry breaking top quark sector has the rather model-independent tendency to determine +the Higgs bosons mass and quartic coupling, thus leading to the identification of the su- +persymmetric top scalar quark, most often called “stop squark”, as the main player setting +the Higgs boson potential. In the Minimal Supersymmetric Standard Model (see (1) for an +extensive review) this is represented by equations for the constraints on the minimization +of the Higgs boson potential +m2 +Z = +��m2 +Hd − m2 +Hu +�� +� +1 − sin2(2β) +− m2 +Hu − m2 +Hd − 2 |µ|2 , +sin(2β) = +2b +m2 +Hu + m2 +Hd + 2 |µ|2 , +coupled with the 1-loop effect of the top quark and top squark on the bilinear terms of the +2 Higgs doublets Hu and Hd . In particular, for the Higgs doublet Hu that interacts with +up-type quarks, hence feels the top quark sector, the RGE equations is +d +d log Qm2 +Hu = 3Xt − 6g2 |M2|2 − 6 +5g2 +1 |M1|2 + 3 +5g2 +1S , +1. +where Xt = 2 |yt|2 � +m2 +Hu + m2 +Q3 + m2 +¯u3 +� ++ 2 |at|2, M1,2 are the U(1) and SU(2) gaugino +mass terms, and S = Tr[Yjm2 +φj]. +These equations naturally lead to possibility that the supersymmetry breaking stop +masses m2 +Q3 and m2 +¯u3 or a large A-term |at| might induce a large Xt, which in turn drives +m2 +Hu < 0 as log Q diminishes from some high-scale down to the weak scale. This possibility +has made the role of stop squarks a very central one in supersymmetric models. In essence, +www.annualreviews.org • +3 + +the supersymmetric partner of the top quark is responsible for breaking the electroweak +symmetry, by making m2 +Hu < 0 hence making the Higgs boson potential unstable at the +origin of the Hd, Hu fields space, and setting the value of the masses that set the weak scale, +e.g. mZ from the above equation or the mass of the Higgs boson that receives the above +mentioned large radiative corrections from the stop squark. +As a matter of fact, once the Higgs boson was discovered and its mass was known, +a number of works tried to determine the impact of this measurement on the properties +of the stop squark (e.g. see Ref. (2) for the MSSM and some extensions). In turn, the +necessity for peculiar supersymmetry breaking to accommodate the Higgs mass has spurred +investigations on the possible supersymmetry breaking models that can lead to such peculiar +stop squarks (see e.g. (3–8) for some examples of supersymmetry breaking models emerged +or re-emerged to address the null searches of supersymmetry and the Higgs discovery). +2.1.1. Phenomenology. The phenomenology of the supersymmetric partners of the top quark +is largely dictated by one feature of the supersymmetric models: the existence of a conserved +quantum number that distinguishes SM states from their supersymmetric partners. The +standard choice for such quantity is called R-parity, a Z2 symmetry under which all SM +states are even and all partners states are odd. The conservation of this symmetry implies +that partners states can appear in interaction vertexes only in even number, e.g. +one +SM states can interact with two supersymmetric states and it is not possible for a single +supersymmetric state to interact with a pair of SM states. For particle colliders this implies +that the lowest order process to produce supersymmetric states in collisions is +SM SM → SUSY SUSY, +and the decay of supersymmetric particles to any number of SM states is forbidden unless +there is at least one supersymmetric particle (or an odd number of them), e.g. +SUSY → SUSY SM . +When R-parity is exact a most copious production mechanism for stop squarks at the LHC +is +gg → ˜ti˜t∗ +j, +2. +where we denoted ˜tk for k = 1, 2 the two stop squarks mass eigenstates 1. Other production +mechanisms are possible, e.g. in decays of supersymmetric partners heavier than the stops +or via production of stops in association with other supersymmetric states. +Once produced, the stop squark can decay in a number of possible channels, depending +on which supersymmetric states are lighter than the state ˜tk at hand. Most studied 2-body +decay modes are +˜t → tχ0, ˜t → bχ+ , +3. +which feature fermions χ that are mixtures of supersymmetric partners of gauge bosons of +the electroweak interactions and of the Higgs bosons of the model. The motivation for the +1The definition of mass eigenstate as “stops” assumes that flavor labels we give in the SM are +the same for the partners states. It must be stressed that the fate of flavor in the supersymmetric +partners sector is largely model dependent and it is possible to use flavor mixing to change the +phenomenology of stop squarks, see e.g. +(9). +See (1) for more details on the gauge and flavor +structure of the squark sector. +4 + +prevalence of these decay modes is that, by the rules of unbroken supersymmetry, these +decays are mediated by couplings given by gauge and Yukawa couplings of the SM, hence +they are pretty much impossible to switch off unless m˜t − mχ < 0. As a matter of fact +the quantity m˜t − mχ plays a major role in determining the stop phenomenology. When +m˜t − mχ → 0 it becomes necessary to consider multi-body processes are also possible and +may be phenomenologically relevant, e.g. +˜t → bW +χ0, ˜t → b ¯ff +′χ0, +4. +as well as possible flavor violating decays that may be induced at loop level, such as +˜t → cχ0 . +5. +In the above discussion the particle χ0 is considered as the lightest supersymmetric state +(LSP), so that, by the conservation of R-parity, it is absolutely stable. As χ0 is not elec- +trically charged and it is color neutral, pretty much like neutrinos it does not leave directly +observables traces in detectors. For this reason the presence of χ0 can be detected only as +momentum missing in the overall momentum conservation in each collision. As we cannot +reliably measure the fractions of the longitudinal momentum of the colliding protons taken +by the partons initiating the production of stops, e.g. the gluons entering in eq.(2), and +the fraction taken by the rest of the partons, the longitudinal momentum conservation is +usually not exploited in hadron colliders, therefore the presence of χ0 is usually sought for +as missing transverse momentum, most often (mis)named missing transverse energy mET. +Being an electrically neutral stable particle charged only under supersymmetric Yukawa +and electroweak gauge interactions, χ0 qualifies as perfect candidate for a WIMP Dark Mat- +ter. The possibility to have a Dark Matter candidate stemming out of supersymmetry has +given formidable motivation to pursue this scenario for the past decades. So much so, that +missing transverse energy searches have becomes synonymous of searches for supersymme- +try. It must be said, however, that the null searches of supersymmetric particles, as well as +WIMP Dark Matter in the mass range suitable for χ0(10), has put this idea under great +pressure lately (11,12). +Given these experimental results, and the vast range of possible models for supersym- +metry breaking, it must be recalled that in general it is possible to have other states than χ0 +as lightest supersymmetric particles. For instance the supersymmetric partner of a neutrino +or even top sector squarks. The latter leads to peculiar phenomena due to the formation of +hadrons containing supersymmetric states(13)(14), but these models typically suffer from +quite stringent limits (15–17). Therefore the majority of the searches for supersymmetric +states in the top quark sector are carried out in the χ0 LSP setting. +Wholly alternative phenomenological scenarios for supersymmetric top quark partners +are possible and are actively pursued in experimental searches. The main possible alter- +native has to do with the non-conservation of R-parity (18). +With broken R-parity all +supersymmetric particles can in principle be produced singly and can decay into just SM +states, e.g. +SM SM → SUSY and SUSY → SM SM , +are now possible processes. In this situation there is no longer an absolutely stable weak +scale particle to purse the idea of Dark Matter as a WIMP2 and the phenomenology of +2Alternative DM candidates can be found in these models, see e.g. (19) for a possible gravitino +dark matter scenarios and issues related to this possibility. +www.annualreviews.org • +5 + +supersymmetric states linked to the top quark is now greatly different from the picture +given above (20). For instance R-parity violating couplings, still respecting the full gauge +symmetry of the SM, allow, among other possibilities, the decays +˜t → bs or ˜t → ℓd . +As the final states of stop decay can now be made entirely of SM particles it is possible +to detect stop squarks as resonances, a very powerful signature, that is not possible to +pursue when χ0 is forced to appear among the decay products. Furthermore these decays, +being mediated by R-parity breaking couplings, that need to be small for a number of +constraints (18), can lead to meta-stable supersymmetric states, which can live measurable +lengths in experiments. +2.1.2. Experimental searches. In a detailed model it is possible to derive very specific signals +from top sector supersymmetric partners, including both signatures at collider experiments +and as well as low energy precision ones. The latter, however, turn out to be usually very +much dependent on the model considered for low energy precision experiments (21). A +similar issue exists with early universe physics, on top of the signals being quite difficult +to detect. +For this reason collider searches are the prime way to search for top sector +supersymmetric partners. +Before listing relevant searches it is necessary to clarify a point on their scope. The +above searches are sensitive in principle to any sign of new physics related to the top quark +sector involving mET or some kind of pair produced resonances. +Although the search +is optimized for supersymmetric partners, it can indeed be used to set bounds on other +models. The interested reader can refer for instance to Ref. (22) for an interpretation of the +“supersymmetry searches” in the context of fermionic top partners to be discussed in later +Section 2.2.2. +The searches for top sector supersymmetric partners can be divided into two main +categories: +• searches in large momentum transfer signals, which feature detector objects (jets, +leptons, photons, ...) with energy and transverse momentum greater than the typical +SM events; +• searches in low momentum transfer signal, in which the detector objects arising from +top sector supersymmetric partners production are not very different from that of +typical SM events. +The large momentum transfer ones are “classic” searches for new physics, and were envisaged +already at the time of design of the experiments (23,24). Currently these searches can probe +supersymmetric top partners up to a mass around 1.2 TeV, although not in full generality. +Indeed it is quite hard to probe in full generality even a model as “minimal” as one having the +full freedom to vary the branching ratios of decays eqs.(3)-(5). For a complete assessment is +then necessary to test very accurately a large number of searches at once, often relying on a +“phenomenological” incarnation of a sufficiently general supersymmetric model, as studied +for instance in Ref. (25). +The interpretation of these results is quite difficult, as many +constraints on the model are imposed at once, e.g. the top partners states are required to +“fix” the mass of the SM Higgs boson to its measured value by the dynamics of radiative +corrections embodied in eq.(1). This requirement, while being a sensible one in the context +of the specific model, can significantly alter the conclusion of that study. +Therefore it +6 + +remains difficult to answer questions as simple as finding the lightest not excluded values +of the mass of stop-like top partners 3. +Further difficulties can arise and make nearly impossible to probe experimentally su- +persymmetric top partners, e.g when special kinematical configurations become the typical +configuration of top partners decay products. In these cases the search in low momentum +transfer signatures can help. Indeed, these searches have been developed to overcome the +difficulty that arise in the limit m˜t − mχ → 0. The shortcomings of the large momentum +transfer searches can be clearly seen in Figure 1, as the excluded stop mass for large m˜t−mχ +is much larger than for small values of this mass difference. In addition, when the stop-LSP +mass gap is small and the stop becomes lighter, its production and decay cannot be reliably +distinguished from other SM processes, e.g. the SM top quark production. This observa- +tion motivates a zoom inset in the figure to display how these peculiar cases are covered. +The most useful strategies to attack these difficult signatures have turned out to be the +studies of angular observables and fiducial rates of top-like final states (27–31). Especially +in angular observables there are modest, but persistent disagreement between the measure- +ments in the top quark sample (32) and theoretical predictions. These disagreement are +also accompanied by other disagreements of small entity, but persisting from Run1 LHC +through Run2, in the kinematics of the reconstructed top quarks e.g. in Refs. (33,34). The +possibility to see effects of BSM related to the top quark and the precision in measurements +afforded by the LHC and the HL-LHC has motivated the great improvement of predictions +for top quark SM observables, e.g. (35) for a seamless description of fixed NLO and PS cal- +culations of top quark resonant and non-resonant rates, (36,37) for specific NNLO and EW +corrections to the BSM sensitive rates and more in general drawing attention on possibly +BSM-sensitive high energy top quarks (see e.g. (38)) and other production modes which +may be of interest for both SM studies and BSM searches (see e.g. (39,40)). +The searches mentioned above, though motivated and sometimes optimized on super- +symmetry searches, are rather general. Thus it is important to stress that the observation +of an excess in one of these “supersymmetry searches” would not at all prove the supersym- +metric nature of the discovered state. A reliable statement on the supersymmetric nature +of the newly discovered object would require several measurements. For some proposal at +the LHC the interested reader can look for instance at (41). In general it is believed that a +machine cleaner than a hadron collider, e.g. an e+e− collider, capable of producing the new +particle would be needed to truly confer it the status of “supersymmetric partner” state of +some SM state. +At the time of writing there are no statistically significant and convincing signs of new +physics in searches for new physics, the searches for supersymmetric top partners being no +exception. Despite the absence of signals for top sector supersymmetric partners these are +still believed to one our best chances to find new physics. Looking at the glass as “half +full” one could even argue that in the minimal model of supersymmetry the relatively large +observed Higgs boson mass requires large loop level corrections from contributions of the +kind of eq.(1). These large loop corrections point towards a stop squarks mass scale at the +TeV or larger, thus perfectly compatible with the present limits and possibly awaiting us +for a next discovery at one of the next updates of the searches as more data is collected at +the LHC. +3One possible answer in the context of (25) is offered in the supplementary material of that +analysis(26). +www.annualreviews.org • +7 + +Observed limits +Expected limits + + -1 + = 13 TeV, 139 fb +s +Data 15-18, +0 +1 +χ∼ + bff' +→ + +1t~ +monojet, +[2102.10874] +0 +1 +χ∼ + bff' +→ + +1t~ + / +0 +1 +χ∼ + bW +→ + +1t~ + / +0 +1 +χ∼ + t +→ + +1t~ +0L, +[2004.14060] +0 +1 +χ∼ + bff' +→ + +1t~ + / +0 +1 +χ∼ + bW +→ + +1t~ + / +0 +1 +χ∼ + t +→ + +1t~ +1L, +[2012.03799] +0 +1 +χ∼ + bff' +→ + +1t~ + / +0 +1 +χ∼ + bW +→ + +1t~ + / +0 +1 +χ∼ + t +→ + +1t~ +2L, +[2102.01444] + + -1 + = 13 TeV, 36.1 fb +s +Data 15-16, +0 +1 +χ∼ + bff' +→ + +1t~ + / +0 +1 +χ∼ + bW +→ + +1t~ + / +0 +1 +χ∼ + t +→ + +1t~ +[1709.04183, 1711.11520, + 1708.03247, 1711.03301] +0 +1 +χ∼ + t +→ + +1t~ +, +tt +[1903.07570] + + -1 + = 8 TeV, 20.3 fb +s +Data 12, +0 +1 +χ∼ + bff' +→ + +1t~ + / +0 +1 +χ∼ + bW +→ + +1t~ + / +0 +1 +χ∼ + t +→ + +1t~ +[1506.08616] +200 +400 +600 +800 +1000 1200 +) [GeV] +1 +t~ +m( +100 +200 +300 +400 +500 +600 +700 +800 +900 +) [GeV] + 0 + 1 +χ∼ +m( + -1 + = 8,13 TeV, 20.3-139 fb +s +March 2021 +ATLAS Preliminary + production +1t~ +1t~ +Limits at 95% CL +180 +200 +220 +0 +10 +20 +30 +40 +50 +60 +70 +) = 0 +0 +1 +χ∼ +, +1t~ + m( +∆ +W + + m +b +) = m +0 +1 +χ∼ +, +1t~ + m( +∆ +t +) = m +0 +1 +χ∼ +, +1t~ + m( +∆ +Figure 1 +Searches for top sector supersymmetric partners in the Stop-LSP mass plane. +As the mass scale of top quark supersymmetric partners is not entirely fixed it often +considered that these particles may be too heavy for the LHC to discover them. Therefore +the discovery reach for these particles is often considered in the evaluation of the physics case +of future particle accelerators. Projections for a 100 TeV pp collider (42, 43) usually cover +a mass range 5-8 times larger than what can be probed at the LHC, while the expectation +for a high energy lepton collider, such as multi-TeV muon collider(44–48), is to probe the +existence of top partners up to the kinematic limits at √s/2. +2.2. Composite and pNGB/Little Higgs +2.2.1. Models . New physics associated to the top quark sector has been motivated also from +a series of model building activities aimed at explaining the origin of the electroweak scale +through the Goldstone boson nature of the agent of its breaking, resulting in theories of +the Higgs boson as a pseudo Nambu-Goldstone boson. From a low energy effective point of +view these theories can be put in the language of a composite Higgs boson, whose lightness +compared to its scale of compositeness is justified by its goldstonian nature. Models built +in this family are reviewed in Refs. (49–52) and they all share the need to enlarge the +symmetries of the SM by a new global symmetry, that is broken at some scale above the +TeV to a smaller symmetry, with the associated Nambu-Goldstone bosons, which will host +the yet smaller symmetry group of the SM at even lower energies. The minimal model of +this type (53) that is able to pass bounds from electroweak precision tests including Zb¯b +8 + +couplings assumes an SO(5) global symmetry, broken to SO(4) ≃ SU(2) × SU(2) which +contain the weak interactions gauged SU(2). +The enlargement of the symmetry of the SM motivates appearance of matter repre- +sentations in multiplets that are necessarily larger than the usual doublets and singlets of +the SM. In particular, in order to obtain Yukawa interactions the constructions of pNGB +and little Higgs model converges in the existence of “partner” states for the top quark, the +bottom quark and in principle for all the fermions of the SM. The precise phenomenological +manifestation of the “partner” states is highly model dependent, as it depends on the choice +the new global symmetry group that one has in building this type of models, the repre- +sentation of this symmetry group that one chooses for the new matter and the imagined +mechanism to originate the SM fermion masses at the most microscopic level. +One possible limitation to the model building choices may comes from the requirement of +not introducing large deviations in well known couplings, e.g. the Zbb couplings (54), still a +large set of possibilities exists. For this review we focus on a unifying feature of many models, +that is the presence of “partner” states directly connected to the SM top quark sector via +Yukawa and gauge interactions with relatively universal decay patterns (55–57), although +other decay modes and more “exotic” partners may exist including possible couplings to +scalar states accompanying the Higgs boson in some models (58–60). +2.2.2. Phenomenology. At the core of the experimental tests of the idea of fermion top part- +ners lies the assumption that the main interaction leading to the decay of these top partners +into SM states is the Yukawa of the top quark, in which the Higgs boson or longitudinal +components of the weak gauge bosons appear. For this reason the large majority of the +searches are presented in terms of exclusions for branching fractions of the top partners +states into the following pairs of SM states +T → tZ, th, Wb, +where T is a charge 2/3 top partner and +B → bZ, bh, Wt, +where B is a charge -1/3 partner of the bottom quark, whose existence is consequence of +the SU(2) weak isospin symmetry that must hold in the theory that supersedes the SM at +high energies. In models with a symmetry larger than SU(2), e.g. (54)(53), it is typical +to have further partners states that appear as necessary to furnish full representations of +the larger symmetry. A much studied case is the state of charge 5/3 that leads to a very +characteristic decay +X5/3 → W +t , +which in turn gives a characteristic same-sign di-lepton signal (61). For little Higgs models +the appearance of this type of exotic partners requires the formulation of somewhat more +involved models, but it is definitively a possibility(58,62). +2.2.3. Experimental searches at colliders. Experimental searches for new states are carried +out at the LHC exploiting the color charge of the top partners in processes such as +gg → TT , +www.annualreviews.org • +9 + +that are analogous to previous processes for supersymmetric partners and depend only on +the QCD charge of T. Unlike for supersymmetric partners, for which the conservation of +R-parity plays a crucial role, the single production of top partners +gq → q′Tb , +is possible in the most minimal models and can in principle lead to a deeper understanding of +the BSM physics, as this process involves directly new physics couplings for the production +of the top partners state (63). For instance the rate of single production of top partners +states can be a discriminant with respect to so-called “vector-like” quarks, whose couplings +are not dictated by Goldstone property of the Higgs (see Ref. (64) for a more in-depth +discussion). +A great difference in the search for the partners discussed in this section is that they +can in principle give rise to resonant signals, e.g. in the invariant mass of an hadronic top +and one hadronic Higgs boson in the decay T → th and other signals discussed for instance +in the search of Ref.(65). +Another consequence of the top partner decaying in purely SM final states is that even +the “heavy” SM particles, such as t, Z, W, h, are produced with significant boost in the +majority of the events. This motivates the use of special experimental techniques for the +identification of those detector objects (66) as for instance in the search of Ref.(67). +The search strategies mentioned above are combined by the experimental collaborations, +that present results in a plane with axes spanning the possible values of two decays, e.g. if +figure 2 an example is shown for T → Ht and T → Wb. The underlying assumption of this +presentation of the results is that the top partner does not decay in any BSM states, hence +the branching ratio of T → Zt is determined by the two branching rations displayed. The +right panel of the same figure shows how the different searches have different sensitivity to +each decay mode and can be patched together to better exclude top partners of a given +mass. For more exotic signals from X5/3 searches are carried out as well, e.g. in Ref. (68). +Results of searches at LHC collected in figure 2 and newer results (67, 69) on the kinds of +top partners described so far put bounds on the top partners mass at around 1.2 TeV. +As mentioned above it is possible to have larger groups and larger representations in the +symmetry breaking pattern. For instance if the large global symmetry of which breaking +the Higgs is a pNGB is chosen to be SO(6) broken to SO(5) and top quark partners states +are chosen to furnish a 6-dimensional representation there is one extra top partners state +compared to the case of top quark partners in the 5 of SO(5) considered for the minimal +model of Ref. (54). If we call this new top partner state Ψ1, the name signals the fact +that it is a singlet under the remnant SO(5) symmetry, we can have new signals from +its production via QCD interactions and decay that do not fit into any of the previously +considered categories e.g. +Ψ1 → th, tZ, tη, Wb , +where η is an extra pNGB that arises due to the larger number of broken generators in the +breaking SO(6) → SO(5) → SO(4) ≃ SU(2) × SU(2) . +In general the extensions of pNGB models can include possible FCNC of top quarks +with new physics states, e.g. Ref. (73) has considered decays of the SM top quark that +violate flavor +t → cη +as a consequence of underlying flavor-changing dynamics in the top partners by a coupling +Tcη which would also yield a new possible search channel for a top partner T → cη. Other +10 + +Figure 2 +Searches for top fermionic partners (70,71) in the plane BR(T → Ht) vs BR(T → Wb) with the +constraint B(bW) + B(th) + B(tZ) = 1. For reference, some model-dependent choices of the +branching ratios introduced in Ref.(72) are shown. +exotic possibilities are covered in the literature, e.g. T → tg, tγ, X5/3 → tφ+ and more +exotics ones are presented in Refs. (74–76) and can in principle lead to new signals for top +quark partners. +3. EFT at current and future colliders +The previous sections dealt with explicit models of new physics giving rise to signals from +direct production of particles beyond those of the Standard Model. As these searches have +so far yield no evidence of new physics a growing interest and motivation have risen for +the description of new physics in Effective Field Theories. The effective character of these +theories is due to the fact that they arise by the removal of heavy states from a theory more +microscopic than the SM and they lead to a set of BSM interactions, that is usually in overlap +with the set generated by other microscopic theories. Therefore it has been done a great +work in identifying the most general sets of interactions under given assumptions (77,78), +so that new physics studies can be carried in a “model-independent” fashion, e.g. searching +for very characteristic interactions involving four top quarks (79–82) or other four-fermion +operators involving top quarks, or other kinds of contact interactions independently of their +microscopic origin. +The plus side of the EFT approach is that it is very comprehensive. The converse of +this comprehensiveness is the possible loss of contact with the microscopic origin of physics +beyond the Standard Model which gives rise to specific patterns and organization principles +for the size of each contact interaction. Thus it is necessary to strike a balance between a +fully general EFT and a “physically efficacious” effective theory. Where this balance lies is +very much dependent on the amount of data that one can use in constraining the couplings +of the effective interactions, as well as the theoretical prejudice on what effects are worth +being considered, e.g. pure top sector effects (78,83–87), or effects involving EW and Higgs +physics as well (88,89) or exploring flavor changing effects (90–95). +As the effect of BSM contact interactions from the EFT affects precision measurements +of SM processes, this enhanced attention towards signals of BSM associated to top quarks +www.annualreviews.org • +11 + +(↑H +1 +m, = 800 Gev +m, = 900 GevV +ATLAS +0.8 +ATLAS Preliminary +1420 +Vs = 13 TeV, 36.1 fb-1 +BR(T +0.6 +0.9 +个 +..Exp.exclusion Obs.exclusion +Vs = 13 TeV, 36.1 fb' + limit [ +0. +W(lv)b+X [arxiv:1707.03347] +1400 +0.8 +VLQ combination +1400 +0.2 +H(bb)+X [arxiv:1803.09678] +R +Z(vv)t+X [ariv:1705.10751] +B +0.7 +Observed limit +nass +m = 950 Gev +m = 1000 Gev +1375 +0.8 +Trilep./same-sign [CERN-EP-2018-171] +0.6 +Z(I)/b+X [arxiv:1806.10555] +0.6 +1380 +m +0.4 +All-had [CERN-EP-2018-176] +★ SU(2) doublet +★ sU(2) doublet ● sU(2) singlet +O sU(2) singlet +1360 +95% +1320 +m = 1050 Gev +m = 1100 GeV +m = 1150 Gev +0.4 +0.8F +0.6 +0.3 +1340 +0.4 +0.2 +0.2 +1320 +m = 1200 GeV ± +m = 1300 GeV +m, = 1400 GeV +0.8 +0.1 +★ +1300 +0.4 +0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 +0.2 +. +BR(T → Wb) +0.2 0.4 0.6 0.8 +00.2 0.4 0.6 0.8 +0.2 0.4 0.6 0.8 +0 +BR(T →> Wb)Figure 3 +Summary from Ref. (101,102) of the constraints on contact interactions involving the top quark. +The left panel shows the effect of HL-LHC compared to present constraints. The right panel +shows the effect of future e+e− machines. The taller (and lighted) bars for each case represent the +looser bounds that are obtained when the coupling of interest is bound while the others are +allowed to float, see (101,102) and references therein for details. +has produced activity on the improvement of the description of several processes that are +either backgrounds or serve as SM reference on top of which search for signs of BSM, +e.g. see recent Ref. (96) for four-top production, recent ttV results discussed in Refs. (97– +99), tth results in Ref. (100) and references therein. For an up to date snapshot of the +characterization of the top quark electroweak interactions and possible BSM in deviations +from the SM we refer the reader to Refs. (83–87). The upshot of the work is that present +measurements, also thanks to the availability of differential measurements and trustable +computations in the same phase-space regions, can put bounds on generic new physics in +the top quark sector in the TeV ballpark. +The possibility to identify indirect signs of new physics in signatures related to the top +quark has become a commonly used benchmark in the evaluation of performances of future +colliders, especially clean e+e− machines, whose best chance to see new physics in the top +sector is through indirect effects. Works such as (101–104) have studied the outcome of +analyses to be carried out at future colliders and the interplay between present and future +colliders probes of new physics in top quark effective field theory. The results are summa- +rized in Figure 3, which shows the significant improvement that will be attained by the +HL-LHC, especially on single-couplings effects. The figure also shows the strong tightening +of the bounds with the addition of data from future e+e− data at the Zh threshold, the t¯t +threshold and above, which will make the global EFT constraints particularly robust by the +removal of possible flat directions in couplings-space and providing new data in channels +that can be probed best at clean e+e− machines. +4. Top quark and BSM related to Flavor Dynamics or Dark Matter (or both) +4.1. Top quark and BSM related to flavor +The top quark flavor remains a special one in the SM. Indeed the top quark is so heavy that +one can easily single out the third generation of quarks as a peculiars source of breaking of +12 + +102 +FIT +I LHC Run 2 + Tevatron + LEP + +HL-LHC S2 +HL-LHC +CEPC +HL-LHC + FCCee +HL-LHC +ILC + HL-LHC + CLIC +FIT +HEPfit +101 +HEPfit +101 +val (TeV-2) +100 +100 +95% Interv +10-1 +10-2 +10-2 +c +10-3 +Ctw +Cot +CoQ +Ctz +Cb +Ctp +Ctw +Cot +Cpb +Ceb +CeQ +Clb +Cet +Cit +Cio +Operator Coefficients +Operator Coefficientsthe flavor symmetry +GF = U(3)qL × U(3)uR × U(3)dR +that the SM would enjoy if all quark masses were zero. A hierarchy of breaking dominated by +the third generation can be accommodated easily, thanks to the freedom about the possible +symmetry breaking patterns and possible mechanisms for breaking the flavor symmetry of +the SM that one can consider. In addition, this way of organizing the breaking of flavor +symmetry is most compatible with experimental bounds. In fact, bounds on first and second +generation flavor changing processes are the most tight, whereas there is a relative lack of +constraints on the third generation. If the sole breaking of the symmetry GF arises from +the Yukawa couplings of the SM, or new sources are aligned with the Yukawa matrices, +the breaking is said to comply with “Minimal Flavor Violation” (MFV) (105–107). In this +setting the bounds from flavor observables are most easily accommodated, but it is not the +only possibility to comply with observations. The fact that the top quark Yukawa coupling +is a possible large source of flavor symmetry breaking motivates to consider BSM related +to the top flavor, but this conclusion holds also in other settings. +A classification of possible states that can couple to quark bilinears charged under the +flavor symmetry, e.g. a new scalar coupled as φtutu, has proven useful in the past to assess +the possibility of flavorful signs of new physics. For a recent listing of the possible states +one can read tables on Ref. (108). From a phenomenological point of view these models +give rise to transitions in four-quark scatterings that do not conserve the flavor charge. For +instance the scattering +uu → tt +can arise via a t-channel exchange of a flavored boson. This can alter the kinematic of +top quark production as well as the net charge of the top quark sample at hadron colliders. +Indeed new flavorful boson of this kind were advocated in response to TeVatron experiments +claiming disagreements between the SM predictions and measured top quark properties, +such as the forward-backward asymmetry in the production of top quarks (109–111). In +addition these new flavored states coupled to the top quark can give rise to transitions +f ¯f → tφtjuj , +that can be observed quite easily at e+e− colliders in multi-jet final states, the detailed +final state depending on the model-dependent decay of the flavored state φ. +The possibility that a flavored state connected to the top quark might be among the +lightest new states from the new physics sector has appeared also in models of gauged +flavor symmetries. +In these models the flavor symmetry GF is gauged, as to not have +to deal with unobserved massless Goldstone bosons. +For instance Refs. (112, 113) have +proposed a new set of states that would have the notable property to make the GF gauging +free from triangular anomalies by the addition of vector-like new quarks. In this kind of +models the new quarks are charged under the SM flavor symmetry and can be arranged as +to have top-flavor new states to be the lightest ones. Indeed in these models the masses of +the the SM quarks would be explained by a see-saw-like mechanism in which the lightest +SM fermions are mixed with a very heavy new state, whereas the heaviest SM states are +mixed with the lightest of the new physics states. In this case the SM top quark would be +the state coupled to the lightest of the new physics states, named t′, possibly accompanied +by a partner state for the bottom quark, named b′. Remarkably this type of model gives +www.annualreviews.org • +13 + +Λ �� +(���) +�������� ����� +������ +��� ������ +������ ����� +��� ������ +������ ����� +�������� +Δ�� +ϵ� +�� → μ+μ- +������� ��� +�������� ��� +μ → � γ +�� +�� +�� +��� +� +� +�� +�� +Figure 4: Lower bounds on ⇤IR on the various flavor scenarios. The first set of bounds corresponds +to our scenario with multiple flavor scales, the second and third sets assume partial compositeness +at ⇤IR for the whole third and second family respectively, while the last set gives the bounds for the +anarchic flavor scenario. To derive the numerical values we have taken g⇤ ' 3, xt ' xc ' 0.5, and +set all free ↵L,R parameters to one. +where +gij ⌘ Ytxt(V † +CKM)i3(VCKM)3j , +(7.2) +and dLi denotes the left-handed down-type quark component in the i-th family. A remarkable +feature of these corrections is the fact that they automatically follow a MFV structure. The first +operator contributes to �F = 2 transitions and generates correlated e↵ects in the ✏K, �MBd and +�MBs observables, which are of the order of the present experimental sensitivity if we take ⇤IR ⇠ +TeV and we allow for a slight reduction of the left-handed top compositeness, xt < 1. The second +operator of Eq. (7.1) gives flavor-changing Z-couplings. At present it only pushes the ⇤IR scale in +the few TeV range. In the future it can be seen either in deviations in the decays K ! µµ or +B ! (X)``. This contribution can however be significantly smaller if the strong sector is invariant +under a custodial PLR symmetry, which protects the down-type quark couplings to the Z boson [30]. +Additional contributions to �F = 2 operators can also be generated at the scales ⇤c,s,d at +which the second and first family quarks get their masses. These corrections however only give +a sizable e↵ect on ✏K, that pushes the ⇤IR scale in the multi-TeV range (⇤IR & 6 TeV), which +is still a milder bound with respect to the anarchic one. It must however be stressed that these +bounds depend on the coe�cients of the e↵ective operators which are a↵ected by some degree +of uncertainty. These contributions to ✏K severely constrain the maximal dimension of the OH +operator, requiring dH . 2. +We also considered possible variations of the framework described above. For example, a more +economical scenario has been proposed in which each family is associated to a single flavor scale +at which the bilinear mass operators are generated. A few additional new-physics flavor e↵ects +20 +Figure 4 +Lower bound on the scale of new physics related to the SM fermion mass generation in a composite Higgs scenario (117) +under different assumptions on the compositeness of SM fermions. +phenomenological signatures very similar to those of top partners states of composite and +little Higgs, e.g. the partner states can be produced by strong interactions and decay as +b′ → bh, bZ, tW +and +t′ → th, tZ, bW . +These ideas also lend themselves to be paired with supersymmetry. Although super- +symmetry is not necessary for the idea of gauged flavor symmetries in general, these models +can provide a setup to originate R-parity breaking with an underlying structure for the +flavor structure of the RPV couplings (114,115), that for instance would motivate +˜t → bs +as the main channel to search RPV stops (116). +A solution with a hierarchy of flavored new physics scales inverted with respect to that of +the SM quarks has been proposed also for composite Higgs models (117–120), which would +otherwise suffer from severe bounds from high-pT and flavor observables (see e.g. (121–123)), +even in presence of some degree of model building (124–126) aimed at keeping all the new +physics at a common low-scale and still survive flavor tests thanks to a friendly, possibly +MFV-like structure, of the flavor origin in the microscopic completion of the composite +Higgs model. As it can be appreciated in Fig. 4 the top quark sector emerges still as a less +constrained one and further motivates to consider BSM physics related to the top quark, +and possibly exclusively to the top quark or to the third generation of SM fermions. +14 + +Observables of interests include indirect probes such as electric dipoles moments (see +e.g. (127)), meson oscillations and decays, and in principle rare Z and Higgs bosons flavor- +violating decays which usually receive important contributions from the top quark sec- +tor (117). In addition, it is possible to have phenomena more directly related to the top +quark such as +t → cV, +where V = γ, Z, g(128,129) and deviations from Vtb = 1 in the CKM matrix (64,130–132). +4.2. Flavored dark matter models +Given the strength of the bounds from direct searches of dark matter scattering on heavy +nuclei it has become interesting to consider dark matter models in which the flavor of SM +quarks and leptons plays a role, as the strongest bounds hinge on effective couplings of the +dark matter to first and, to a slightly lesser extent, to second generation quarks and gluons. +Rather interestingly the flavor puzzle of the SM comes equipped with a symmetry, +which, though not exact, can be used to stabilize the dark matter if it is broken according to +Minimal Flavor Violation (133,134) and even with more general patterns of flavor symmetry +and its breaking (135). As a dark matter coupling sensitive to flavor could mediate flavor +changing transitions the option of the MFV structure, or slight departures from it, has been +so far been a main route in model building aimed at removing possible tensions with flavor +observables. +Among the possible flavor structures that the Dark Matter and the SM can fields can +be cast in, for our work here we focus on the possibility that the top quark flavor has a +special role. Explicit models have appeared in the context of possible explanations of the +CDF AF B anomaly (109–111), e.g. see the model built in Ref. (136), but the idea stands +out on itself even without anomalies in top quark physics. Indeed if one considers that the +complexity of the SM may be replicated in the sector of dark matter it is natural to consider +multiple species of dark matter, that are “flavors” of dark matter (137–139). These flavors +can be separated from our own SM flavors or can be related to our species of fermions. In +case some relation exists between flavors of the SM and of the dark sector the possibility +that the top quark flavored dark matter is the lightest state is at least as probable as any +other flavor assumption. For example, when Minimal Flavor Violation is advocated one can +explicitly write a mass term for the dark-flavor fermion multiplet χ which in general has +the form +¯χ (m0 + Υ(Y Y )) χ , +where Υ is a function of combinations of the Yukawa matrices of the SM that form singlets +under the flavor group that is dominated by the piece proportional to Y † +u Yu, hence the top +quark flavor tends to be special just from the principle of MFV itself. In a concrete case +we can have interactions of SM fermions u(i) +R +and mass terms for the dark matter flavor +multiplet χ +φ¯χ +� +g0 + g1Y † +u Yu +� +u(i) +R + h.c. + ¯χ +� +m0 + m1Y † +u Yu + ... +� +χ , +6. +where φ is a suitable representation of GSM ⊗ GF . +In Ref. (136) for instance φ ∼ +(3, 1, 2/3)SM ⊗ (1, 1, 1)F , χ ∼ (1, 1, 0)SM ⊗ (1, 3, 1)F and the Yukawa matrices, as in +general in MFV, transform as spurions Yu ∼ (3, ¯3, 1)F and Yd ∼ (3, 1, ¯3)F . We see that +it is possible to pick m1 as to partly cancel the flavor universal m0 term, making χt the +www.annualreviews.org • +15 + +lightest particle of the χ multiplet while retaining full freedom to pick the combinations of +g0 and g1 that corresponds to the couplings of the mass eigenstates χi. +In absence of a field φ one can imagine contact operators to couple the Dark Matter +and the SM flavors i and j, e.g. operators of the type +(¯χΓSχ) +� +¯ψ(i)ΓSψ(j)� +7. +for some Lorentz structure ΓS have been considered as low energy remnants of flavored +gauge bosons (137) or other heavy scalar and fermion states charged under a MFV-broken +flavor symmetry or in a horizontal symmetry model (138). Operators involving the SM +Higgs boson, e.g. +� ¯Qχ +� +(χ∗Hu) +have also been considered in (133) for a scalar χ ∼ (1, 1, 0)SM ⊗ (3, 1, 1)F . A variation of +the model of Ref. (137) could lead to top quark flavor being singled out, the other referred +works already consider the third generation, hence the top quark and/or the bottom quark, +as special due to either the MFV structure or as a result of the horizontal symmetry. +The phenomenology of top flavored dark matter is very rich as it comprises both possible +signals in dark matter searches and in precision flavor observables as well as in high energy +collider searches. Flavor observables put in general stringent bounds on flavored dark matter +models, the case of top-flavored dark matter being significantly less constrained due to +majority of data belonging to u, d, s, c, b quark systems. Dark matter direct detection is also +in general suppressed because nucleons involved in dark matter scattering do not contain +top flavor, hence the interactions are usually originated at loop level or via breaking of the +flavor alignments, i.e. the dark matter interacts almost exclusively with top quark flavor, +but it may have a small, though not completely negligible coupling to light flavors. The +existence of such coupling depends on the model. A specific analysis for a case in which only +top quark flavor interacts with the DM in the model eq.(6) is presented in Ref. (140) for both +dark matter direct detection and collider prospects in a MFV scenario. The annihilation +rate for the thermal freeze-out is set by the scattering +χχ → tt +8. +mediated by a mediator φ (other scatterings are discussed in detail for instance in (141)). +In this specific case the direct detection scattering on nucleons +χN → χN +is mediated by a loop induced couplings of Z, γ to χ from a bubble loop of t and φ from +eq.(6). +Despite the smallness of these couplings the reach of current and future large +exposure experiments, e.g. +see (142), could probe such low level of scattering rates for +exposure around 1 ton year, that means the model can be tested with presently available +data (10). +A more recent analysis (143) considered flavor, direct dark matter detection and collider +searches for a model featuring a top-flavored dark matter χ and a new state φ. In this work +a “Dark Minimal Flavor Violation” flavor structure that extends MFV, but can recover it as +a limit, is considered and allows for a more generic structure in flavor space for the vertex +λij ¯u(i) +R φχ + h.c. . +16 + +In this context it is possible to delay the observation of χ in direct detection experiments, +as new contributions to the direct detection rate appear compared to the MFV case and +it is possible to arrange for cancellations among scattering amplitudes. It remains an open +questions if it is going to be possible to claim an observation in spite of the so-called +“neutrino fog” that future Xenon experiments (142) face when probing rates so small that +neutrinos from the Sun, supernovae and other natural sources are expected to contribute +an event rate comparable or larger than that of the dark matter. +In principle it is possible to have mχ < mt so that the thermal freeze-out is controlled by +other processes than the simple tree-level exchange of eq.(8). Reference (143) experimented +with this possibility in Dark Minimal Flavor Violations, but it appears in tension with the +direct detection experiments. This conclusion concurs with what can be extrapolated from +the earlier MFV analysis of (136). +The search for models with mediators, that are colored in all models considered so far, +can be carried out very effectively at hadron colliders searching for signals +pp → φφ → tχtχ , +that very much resemble the search for supersymmetric top partners. Depending on the +model there can be more general combinations of flavors of quarks +pp → φφ → qjχqiχ . +Therefore it is in general useful to consider the whole list of squark searches to put bounds on +this type of models. References (143,144) reports bounds in the TeV ballpark which inherit +the strengths and weaknesses discussed for the search of supersymmetric quark partners. +Other possible signals at hadron collider are the +pp → tχχ +scattering, which can arise from interactions such as eq.(7), studied in (138), or associated +production φχ, followed by φ → tχ studied for instance in (144). +It is also possible to consider models that go beyond what we have considered here +starting from the notable feature that MFV and some extensions may render the DM +stable. In a model of such “top-philic” dark matter model on can have (145) scalars that +couple to tχ as well as to light quark bilinears, e.g. from RPV supersymmetry, so that they +mediate scatterings of the type +qi¯qj → Sij → tχ . +Other potentially interesting signals possible flavored gauge bosons with couplings ρijqiqj +can appear, replacing Sij with ρij in the above process. Further signals in this type of +models arise, e.g. +qig → tρti +possibly followed by ρ → χt, and similarly for S. A model with a flavored gauge boson +has been studied in (146) with the goal of pinning down the flavor of light quark that +interacts with the top quark and the dark matter leveraging charm-tagging and lepton +charge asymmetry at the LHC. +Though many general issues follow the same path for scalar and fermionic dark matter +it is worth mentioning that references (147, 148) contain a full study of the case in which +the partner and the dark matter are a fermion and a scalar, respectively, at the converse of +most of what we discussed above. Further studies of top and dark matter related matters +can be found in the context of simplified models building (141,149,150). +www.annualreviews.org • +17 + +5. Conclusions +The connection between new physics and the top quark sector is well established and has +lead to a large amount of model building and phenomenological studies. +Here we have +presented supersymmetric top partners, motivated by supersymmetry as the symmetry that +stabilizes the weak scale, and top partners states motivated by the possible compositeness +and pseudo-Nambu-Goldstone boson nature of the Higgs boson. +The phenomenological +relevance of these incarnations of “BSM in the top quark sector” is tightly tied to the +motivations of the models to which the top partners states belong. +As the models in +question are themselves in a “critical” phase at the moment, so is the situation for this +type of new physics in the top quark sector. We say this in the sense that on one hand +we have reached a point at which the expectation was to have already discovered signs +of new physics, especially in the top quark sector in the mass range explored by current +experiments, hence we should start to dismiss these ideas, while on the other hand we are +still largely convinced of the validity of the arguments that lead to the formulation of these +models. Furthermore no serious alternatives have appeared in the model building landscape +and we still have plenty of evidence for the existence of physics beyond the Standard Model. +Thus one can be lead to reconsider if the entire motivational construction for these models +was somewhat wrong or at least biased towards a “close-by” and experimentally friendly +solution. +The way out of this crisis, in absence of experimental results changing the situation, +is for everyone to decide. A possibility is to conclude that we need to update our beliefs +about “where” (151) new physics can appear in the top quark sector and more in general +in going beyond the SM. In this sense top partner searches are a gauge of our progress on +testing well established ideas on new physics. +It should be remarked that the top quark sector remains central also in the formulation +of new physics models that try alternatives to the more well established ideas, see e.g. +Refs. (152,153) on possible ways the top quark can lead the way to construct new physics +models of a somewhat different kind that the two mainstream ideas discussed here. +Given the absence of clear signs and directions in model building into which entrust our +hopes for new physics we have discussed the power of general effective field theory analyses +that can be used to search for new physics in precise SM measurements. These tools have +become the weapon of choice in a post-LHC epoch for the so-called model-independent +search of new physics. We have presented the power of current LHC and future HL-LHC +analyses to see deviations from the SM due to top quark interactions. Overall the LHC +has a chance to see deviation in some more friendly observables for a new physics scale in +the TeV range. In order to secure this result and avoid possible blind-spots a new particle +accelerator is needed, a most popular option being an e+e− capable of operating at or above +the t¯t threshold with the luminosity to produce around 106 top quark pairs. +Other great mysteries beyond the origin of the electroweak scale remain unsolved in +the Standard Model. We have looked at possible solutions of the flavor puzzle in which +the top quark flavor plays a special role. The phenomenology of models with lowest lying +new physics states charged under top flavor has some similarity with that of top quark +partners at colliders, but there is also the possibility to generate observable flavor violations +as further distinctive experimental signatures. +We have examined the possibility that the top quark may be a key to solve the mystery +of dark matter of the Universe. We have presented scenarios in which the dark matter +interacts predominantly or exclusively with the top quark flavor, possibly ascribing the +18 + +stability of the dark matter to the same flavor structure that makes the top quark flavor +special among the SM flavors. Such possibility appears very well motivated as a way to +reduce otherwise intolerably large couplings of dark matter with lighter generations and +explain the stability of dark matter. The flavor dependence of the couplings has motivated +efforts to build models for the realization of this idea in a coherent, though maybe still +effective, theory of favor of which we have presented a few instances. We remarked how +in these scenarios the dark matter phenomenology is quite different from other types of +thermal dark matter and we have summarized dedicated analyses that have been carried +out to identify the relevant bounds and constraints. The upshot is that idea can be broadly +tested with current and future direct detection dark matter experiments. At the same time +the new states associated with the dark matter may be observed on-shell at colliders, which +can in principle also probe contact interactions that originate from off-shell states associated +with the dark matter. 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Kim, Running top yukawa for the naturalness, GGI Workshop "Beyond Standard +Model: Where do we go from here?" . +www.annualreviews.org • +25 + diff --git a/3tE3T4oBgHgl3EQfPwn9/content/tmp_files/load_file.txt b/3tE3T4oBgHgl3EQfPwn9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c7fe90ebe2dc6a011805ac842d64f820cd638d7 --- /dev/null +++ b/3tE3T4oBgHgl3EQfPwn9/content/tmp_files/load_file.txt @@ -0,0 +1,1989 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf,len=1988 +page_content='Beyond-Standard-Model Physics Associated with the Top Quark Roberto Franceschini Università degli Studi and INFN Roma Tre, Via della Vasca Navale 84, I-00146, Roma email: roberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='franceschini@uniroma3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='it xxxxxx 0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 00:1–25 Copyright © 0000 by Annual Reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' All rights reserved Keywords top quark, beyond the standard model, hierarchy problem, flavor, dark matter, new physics Abstract We review scenarios of physics beyond the Standard Model in which the top quark plays a special role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Models that aim at the stabilization of the weak scale are presented together with the specific phenomenology of partner states that are characteristic of this type of model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Fur- ther, we present models of flavor in which the top quark is singled out as a special flavor among the SM ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The flavor and collider phe- nomenology of these models is broadly presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Finally, we discuss the possibility that dark matter interacts preferably with the top quark flavor and broadly present the dark matter phenomenology of these scenarios, as well as collider and flavor signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='04407v1 [hep-ph] 11 Jan 2023 Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Introduction The top quark is a singular object amidst the fermions of the Standard Model as it is the heaviest among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This entails several peculiar properties: in the domain of QCD it stands out as it is the only quark never to be observed into a hadron, as its decay is much faster than the hadron formation time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' after the discovery of the Higgs boson, and for all the time before in which the Higgs mechanism has dominated the landscape of model building for the electroweak sector of the SM, the top quark stands out as the only one with a “normal” size coupling with the Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This latter property has made the top quark very interesting both for the question about the origin of the structure of flavor in the SM and for the origin of the electroweak scale itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The special interest about top flavor has to do with its strong preference to decay into bottom quarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='e not involving other flavor families, which in the CKM picture results in Vtb = 1 up to small corrections, and its large mass, which can possibly act as a magnifier of the effects of physics beyond the Higgs boson as origin of flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For electroweak physics the top quark plays a crucial role in that it affects the properties of the Higgs boson, and by the Higgs mechanism for weak bosons mass generation, also in the physics of weak gauge bosons: its effect can be seen in their masses and decay rates, which are sensitive to the strength of the top quark gauge and Yukawa couplings and to its mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Deviations of these properties from the SM predictions can be signs of new physics related to the top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' While the importance of the top quark can be appreciated already from these general facts, the detailed role played by the top quark can be better understood going closer to explicit new physics models, which will pace the exposition of the greatest part of the following material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 we discuss models in which the top quark plays a special role for the origin of the electroweak symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The discussion is further extended in section 3 in a more model independent direction using a flavor-conserving effective field theory of the top quark sector, which also allow to discuss prospects for top quark physics at future colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1 we attack a different problem, that of the origin of the flavors of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 we extend the discussion to the possibility that SM flavor plays a part in the stabilization of the dark matter in a way that makes the dark matter interact preferably with the top quark flavor and discuss the phenomenology of dark matter in these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Finally in section 5 we offer some conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Being the subjects list rather large, the discussion is necessarily kept free from some details, which are available in the provided references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This review is conceived so that it 2 can also be useful for younger graduate students seeking an high-level introduction to the subject(s) discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Hopefully the readers can start here their own exploration on topics that would otherwise require to go through a large stack of literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' References are kept to a minimum of key works as to encourage the reader to actually study these selected works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Top quark and BSM related to the Higgs boson and the origin of the weak scale 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Supersymmetry Supersymmetry has been proposed as a space-time symmetry involving fermionic genera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Unlike in gauge symmetries, this makes possible to involve spin and momentum in the definition of the symmetry algebra, which, up to violations of the symmetry itself, would require interactions and masses of bosonic and fermionic particles to be tightly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' One such relation would require the electron to be accompanied by exactly mass degener- ate states of spin-0, pretty much the same as Lorentz symmetry of space-time built-in the Dirac equation implies the existence of exactly mass degenerate anti-particles of the elec- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The absence of any evidence in experiments for spin-0 electron-like state motivates to consider supersymmetry as an approximate symmetry, broken at some unknown scale so that all the supersymmetric partners of the SM states are pushed beyond the mass scale presently probed by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The mechanism for supersymmetry breaking is a subject for model building, which is outside of the scope of this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For our purpose it is key to recall that the supersym- metry breaking top quark sector has the rather model-independent tendency to determine the Higgs bosons mass and quartic coupling, thus leading to the identification of the su- persymmetric top scalar quark, most often called “stop squark”, as the main player setting the Higgs boson potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In the Minimal Supersymmetric Standard Model (see (1) for an extensive review) this is represented by equations for the constraints on the minimization of the Higgs boson potential m2 Z = ��m2 Hd − m2 Hu �� � 1 − sin2(2β) − m2 Hu − m2 Hd − 2 |µ|2 , sin(2β) = 2b m2 Hu + m2 Hd + 2 |µ|2 , coupled with the 1-loop effect of the top quark and top squark on the bilinear terms of the 2 Higgs doublets Hu and Hd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In particular, for the Higgs doublet Hu that interacts with up-type quarks, hence feels the top quark sector, the RGE equations is d d log Qm2 Hu = 3Xt − 6g2 |M2|2 − 6 5g2 1 |M1|2 + 3 5g2 1S , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' where Xt = 2 |yt|2 � m2 Hu + m2 Q3 + m2 ¯u3 � + 2 |at|2, M1,2 are the U(1) and SU(2) gaugino mass terms, and S = Tr[Yjm2 φj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' These equations naturally lead to possibility that the supersymmetry breaking stop masses m2 Q3 and m2 ¯u3 or a large A-term |at| might induce a large Xt, which in turn drives m2 Hu < 0 as log Q diminishes from some high-scale down to the weak scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This possibility has made the role of stop squarks a very central one in supersymmetric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In essence, www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='annualreviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='org • 3 the supersymmetric partner of the top quark is responsible for breaking the electroweak symmetry, by making m2 Hu < 0 hence making the Higgs boson potential unstable at the origin of the Hd, Hu fields space, and setting the value of the masses that set the weak scale, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' mZ from the above equation or the mass of the Higgs boson that receives the above mentioned large radiative corrections from the stop squark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As a matter of fact, once the Higgs boson was discovered and its mass was known, a number of works tried to determine the impact of this measurement on the properties of the stop squark (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (2) for the MSSM and some extensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In turn, the necessity for peculiar supersymmetry breaking to accommodate the Higgs mass has spurred investigations on the possible supersymmetry breaking models that can lead to such peculiar stop squarks (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (3–8) for some examples of supersymmetry breaking models emerged or re-emerged to address the null searches of supersymmetry and the Higgs discovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The phenomenology of the supersymmetric partners of the top quark is largely dictated by one feature of the supersymmetric models: the existence of a conserved quantum number that distinguishes SM states from their supersymmetric partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The standard choice for such quantity is called R-parity, a Z2 symmetry under which all SM states are even and all partners states are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The conservation of this symmetry implies that partners states can appear in interaction vertexes only in even number, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' one SM states can interact with two supersymmetric states and it is not possible for a single supersymmetric state to interact with a pair of SM states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For particle colliders this implies that the lowest order process to produce supersymmetric states in collisions is SM SM → SUSY SUSY, and the decay of supersymmetric particles to any number of SM states is forbidden unless there is at least one supersymmetric particle (or an odd number of them), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' SUSY → SUSY SM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' When R-parity is exact a most copious production mechanism for stop squarks at the LHC is gg → ˜ti˜t∗ j, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' where we denoted ˜tk for k = 1, 2 the two stop squarks mass eigenstates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Other production mechanisms are possible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' in decays of supersymmetric partners heavier than the stops or via production of stops in association with other supersymmetric states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Once produced, the stop squark can decay in a number of possible channels, depending on which supersymmetric states are lighter than the state ˜tk at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Most studied 2-body decay modes are ˜t → tχ0, ˜t → bχ+ , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' which feature fermions χ that are mixtures of supersymmetric partners of gauge bosons of the electroweak interactions and of the Higgs bosons of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The motivation for the 1The definition of mass eigenstate as “stops” assumes that flavor labels we give in the SM are the same for the partners states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' It must be stressed that the fate of flavor in the supersymmetric partners sector is largely model dependent and it is possible to use flavor mixing to change the phenomenology of stop squarks, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' See (1) for more details on the gauge and flavor structure of the squark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 4 prevalence of these decay modes is that, by the rules of unbroken supersymmetry, these decays are mediated by couplings given by gauge and Yukawa couplings of the SM, hence they are pretty much impossible to switch off unless m˜t − mχ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As a matter of fact the quantity m˜t − mχ plays a major role in determining the stop phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' When m˜t − mχ → 0 it becomes necessary to consider multi-body processes are also possible and may be phenomenologically relevant, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' ˜t → bW +χ0, ˜t → b ¯ff ′χ0, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' as well as possible flavor violating decays that may be induced at loop level, such as ˜t → cχ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In the above discussion the particle χ0 is considered as the lightest supersymmetric state (LSP), so that, by the conservation of R-parity, it is absolutely stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As χ0 is not elec- trically charged and it is color neutral, pretty much like neutrinos it does not leave directly observables traces in detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For this reason the presence of χ0 can be detected only as momentum missing in the overall momentum conservation in each collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As we cannot reliably measure the fractions of the longitudinal momentum of the colliding protons taken by the partons initiating the production of stops, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' the gluons entering in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (2), and the fraction taken by the rest of the partons, the longitudinal momentum conservation is usually not exploited in hadron colliders, therefore the presence of χ0 is usually sought for as missing transverse momentum, most often (mis)named missing transverse energy mET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Being an electrically neutral stable particle charged only under supersymmetric Yukawa and electroweak gauge interactions, χ0 qualifies as perfect candidate for a WIMP Dark Mat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The possibility to have a Dark Matter candidate stemming out of supersymmetry has given formidable motivation to pursue this scenario for the past decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' So much so, that missing transverse energy searches have becomes synonymous of searches for supersymme- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' It must be said, however, that the null searches of supersymmetric particles, as well as WIMP Dark Matter in the mass range suitable for χ0(10), has put this idea under great pressure lately (11,12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Given these experimental results, and the vast range of possible models for supersym- metry breaking, it must be recalled that in general it is possible to have other states than χ0 as lightest supersymmetric particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For instance the supersymmetric partner of a neutrino or even top sector squarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The latter leads to peculiar phenomena due to the formation of hadrons containing supersymmetric states(13)(14), but these models typically suffer from quite stringent limits (15–17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Therefore the majority of the searches for supersymmetric states in the top quark sector are carried out in the χ0 LSP setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Wholly alternative phenomenological scenarios for supersymmetric top quark partners are possible and are actively pursued in experimental searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The main possible alter- native has to do with the non-conservation of R-parity (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' With broken R-parity all supersymmetric particles can in principle be produced singly and can decay into just SM states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' SM SM → SUSY and SUSY → SM SM , are now possible processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In this situation there is no longer an absolutely stable weak scale particle to purse the idea of Dark Matter as a WIMP2 and the phenomenology of 2Alternative DM candidates can be found in these models, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (19) for a possible gravitino dark matter scenarios and issues related to this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='annualreviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='org • 5 supersymmetric states linked to the top quark is now greatly different from the picture given above (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For instance R-parity violating couplings, still respecting the full gauge symmetry of the SM, allow, among other possibilities, the decays ˜t → bs or ˜t → ℓd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As the final states of stop decay can now be made entirely of SM particles it is possible to detect stop squarks as resonances, a very powerful signature, that is not possible to pursue when χ0 is forced to appear among the decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Furthermore these decays, being mediated by R-parity breaking couplings, that need to be small for a number of constraints (18), can lead to meta-stable supersymmetric states, which can live measurable lengths in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Experimental searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In a detailed model it is possible to derive very specific signals from top sector supersymmetric partners, including both signatures at collider experiments and as well as low energy precision ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The latter, however, turn out to be usually very much dependent on the model considered for low energy precision experiments (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A similar issue exists with early universe physics, on top of the signals being quite difficult to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For this reason collider searches are the prime way to search for top sector supersymmetric partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Before listing relevant searches it is necessary to clarify a point on their scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The above searches are sensitive in principle to any sign of new physics related to the top quark sector involving mET or some kind of pair produced resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Although the search is optimized for supersymmetric partners, it can indeed be used to set bounds on other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The interested reader can refer for instance to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (22) for an interpretation of the “supersymmetry searches” in the context of fermionic top partners to be discussed in later Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The searches for top sector supersymmetric partners can be divided into two main categories: searches in large momentum transfer signals, which feature detector objects (jets, leptons, photons, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=') with energy and transverse momentum greater than the typical SM events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' searches in low momentum transfer signal, in which the detector objects arising from top sector supersymmetric partners production are not very different from that of typical SM events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The large momentum transfer ones are “classic” searches for new physics, and were envisaged already at the time of design of the experiments (23,24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Currently these searches can probe supersymmetric top partners up to a mass around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 TeV, although not in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Indeed it is quite hard to probe in full generality even a model as “minimal” as one having the full freedom to vary the branching ratios of decays eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='(3)-(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For a complete assessment is then necessary to test very accurately a large number of searches at once, often relying on a “phenomenological” incarnation of a sufficiently general supersymmetric model, as studied for instance in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The interpretation of these results is quite difficult, as many constraints on the model are imposed at once, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' the top partners states are required to “fix” the mass of the SM Higgs boson to its measured value by the dynamics of radiative corrections embodied in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This requirement, while being a sensible one in the context of the specific model, can significantly alter the conclusion of that study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Therefore it 6 remains difficult to answer questions as simple as finding the lightest not excluded values of the mass of stop-like top partners 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Further difficulties can arise and make nearly impossible to probe experimentally su- persymmetric top partners, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g when special kinematical configurations become the typical configuration of top partners decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In these cases the search in low momentum transfer signatures can help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Indeed, these searches have been developed to overcome the difficulty that arise in the limit m˜t − mχ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The shortcomings of the large momentum transfer searches can be clearly seen in Figure 1, as the excluded stop mass for large m˜t−mχ is much larger than for small values of this mass difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In addition, when the stop-LSP mass gap is small and the stop becomes lighter, its production and decay cannot be reliably distinguished from other SM processes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' the SM top quark production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This observa- tion motivates a zoom inset in the figure to display how these peculiar cases are covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The most useful strategies to attack these difficult signatures have turned out to be the studies of angular observables and fiducial rates of top-like final states (27–31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Especially in angular observables there are modest, but persistent disagreement between the measure- ments in the top quark sample (32) and theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' These disagreement are also accompanied by other disagreements of small entity, but persisting from Run1 LHC through Run2, in the kinematics of the reconstructed top quarks e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (33,34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The possibility to see effects of BSM related to the top quark and the precision in measurements afforded by the LHC and the HL-LHC has motivated the great improvement of predictions for top quark SM observables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (35) for a seamless description of fixed NLO and PS cal- culations of top quark resonant and non-resonant rates, (36,37) for specific NNLO and EW corrections to the BSM sensitive rates and more in general drawing attention on possibly BSM-sensitive high energy top quarks (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (38)) and other production modes which may be of interest for both SM studies and BSM searches (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (39,40)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The searches mentioned above, though motivated and sometimes optimized on super- symmetry searches, are rather general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Thus it is important to stress that the observation of an excess in one of these “supersymmetry searches” would not at all prove the supersym- metric nature of the discovered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A reliable statement on the supersymmetric nature of the newly discovered object would require several measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For some proposal at the LHC the interested reader can look for instance at (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In general it is believed that a machine cleaner than a hadron collider, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' an e+e− collider, capable of producing the new particle would be needed to truly confer it the status of “supersymmetric partner” state of some SM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' At the time of writing there are no statistically significant and convincing signs of new physics in searches for new physics, the searches for supersymmetric top partners being no exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Despite the absence of signals for top sector supersymmetric partners these are still believed to one our best chances to find new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Looking at the glass as “half full” one could even argue that in the minimal model of supersymmetry the relatively large observed Higgs boson mass requires large loop level corrections from contributions of the kind of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' These large loop corrections point towards a stop squarks mass scale at the TeV or larger, thus perfectly compatible with the present limits and possibly awaiting us for a next discovery at one of the next updates of the searches as more data is collected at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 3One possible answer in the context of (25) is offered in the supplementary material of that analysis(26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='annualreviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content="org • 7 Observed limits Expected limits 1 = 13 TeV, 139 fb s Data 15-18, 0 1 χ∼ bff' → 1t~ monojet, [2102." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content="10874] 0 1 χ∼ bff' → 1t~ / 0 1 χ∼ bW → 1t~ / 0 1 χ∼ t → 1t~ 0L, [2004." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content="14060] 0 1 χ∼ bff' → 1t~ / 0 1 χ∼ bW → 1t~ / 0 1 χ∼ t → 1t~ 1L, [2012." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content="03799] 0 1 χ∼ bff' → 1t~ / 0 1 χ∼ bW → 1t~ / 0 1 χ∼ t → 1t~ 2L, [2102." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='01444] 1 = 13 TeV, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content="1 fb s Data 15-16, 0 1 χ∼ bff' → 1t~ / 0 1 χ∼ bW → 1t~ / 0 1 χ∼ t → 1t~ [1709." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='04183, 1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='11520, 1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='03247, 1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='03301] 0 1 χ∼ t → 1t~ , tt [1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='07570] 1 = 8 TeV, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content="3 fb s Data 12, 0 1 χ∼ bff' → 1t~ / 0 1 χ∼ bW → 1t~ / 0 1 χ∼ t → 1t~ [1506." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='08616] 200 400 600 800 1000 1200 ) [GeV] 1 t~ m( 100 200 300 400 500 600 700 800 900 ) [GeV] 0 1 χ∼ m( 1 = 8,13 TeV, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='3-139 fb s March 2021 ATLAS Preliminary production 1t~ 1t~ Limits at 95% CL 180 200 220 0 10 20 30 40 50 60 70 ) = 0 0 1 χ∼ , 1t~ m( ∆ W + m b ) = m 0 1 χ∼ , 1t~ m( ∆ t ) = m 0 1 χ∼ , 1t~ m( ∆ Figure 1 Searches for top sector supersymmetric partners in the Stop-LSP mass plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As the mass scale of top quark supersymmetric partners is not entirely fixed it often considered that these particles may be too heavy for the LHC to discover them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Therefore the discovery reach for these particles is often considered in the evaluation of the physics case of future particle accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Projections for a 100 TeV pp collider (42, 43) usually cover a mass range 5-8 times larger than what can be probed at the LHC, while the expectation for a high energy lepton collider, such as multi-TeV muon collider(44–48), is to probe the existence of top partners up to the kinematic limits at √s/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Composite and pNGB/Little Higgs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' New physics associated to the top quark sector has been motivated also from a series of model building activities aimed at explaining the origin of the electroweak scale through the Goldstone boson nature of the agent of its breaking, resulting in theories of the Higgs boson as a pseudo Nambu-Goldstone boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' From a low energy effective point of view these theories can be put in the language of a composite Higgs boson, whose lightness compared to its scale of compositeness is justified by its goldstonian nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Models built in this family are reviewed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (49–52) and they all share the need to enlarge the symmetries of the SM by a new global symmetry, that is broken at some scale above the TeV to a smaller symmetry, with the associated Nambu-Goldstone bosons, which will host the yet smaller symmetry group of the SM at even lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The minimal model of this type (53) that is able to pass bounds from electroweak precision tests including Zb¯b 8 couplings assumes an SO(5) global symmetry, broken to SO(4) ≃ SU(2) × SU(2) which contain the weak interactions gauged SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The enlargement of the symmetry of the SM motivates appearance of matter repre- sentations in multiplets that are necessarily larger than the usual doublets and singlets of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In particular, in order to obtain Yukawa interactions the constructions of pNGB and little Higgs model converges in the existence of “partner” states for the top quark, the bottom quark and in principle for all the fermions of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The precise phenomenological manifestation of the “partner” states is highly model dependent, as it depends on the choice the new global symmetry group that one has in building this type of models, the repre- sentation of this symmetry group that one chooses for the new matter and the imagined mechanism to originate the SM fermion masses at the most microscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' One possible limitation to the model building choices may comes from the requirement of not introducing large deviations in well known couplings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' the Zbb couplings (54), still a large set of possibilities exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For this review we focus on a unifying feature of many models, that is the presence of “partner” states directly connected to the SM top quark sector via Yukawa and gauge interactions with relatively universal decay patterns (55–57), although other decay modes and more “exotic” partners may exist including possible couplings to scalar states accompanying the Higgs boson in some models (58–60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' At the core of the experimental tests of the idea of fermion top part- ners lies the assumption that the main interaction leading to the decay of these top partners into SM states is the Yukawa of the top quark, in which the Higgs boson or longitudinal components of the weak gauge bosons appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For this reason the large majority of the searches are presented in terms of exclusions for branching fractions of the top partners states into the following pairs of SM states T → tZ, th, Wb, where T is a charge 2/3 top partner and B → bZ, bh, Wt, where B is a charge -1/3 partner of the bottom quark, whose existence is consequence of the SU(2) weak isospin symmetry that must hold in the theory that supersedes the SM at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In models with a symmetry larger than SU(2), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (54)(53), it is typical to have further partners states that appear as necessary to furnish full representations of the larger symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A much studied case is the state of charge 5/3 that leads to a very characteristic decay X5/3 → W +t , which in turn gives a characteristic same-sign di-lepton signal (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For little Higgs models the appearance of this type of exotic partners requires the formulation of somewhat more involved models, but it is definitively a possibility(58,62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Experimental searches at colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Experimental searches for new states are carried out at the LHC exploiting the color charge of the top partners in processes such as gg → TT , www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='annualreviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='org • 9 that are analogous to previous processes for supersymmetric partners and depend only on the QCD charge of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Unlike for supersymmetric partners, for which the conservation of R-parity plays a crucial role, the single production of top partners gq → q′Tb , is possible in the most minimal models and can in principle lead to a deeper understanding of the BSM physics, as this process involves directly new physics couplings for the production of the top partners state (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For instance the rate of single production of top partners states can be a discriminant with respect to so-called “vector-like” quarks, whose couplings are not dictated by Goldstone property of the Higgs (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (64) for a more in-depth discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A great difference in the search for the partners discussed in this section is that they can in principle give rise to resonant signals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' in the invariant mass of an hadronic top and one hadronic Higgs boson in the decay T → th and other signals discussed for instance in the search of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Another consequence of the top partner decaying in purely SM final states is that even the “heavy” SM particles, such as t, Z, W, h, are produced with significant boost in the majority of the events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This motivates the use of special experimental techniques for the identification of those detector objects (66) as for instance in the search of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The search strategies mentioned above are combined by the experimental collaborations, that present results in a plane with axes spanning the possible values of two decays, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' if figure 2 an example is shown for T → Ht and T → Wb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The underlying assumption of this presentation of the results is that the top partner does not decay in any BSM states, hence the branching ratio of T → Zt is determined by the two branching rations displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The right panel of the same figure shows how the different searches have different sensitivity to each decay mode and can be patched together to better exclude top partners of a given mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For more exotic signals from X5/3 searches are carried out as well, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Results of searches at LHC collected in figure 2 and newer results (67, 69) on the kinds of top partners described so far put bounds on the top partners mass at around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As mentioned above it is possible to have larger groups and larger representations in the symmetry breaking pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For instance if the large global symmetry of which breaking the Higgs is a pNGB is chosen to be SO(6) broken to SO(5) and top quark partners states are chosen to furnish a 6-dimensional representation there is one extra top partners state compared to the case of top quark partners in the 5 of SO(5) considered for the minimal model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' If we call this new top partner state Ψ1, the name signals the fact that it is a singlet under the remnant SO(5) symmetry, we can have new signals from its production via QCD interactions and decay that do not fit into any of the previously considered categories e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Ψ1 → th, tZ, tη, Wb , where η is an extra pNGB that arises due to the larger number of broken generators in the breaking SO(6) → SO(5) → SO(4) ≃ SU(2) × SU(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In general the extensions of pNGB models can include possible FCNC of top quarks with new physics states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (73) has considered decays of the SM top quark that violate flavor t → cη as a consequence of underlying flavor-changing dynamics in the top partners by a coupling Tcη which would also yield a new possible search channel for a top partner T → cη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Other 10 Figure 2 Searches for top fermionic partners (70,71) in the plane BR(T → Ht) vs BR(T → Wb) with the constraint B(bW) + B(th) + B(tZ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For reference, some model-dependent choices of the branching ratios introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (72) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' exotic possibilities are covered in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' T → tg, tγ, X5/3 → tφ+ and more exotics ones are presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (74–76) and can in principle lead to new signals for top quark partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' EFT at current and future colliders The previous sections dealt with explicit models of new physics giving rise to signals from direct production of particles beyond those of the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As these searches have so far yield no evidence of new physics a growing interest and motivation have risen for the description of new physics in Effective Field Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The effective character of these theories is due to the fact that they arise by the removal of heavy states from a theory more microscopic than the SM and they lead to a set of BSM interactions, that is usually in overlap with the set generated by other microscopic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Therefore it has been done a great work in identifying the most general sets of interactions under given assumptions (77,78), so that new physics studies can be carried in a “model-independent” fashion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' searching for very characteristic interactions involving four top quarks (79–82) or other four-fermion operators involving top quarks, or other kinds of contact interactions independently of their microscopic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The plus side of the EFT approach is that it is very comprehensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The converse of this comprehensiveness is the possible loss of contact with the microscopic origin of physics beyond the Standard Model which gives rise to specific patterns and organization principles for the size of each contact interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Thus it is necessary to strike a balance between a fully general EFT and a “physically efficacious” effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Where this balance lies is very much dependent on the amount of data that one can use in constraining the couplings of the effective interactions, as well as the theoretical prejudice on what effects are worth being considered, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' pure top sector effects (78,83–87), or effects involving EW and Higgs physics as well (88,89) or exploring flavor changing effects (90–95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As the effect of BSM contact interactions from the EFT affects precision measurements of SM processes, this enhanced attention towards signals of BSM associated to top quarks www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='annualreviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='org • 11 (↑H 1 m, = 800 Gev m, = 900 GevV ATLAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='8 ATLAS Preliminary 1420 Vs = 13 TeV, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1 fb-1 BR(T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='9 个 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='.Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='exclusion Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='exclusion Vs = 13 TeV, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content="1 fb' limit [ 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' W(lv)b+X [arxiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='03347] 1400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='8 VLQ combination 1400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 H(bb)+X [arxiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='09678] R Z(vv)t+X [ariv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='10751] B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='7 Observed limit nass m = 950 Gev m = 1000 Gev 1375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='8 Trilep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='/same-sign [CERN-EP-2018-171] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='6 Z(I)/b+X [arxiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='10555] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='6 1380 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='4 All-had [CERN-EP-2018-176] ★ SU(2) doublet ★ sU(2) doublet ● sU(2) singlet O sU(2) singlet 1360 95% 1320 m = 1050 Gev m = 1100 GeV m = 1150 Gev 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='8F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='3 1340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 1320 m = 1200 GeV ± m = 1300 GeV m, = 1400 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1 ★ 1300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' BR(T → Wb) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='8 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='8 0 BR(T →> Wb)Figure 3 Summary from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (101,102) of the constraints on contact interactions involving the top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The left panel shows the effect of HL-LHC compared to present constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The right panel shows the effect of future e+e− machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The taller (and lighted) bars for each case represent the looser bounds that are obtained when the coupling of interest is bound while the others are allowed to float, see (101,102) and references therein for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' has produced activity on the improvement of the description of several processes that are either backgrounds or serve as SM reference on top of which search for signs of BSM, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' see recent Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (96) for four-top production, recent ttV results discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (97– 99), tth results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (100) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For an up to date snapshot of the characterization of the top quark electroweak interactions and possible BSM in deviations from the SM we refer the reader to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (83–87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The upshot of the work is that present measurements, also thanks to the availability of differential measurements and trustable computations in the same phase-space regions, can put bounds on generic new physics in the top quark sector in the TeV ballpark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The possibility to identify indirect signs of new physics in signatures related to the top quark has become a commonly used benchmark in the evaluation of performances of future colliders, especially clean e+e− machines, whose best chance to see new physics in the top sector is through indirect effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Works such as (101–104) have studied the outcome of analyses to be carried out at future colliders and the interplay between present and future colliders probes of new physics in top quark effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The results are summa- rized in Figure 3, which shows the significant improvement that will be attained by the HL-LHC, especially on single-couplings effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The figure also shows the strong tightening of the bounds with the addition of data from future e+e− data at the Zh threshold, the t¯t threshold and above, which will make the global EFT constraints particularly robust by the removal of possible flat directions in couplings-space and providing new data in channels that can be probed best at clean e+e− machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Top quark and BSM related to Flavor Dynamics or Dark Matter (or both) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Top quark and BSM related to flavor The top quark flavor remains a special one in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Indeed the top quark is so heavy that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='one can easily single out the third generation of quarks as a peculiars source of breaking of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='FIT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='I LHC Run 2 + Tevatron + LEP ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='CoQ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='Ctz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='Cb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='Ctp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='Ctw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='Cot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='Cpb ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='Operator Coefficientsthe flavor symmetry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='GF = U(3)qL × U(3)uR × U(3)dR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='that the SM would enjoy if all quark masses were zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A hierarchy of breaking dominated by the third generation can be accommodated easily, thanks to the freedom about the possible symmetry breaking patterns and possible mechanisms for breaking the flavor symmetry of the SM that one can consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In addition, this way of organizing the breaking of flavor symmetry is most compatible with experimental bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In fact, bounds on first and second generation flavor changing processes are the most tight, whereas there is a relative lack of constraints on the third generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' If the sole breaking of the symmetry GF arises from the Yukawa couplings of the SM, or new sources are aligned with the Yukawa matrices, the breaking is said to comply with “Minimal Flavor Violation” (MFV) (105–107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In this setting the bounds from flavor observables are most easily accommodated, but it is not the only possibility to comply with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The fact that the top quark Yukawa coupling is a possible large source of flavor symmetry breaking motivates to consider BSM related to the top flavor, but this conclusion holds also in other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A classification of possible states that can couple to quark bilinears charged under the flavor symmetry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' a new scalar coupled as φtutu, has proven useful in the past to assess the possibility of flavorful signs of new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For a recent listing of the possible states one can read tables on Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' From a phenomenological point of view these models give rise to transitions in four-quark scatterings that do not conserve the flavor charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For instance the scattering uu → tt can arise via a t-channel exchange of a flavored boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This can alter the kinematic of top quark production as well as the net charge of the top quark sample at hadron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Indeed new flavorful boson of this kind were advocated in response to TeVatron experiments claiming disagreements between the SM predictions and measured top quark properties, such as the forward-backward asymmetry in the production of top quarks (109–111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In addition these new flavored states coupled to the top quark can give rise to transitions f ¯f → tφtjuj , that can be observed quite easily at e+e− colliders in multi-jet final states, the detailed final state depending on the model-dependent decay of the flavored state φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The possibility that a flavored state connected to the top quark might be among the lightest new states from the new physics sector has appeared also in models of gauged flavor symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In these models the flavor symmetry GF is gauged, as to not have to deal with unobserved massless Goldstone bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For instance Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (112, 113) have proposed a new set of states that would have the notable property to make the GF gauging free from triangular anomalies by the addition of vector-like new quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In this kind of models the new quarks are charged under the SM flavor symmetry and can be arranged as to have top-flavor new states to be the lightest ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Indeed in these models the masses of the the SM quarks would be explained by a see-saw-like mechanism in which the lightest SM fermions are mixed with a very heavy new state, whereas the heaviest SM states are mixed with the lightest of the new physics states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In this case the SM top quark would be the state coupled to the lightest of the new physics states, named t′, possibly accompanied by a partner state for the bottom quark, named b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Remarkably this type of model gives www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='annualreviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='org • 13 Λ �� (���) �������� ����� ������ ��� ������ ������ ����� ��� ������ ������ ����� �������� Δ�� ϵ� �� → μ+μ- ������� ��� �������� ��� μ → � γ �� �� �� ��� � � �� �� Figure 4: Lower bounds on ⇤IR on the various flavor scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The first set of bounds corresponds to our scenario with multiple flavor scales, the second and third sets assume partial compositeness at ⇤IR for the whole third and second family respectively, while the last set gives the bounds for the anarchic flavor scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=" To derive the numerical values we have taken g⇤ ' 3, xt ' xc ' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='5, and set all free ↵L,R parameters to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' where gij ⌘ Ytxt(V † CKM)i3(VCKM)3j , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2) and dLi denotes the left-handed down-type quark component in the i-th family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A remarkable feature of these corrections is the fact that they automatically follow a MFV structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The first operator contributes to �F = 2 transitions and generates correlated e↵ects in the ✏K, �MBd and �MBs observables, which are of the order of the present experimental sensitivity if we take ⇤IR ⇠ TeV and we allow for a slight reduction of the left-handed top compositeness, xt < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The second operator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='1) gives flavor-changing Z-couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' At present it only pushes the ⇤IR scale in the few TeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In the future it can be seen either in deviations in the decays K !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' µµ or B !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (X)``.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This contribution can however be significantly smaller if the strong sector is invariant under a custodial PLR symmetry, which protects the down-type quark couplings to the Z boson [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Additional contributions to �F = 2 operators can also be generated at the scales ⇤c,s,d at which the second and first family quarks get their masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' These corrections however only give a sizable e↵ect on ✏K, that pushes the ⇤IR scale in the multi-TeV range (⇤IR & 6 TeV), which is still a milder bound with respect to the anarchic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' It must however be stressed that these bounds depend on the coe�cients of the e↵ective operators which are a↵ected by some degree of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' These contributions to ✏K severely constrain the maximal dimension of the OH operator, requiring dH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' We also considered possible variations of the framework described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For example, a more economical scenario has been proposed in which each family is associated to a single flavor scale at which the bilinear mass operators are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A few additional new-physics flavor e↵ects 20 Figure 4 Lower bound on the scale of new physics related to the SM fermion mass generation in a composite Higgs scenario (117) under different assumptions on the compositeness of SM fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' phenomenological signatures very similar to those of top partners states of composite and little Higgs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' the partner states can be produced by strong interactions and decay as b′ → bh, bZ, tW and t′ → th, tZ, bW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' These ideas also lend themselves to be paired with supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Although super- symmetry is not necessary for the idea of gauged flavor symmetries in general, these models can provide a setup to originate R-parity breaking with an underlying structure for the flavor structure of the RPV couplings (114,115), that for instance would motivate ˜t → bs as the main channel to search RPV stops (116).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A solution with a hierarchy of flavored new physics scales inverted with respect to that of the SM quarks has been proposed also for composite Higgs models (117–120), which would otherwise suffer from severe bounds from high-pT and flavor observables (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (121–123)), even in presence of some degree of model building (124–126) aimed at keeping all the new physics at a common low-scale and still survive flavor tests thanks to a friendly, possibly MFV-like structure, of the flavor origin in the microscopic completion of the composite Higgs model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As it can be appreciated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 4 the top quark sector emerges still as a less constrained one and further motivates to consider BSM physics related to the top quark, and possibly exclusively to the top quark or to the third generation of SM fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 14 Observables of interests include indirect probes such as electric dipoles moments (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (127)), meson oscillations and decays, and in principle rare Z and Higgs bosons flavor- violating decays which usually receive important contributions from the top quark sec- tor (117).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In addition, it is possible to have phenomena more directly related to the top quark such as t → cV, where V = γ, Z, g(128,129) and deviations from Vtb = 1 in the CKM matrix (64,130–132).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Flavored dark matter models Given the strength of the bounds from direct searches of dark matter scattering on heavy nuclei it has become interesting to consider dark matter models in which the flavor of SM quarks and leptons plays a role, as the strongest bounds hinge on effective couplings of the dark matter to first and, to a slightly lesser extent, to second generation quarks and gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Rather interestingly the flavor puzzle of the SM comes equipped with a symmetry, which, though not exact, can be used to stabilize the dark matter if it is broken according to Minimal Flavor Violation (133,134) and even with more general patterns of flavor symmetry and its breaking (135).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As a dark matter coupling sensitive to flavor could mediate flavor changing transitions the option of the MFV structure, or slight departures from it, has been so far been a main route in model building aimed at removing possible tensions with flavor observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Among the possible flavor structures that the Dark Matter and the SM can fields can be cast in, for our work here we focus on the possibility that the top quark flavor has a special role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Explicit models have appeared in the context of possible explanations of the CDF AF B anomaly (109–111), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' see the model built in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (136), but the idea stands out on itself even without anomalies in top quark physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Indeed if one considers that the complexity of the SM may be replicated in the sector of dark matter it is natural to consider multiple species of dark matter, that are “flavors” of dark matter (137–139).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' These flavors can be separated from our own SM flavors or can be related to our species of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In case some relation exists between flavors of the SM and of the dark sector the possibility that the top quark flavored dark matter is the lightest state is at least as probable as any other flavor assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' For example, when Minimal Flavor Violation is advocated one can explicitly write a mass term for the dark-flavor fermion multiplet χ which in general has the form ¯χ (m0 + Υ(Y Y )) χ , where Υ is a function of combinations of the Yukawa matrices of the SM that form singlets under the flavor group that is dominated by the piece proportional to Y † u Yu, hence the top quark flavor tends to be special just from the principle of MFV itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In a concrete case we can have interactions of SM fermions u(i) R and mass terms for the dark matter flavor multiplet χ φ¯χ � g0 + g1Y † u Yu � u(i) R + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' + ¯χ � m0 + m1Y † u Yu + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' � χ , 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' where φ is a suitable representation of GSM ⊗ GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (136) for instance φ ∼ (3, 1, 2/3)SM ⊗ (1, 1, 1)F , χ ∼ (1, 1, 0)SM ⊗ (1, 3, 1)F and the Yukawa matrices, as in general in MFV, transform as spurions Yu ∼ (3, ¯3, 1)F and Yd ∼ (3, 1, ¯3)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' We see that it is possible to pick m1 as to partly cancel the flavor universal m0 term, making χt the www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='annualreviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='org • 15 lightest particle of the χ multiplet while retaining full freedom to pick the combinations of g0 and g1 that corresponds to the couplings of the mass eigenstates χi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In absence of a field φ one can imagine contact operators to couple the Dark Matter and the SM flavors i and j, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' operators of the type (¯χΓSχ) � ¯ψ(i)ΓSψ(j)� 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' for some Lorentz structure ΓS have been considered as low energy remnants of flavored gauge bosons (137) or other heavy scalar and fermion states charged under a MFV-broken flavor symmetry or in a horizontal symmetry model (138).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Operators involving the SM Higgs boson, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' � ¯Qχ � (χ∗Hu) have also been considered in (133) for a scalar χ ∼ (1, 1, 0)SM ⊗ (3, 1, 1)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A variation of the model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (137) could lead to top quark flavor being singled out, the other referred works already consider the third generation, hence the top quark and/or the bottom quark, as special due to either the MFV structure or as a result of the horizontal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The phenomenology of top flavored dark matter is very rich as it comprises both possible signals in dark matter searches and in precision flavor observables as well as in high energy collider searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Flavor observables put in general stringent bounds on flavored dark matter models, the case of top-flavored dark matter being significantly less constrained due to majority of data belonging to u, d, s, c, b quark systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Dark matter direct detection is also in general suppressed because nucleons involved in dark matter scattering do not contain top flavor, hence the interactions are usually originated at loop level or via breaking of the flavor alignments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' the dark matter interacts almost exclusively with top quark flavor, but it may have a small, though not completely negligible coupling to light flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The existence of such coupling depends on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A specific analysis for a case in which only top quark flavor interacts with the DM in the model eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (6) is presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (140) for both dark matter direct detection and collider prospects in a MFV scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The annihilation rate for the thermal freeze-out is set by the scattering χχ → tt 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' mediated by a mediator φ (other scatterings are discussed in detail for instance in (141)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In this specific case the direct detection scattering on nucleons χN → χN is mediated by a loop induced couplings of Z, γ to χ from a bubble loop of t and φ from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Despite the smallness of these couplings the reach of current and future large exposure experiments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' see (142), could probe such low level of scattering rates for exposure around 1 ton year, that means the model can be tested with presently available data (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A more recent analysis (143) considered flavor, direct dark matter detection and collider searches for a model featuring a top-flavored dark matter χ and a new state φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In this work a “Dark Minimal Flavor Violation” flavor structure that extends MFV, but can recover it as a limit, is considered and allows for a more generic structure in flavor space for the vertex λij ¯u(i) R φχ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' 16 In this context it is possible to delay the observation of χ in direct detection experiments, as new contributions to the direct detection rate appear compared to the MFV case and it is possible to arrange for cancellations among scattering amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' It remains an open questions if it is going to be possible to claim an observation in spite of the so-called “neutrino fog” that future Xenon experiments (142) face when probing rates so small that neutrinos from the Sun, supernovae and other natural sources are expected to contribute an event rate comparable or larger than that of the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In principle it is possible to have mχ < mt so that the thermal freeze-out is controlled by other processes than the simple tree-level exchange of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='(8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Reference (143) experimented with this possibility in Dark Minimal Flavor Violations, but it appears in tension with the direct detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' This conclusion concurs with what can be extrapolated from the earlier MFV analysis of (136).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The search for models with mediators, that are colored in all models considered so far, can be carried out very effectively at hadron colliders searching for signals pp → φφ → tχtχ , that very much resemble the search for supersymmetric top partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Depending on the model there can be more general combinations of flavors of quarks pp → φφ → qjχqiχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Therefore it is in general useful to consider the whole list of squark searches to put bounds on this type of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' References (143,144) reports bounds in the TeV ballpark which inherit the strengths and weaknesses discussed for the search of supersymmetric quark partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Other possible signals at hadron collider are the pp → tχχ scattering, which can arise from interactions such as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (7), studied in (138), or associated production φχ, followed by φ → tχ studied for instance in (144).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' It is also possible to consider models that go beyond what we have considered here starting from the notable feature that MFV and some extensions may render the DM stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In a model of such “top-philic” dark matter model on can have (145) scalars that couple to tχ as well as to light quark bilinears, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' from RPV supersymmetry, so that they mediate scatterings of the type qi¯qj → Sij → tχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Other potentially interesting signals possible flavored gauge bosons with couplings ρijqiqj can appear, replacing Sij with ρij in the above process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Further signals in this type of models arise, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' qig → tρti possibly followed by ρ → χt, and similarly for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A model with a flavored gauge boson has been studied in (146) with the goal of pinning down the flavor of light quark that interacts with the top quark and the dark matter leveraging charm-tagging and lepton charge asymmetry at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Though many general issues follow the same path for scalar and fermionic dark matter it is worth mentioning that references (147, 148) contain a full study of the case in which the partner and the dark matter are a fermion and a scalar, respectively, at the converse of most of what we discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Further studies of top and dark matter related matters can be found in the context of simplified models building (141,149,150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='annualreviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='org • 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Conclusions The connection between new physics and the top quark sector is well established and has lead to a large amount of model building and phenomenological studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Here we have presented supersymmetric top partners, motivated by supersymmetry as the symmetry that stabilizes the weak scale, and top partners states motivated by the possible compositeness and pseudo-Nambu-Goldstone boson nature of the Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The phenomenological relevance of these incarnations of “BSM in the top quark sector” is tightly tied to the motivations of the models to which the top partners states belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' As the models in question are themselves in a “critical” phase at the moment, so is the situation for this type of new physics in the top quark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' We say this in the sense that on one hand we have reached a point at which the expectation was to have already discovered signs of new physics, especially in the top quark sector in the mass range explored by current experiments, hence we should start to dismiss these ideas, while on the other hand we are still largely convinced of the validity of the arguments that lead to the formulation of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Furthermore no serious alternatives have appeared in the model building landscape and we still have plenty of evidence for the existence of physics beyond the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Thus one can be lead to reconsider if the entire motivational construction for these models was somewhat wrong or at least biased towards a “close-by” and experimentally friendly solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The way out of this crisis, in absence of experimental results changing the situation, is for everyone to decide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' A possibility is to conclude that we need to update our beliefs about “where” (151) new physics can appear in the top quark sector and more in general in going beyond the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In this sense top partner searches are a gauge of our progress on testing well established ideas on new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' It should be remarked that the top quark sector remains central also in the formulation of new physics models that try alternatives to the more well established ideas, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' (152,153) on possible ways the top quark can lead the way to construct new physics models of a somewhat different kind that the two mainstream ideas discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Given the absence of clear signs and directions in model building into which entrust our hopes for new physics we have discussed the power of general effective field theory analyses that can be used to search for new physics in precise SM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' These tools have become the weapon of choice in a post-LHC epoch for the so-called model-independent search of new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' We have presented the power of current LHC and future HL-LHC analyses to see deviations from the SM due to top quark interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Overall the LHC has a chance to see deviation in some more friendly observables for a new physics scale in the TeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' In order to secure this result and avoid possible blind-spots a new particle accelerator is needed, a most popular option being an e+e− capable of operating at or above the t¯t threshold with the luminosity to produce around 106 top quark pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Other great mysteries beyond the origin of the electroweak scale remain unsolved in the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' We have looked at possible solutions of the flavor puzzle in which the top quark flavor plays a special role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The phenomenology of models with lowest lying new physics states charged under top flavor has some similarity with that of top quark partners at colliders, but there is also the possibility to generate observable flavor violations as further distinctive experimental signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' We have examined the possibility that the top quark may be a key to solve the mystery of dark matter of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' We have presented scenarios in which the dark matter interacts predominantly or exclusively with the top quark flavor, possibly ascribing the 18 stability of the dark matter to the same flavor structure that makes the top quark flavor special among the SM flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Such possibility appears very well motivated as a way to reduce otherwise intolerably large couplings of dark matter with lighter generations and explain the stability of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The flavor dependence of the couplings has motivated efforts to build models for the realization of this idea in a coherent, though maybe still effective, theory of favor of which we have presented a few instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' We remarked how in these scenarios the dark matter phenomenology is quite different from other types of thermal dark matter and we have summarized dedicated analyses that have been carried out to identify the relevant bounds and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' The upshot is that idea can be broadly tested with current and future direct detection dark matter experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' At the same time the new states associated with the dark matter may be observed on-shell at colliders, which can in principle also probe contact interactions that originate from off-shell states associated with the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} +page_content=' Low energy flavor observables can also help to restrict the range of possible models of flavored dark matter leading to significant constraints both on MFV and non-MFV scenarios when a thermal relic abundance and a significant suppression of spin-dependent and spin-independent direct detection rates are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfPwn9/content/2301.04407v1.pdf'} 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b/DdE0T4oBgHgl3EQfygJ9/content/tmp_files/2301.02660v1.pdf.txt @@ -0,0 +1,1583 @@ +Title: Decreased serum vitamin D level as a prognostic marker in patients with +COVID-19 + +Ruyi Qu 1#, Qiuji Yang 1#, Yingying Bi 2#, Jiajing Cheng 2, Mengna He 3, Xin Wei +4, Yiqi Yuan 5, Yuxin Yang 6* and Jinlong Qin2* + +1 Department of Geriatrics, Shanghai Fourth People's Hospital, School of Medicine, +Tongji University, Shanghai 200434, China +2 Department of Obstetrics and Gynecology, Shanghai Fourth People’s Hospital, +School of Medicine, Tongji University, Shanghai 200434, China +3 Information Department, Shanghai Fourth People's Hospital, School of Medicine, +Tongji University, Shanghai 200434, China +4 Department of Radiology, Shanghai Fourth People's Hospital, School of Medicine, +Tongji University, Shanghai 200434, China +5 Clinical Laboratory, Shanghai Fourth People's Hospital, School of Medicine, Tongji +University, Shanghai 200434, China +6 Department of Obstetrics and Gynecology, Shanghai Fourth People’s Hospital, +School of Life and Sciences and Technology, Tongji University, Shanghai 200434, +China +* Correspondence: 22310050@tongji.edu.cn (Yuxin Yang); 2180129@tongji.edu.cn +(Jinlong Qin) +# These authors contributed equally to this work. + + + + +Abstract +Background: The corona virus disease 2019 (COVID-19) pandemic, which is caused +by severe acute respiratory syndrome coronavirus 2, is still localized outbreak and has +resulted in a high rate of infection and severe disease in older patients with +comorbidities. The vitamin D status of the population has been found to be an important +factor that could influence outcome of COVID-19. However, whether vitamin D can +lessen the symptoms or severity of COVID-19 still remains controversial. +Methods: A total of 719 patients with confirmed COVID-19 were enrolled +retrospectively in this study from April 13 to June 6, 2022 at Shanghai Forth People’s +Hospital. The circulating levels of 25(OH)D3, inflammatory factors, and clinical +parameters were assayed. Time to viral RNA clearance (TVRC), classification and +prognosis of COVID-19 were used to evaluate the severity of COVID-19 infection. +Results: The median age was 76 years (interquartile range, IQR, 64.5-84.6), 44.1% of +patients were male, and the TVRC was 11 days (IQR, 7-16) in this population. The +median level of 25(OH)D3 was 27.15 (IQR, 19.31-38.89) nmol/L. Patients with lower +serum 25(OH)D3 had prolonged time to viral clearance, more obvious inflammatory +response, more severe respiratory symptoms and higher risks of impaired hepatic and +renal function. Multiple regression analyses revealed that serum 25(OH)D3 level was +negatively associated with TVRC independently. ROC curve showed the serum +vitamin D level could predict the severity classification and prognosis of COVID-19 +significantly. +Conclusions: Serum 25(OH)D3 level is independently associated with the severity of +COVID-19 in elderly, and it could be used as a predictor of the severity of COVID-19. +In addition, supplementation with vitamin D might provide beneficial effects in old +patients with COVID-19. + +Keywords: COVID-19; vitamin D; time to viral RNA clearance; inflammatory +response + + + +Introduction +As of 30 November 2022, the corona virus disease 2019 (COVID-19) pandemic +has resulted in more than 639 million confirmed cases with more than 6.6 million +deaths[1]. China has also experienced several waves of COVID-19 pandemic and has +explored various strategies to protect the susceptible people from infection, especially +the elderly and patients with comorbidities. Therefore, it is of great significance to +analyze the risk factors of COVID-19 and investigate inventions to reduce the risks of +infection or serious illness for the prevention and treatment of COVID-19 in China. +In addition to the injury of alveolar epithelial cells mediated by angiotensin- +converting enzyme 2 (ACE2), severe acute respiratory syndrome coronavirus 2 (SARS- +CoV-2) could activate macrophages via ACE2 receptors [2, 3]. The interleukins 1 (IL- +1), IL-6 and tumor necrosis factor (TNF-α) released by the activated macrophage could +further stimulate the neutrophils and T lymphocytes, followed by the release of a large +number of inflammatory factors. This inflammatory cascade and inflammation storm +are considered to be the main pathogenesis of acute respiratory distress syndrome in +COVID-19 [4, 5]. +In addition to regulating calcium and phosphorus metabolism, vitamin D is also +closely related to immune regulation, cardiovascular diseases, metabolic syndrome, +obesity, diabetes, hypertension, cancer, infection and other diseases [6-10]. The +multiple effects of vitamin D are related to the wide distribution of the vitamin D +receptor (VDR). After binding with VDR, the activated vitamin D acts on the cis-acting +elements in the promoter of target genes and thus regulating the transcription of the +target genes. Most immune cells, including dendritic cells, T lymphocytes, and B +lymphocytes, have high levels of VDR that could modulate the cellular response to +viruses as binding with vitamin D [11, 12]. In addition, there are expressions of VDR +in the lung tissue, which are related with the severity of lung injury. Previous studies +showed that mice with VDR knockout had more serious lung injury induced by LPS +compared with WT mice [13, 14]. The histological study showed increased alveolar +permeability, pulmonary vascular exudation, neutrophil infiltration and inflammatory +factors in the lungs of VDR knockout mice [13, 14]. + +Although vitamin D has multiple beneficial effects, the nutritional status of +vitamin D in the population is unsatisfactory. An epidemiological study in more than +40 countries conducted by Lips P et al. found that vitamin D deficiency was present in +more than 50% of the population, especially among nursing home residents (mainly +elderly) [15]. In Europe, approximately 40% of the population is vitamin D deficient +(< 20 nmol/L), as well as in USA (24%) and Canada (37%) [16, 17]. +Previous study showed there was a close relationship between the risk of +respiratory tract infection and vitamin D [18]. Patients with daily or weekly vitamin D +supplementation, especially those with 25(OH)D3 < 10 nmol/L, were found to have a +reduced risk of respiratory tract infections [18]. Studies in patients with COVID-19 also +demonstrated that vitamin D levels were related with the infectious risk and severity of +COVID-19 [18-20]. A retrospective study by Angelidi et al. found that patients with +low serum 25(OH)D3 levels had increased mortality and risk of invasive mechanical +ventilation. The median 25(OH)D3 level in this population was 28 nmol/L. The +mortality in patients with 25(OH)D3 < 30 nmol/L was 25%, compared with the 9% +mortality in patients with > 30 nmol/L. The results were similar when the cutoff value +was adjusted as 20 nmol/L [19, 20]. Although treatment with vitamin D failed to +improve the survival in critically ill patients with COVID-19 [21], epidemiological +studies showed that patients with high vitamin D levels had low risks of infection with +COVID-19 [22], which demonstrated that vitamin D could prevent people from +COVID-19. Therefore, vitamin D levels have the potential to be a predictor of the risk +of infection and the severity of COVID-19. In addition, it is unclear whether some +subgroups of patients would benefit from treatment with vitamin D. +In this study, we aimed to assess vitamin D levels in patients with COVID-19 +infection, and to investigate the relationship between vitamin D levels and time to +clearance of virus, the classification and progression of the disease, which might +provide evidence for identifying of high-risk patients for COVID-19, and protecting +these vulnerable patients from infection and critical illness of COVID-19. + +Patients and Methods + +Study Population +This is a retrospective cohort study of 719 patients aged 22 to 92 years with +confirmed COVID-19 pneumonia hospitalized at the Shanghai Fourth People's Hospital, +School of Medicine, Tongji University, Shanghai, China. All patients were diagnosed +with COVID-19 pneumonia according to World Health Organization interim guidance. +According to hospital data, patients were admitted from April 13 to June 6, 2022, the +final date of follow-up was June 20, 2022. The study was approved by the ethics +committee of Shanghai Fourth People’s Hospital (No. 2020012) and individual consent +for this retrospective analysis was waived. +Data Collection +The epidemiological, clinical evaluation and outcomes data of all participants +during hospitalization were collected from electronic medical records by a trained team +of physicians. The individual components of clinical outcomes were reviewed +independently and recorded into the computer data base by 2 authors (R.L. and Q.L). +The clinical outcomes (including the time to viral RNA clearance (TVRC), the +classification and progression of COVID-19) were monitored up to June 20, 2022, the +final date of follow-up. The Viral Nucleic Acid Kit (Health) was used to extract nucleic +acids from clinical throat swab samples obtained from all patients at admission. A 2019- +nCoV detection kit (Bioperfectus) was used to detect the ORF1ab gene (nCovORF1ab) +and the Ngene (nCoV-NP) according to the manufacturer’s instructions using real-time +reverse transcriptase–polymerase chain reaction (qPCR). COVID-19 infection was +considered laboratory-confirmed if both the nCovORF1ab and nCoV-NP tests showed +positive results. Liver and kidney function, lipids and electrolytes were measured by +CH930, Atellica Solution, Siemens, Germany. Cytokines were measured by Deflex, +Beckman flow cytometry. Blood Routine and C-reactive protein (CRP) were measured +by BC7500, Mindray, China. Serum 25 hydroxyvitamin D was determined by cobas +8000, Roche. +Statistical Analysis +All statistical analyses were performed using IBM SPSS Statistics (Version 22.0, +147 SPSS, IBM Corp., Armonk, New York, USA), GraphPad Prism 8.0.2 (GraphPad + +Software, Inc., San Diego CA, USA) and R (version 3.4.1, R Foundation for Statistical +Computing, Vienna, Austria). Continuous variables were presented as mean ± Standard +Deviation (SD) or median (quartile), and categorical variables were summarized as +counts (frequency percentages). χ2 or Fisher exact test (for small cell counts) was +applied to compare categorical variables. For continuous variables, normal distribution +was evaluated with Kolmogorov-Smirnov test. Then One-way ANOVA (if +homogeneity of variances was assumed) or Wilcoxon-Mann-Whitney U test (if +homogeneity of variances was not met) was used. Furthermore, receiver operating +characteristics (ROC) curves were performed to investigate the value of serum vitamin +D level in predicting the severity classification and prognosis of COVID-19 in the +population. +All reported values were two-sided and P < 0.05 was considered as statistical +significance. + + +Results +1. Clinical baseline characteristics of enrolled patients +A total of 719 patients with confirmed COVID-19 were enrolled retrospectively +in this study from April 13 to June 6, 2022 at Shanghai Forth People’s Hospital. In these +patients, the median age was 76 years (interquartile range, IQR, 64.5-84.6), 44.1% of +patients were male, and the TVRC was 11 days (IQR, 7-16). The median level of +25(OH)D3 was 27.15 (IQR, 19.31-38.89) nmol/L in these patients. The body mass +index (BMI) was 23.08 ± 2.59 Kg/m2 in these patients. There were slightly increased +levels of mean systolic blood pressure (SBP, 140.85 ± 20.76 mmHg) and respiratory +rate (RR, 19.53 ± 1.39 bpm), but normal levels of mean diastolic blood pressure (DBP, +79.85 ± 11.84 mmHg), heart rate (HR, 87.07 ± 15.11 bpm), temperature (Temp, 36.71 +± 0.47℃) and oxygen saturation (SaO2, 97.29 ± 3.54%). The fraction of inspiration O2 +(FiO2) was 29 (21- 33) %. The CRP levels were 9.45 (3.39 - 27.9) mg/L, but almost +normal levels of white blood cells (WBC, 6.37 ± 3.1×10^9/L), red blood cell (RBC, +4.05 ± 0.70×10^9/L) and hemoglobin (Hb, 121.61 ± 21.07g/L). In addition, the levels + +of serum bilirubin (Bil), albumin (Alb), alanine aminotransferase (ALT)，fasting +glucose (FBG) and renal function were in normal ranges (Table 1). + +Table 1 Clinical baseline characteristics of enrolled patients. +Parameter +Value +Age (years) +76.0 (64.5, 84.6) +Sex (M/F) +317/402 +TVRC (days) +11 (7, 16) +25(OH)D3 (nmol/L) +27.15 (19.31, 38.89) +BMI (Kg/m2) +23.08 ± 2.59 +CRP (mg/L) +9.45 (3.39, 27.9) +Temp (℃) +36.71 ± 0.47 +SBP (mmHg) +140.85 ± 20.76 +DBP (mmHg) +79.85 ± 11.84 +HR (bpm) +87.07 ± 15.11 +RR (bpm) +19.53 ± 1.39 +SaO2 (%) +97.29 ± 3.54 +FiO2 (%) +29 (21, 33) +WBC (10^9/L) +6.37 ± 3.1 +RBC (10^12/L) +4.05 ± 0.70 +FBG (mmol/L) +6.28 ± 2.57 +Hb (g/L) +121.61 ± 21.07 +T-Bil (mmol/L) +13.08 ± 7.89 +ALT (U/L) +19.96 (13.77, 30.87) +T-Pro (g/L) +61.55 ± 6.11 +Alb (g/L) +39.57 (35.81, 42.64) +Pre-Alb (g/L) +183.86 (136.10, 225.50) +BUN (mmol/L) +5.77 (4.57, 7.81) +Cr (umol/L) +57.9 (48.1, 73.7) +UA (umol/L) +288.16 (225.48, 363.44) +Cystatin C (mg/mL) +1.09 (0.91, 1.44) +Abbreviations: M: male; F: female; TVRC: time to viral RNA clearance; BMI: body mass index; +CRP: C-reaction protein; Temp: temperature; SBP: systolic blood pressure; DBP: diastolic blood +pressure; HR: heart rate; RR: respiration rate; SaO2: oxygen saturation; FiO2: fraction of inspiration +O2; WBC: white blood corpuscle; RBC: red blood corpuscle; FBG: fasting blood glucose; Hb: +hemoglobin; T-Bil: total bilirubin; ALT: alanine aminotransferase; T-Pro: total protein; Alb: +albumin; Pre-Alb: prealbumin; BUN: blood urea nitrogen; Cr: crea; UA: uric acid. + +2. Comparison of clinical baseline characteristics and comorbidities among +patients with different levels of vitamin D + +The levels of serum vitamin D were measured in 609 patients with COVID-19. +Then patients were divided into 4 groups according to the quartile values of serum +vitamin D levels: Q1 < 13.14 (9.59, 16.56) nmol/L, 13.14 (9.59, 16.56) nmol/L < Q2 < +23.1 (21.37, 25.13) nmol/L, 23.1 (21.37, 25.13) nmol/L < Q3 < 32.42 (29.9, 35.35) +nmol/L, and 32.42 (29.9, 35.35) nmol/L < Q4 < 49.29 (43.21, 63.29) nmol/L. +Table 2 showed the clinical baseline characteristics among the four groups. +Compared with patients with higher levels of serum vitamin D, patients in Q1 group +were older, and had more severe illness, which manifested as longer TVRC, lower +oxygen saturation, and FiO2. Patients in the Q1 group also had higher levels of +inflammation, which included higher levels of CRP and WBC. There was increased +percentage of neutrophil and decreased percentage of monocyte and lymphocyte. +Patients in Q1 group also had decreased levels of total protein (T-Pro), Alb, and +increased levels of lactate dehydrogenase (LDH), which indicated that patients with +low vitamin D levels had impaired hepatic synthetical function and nutritional state. +Although there was no significant difference in the blood urea nitrogen (BUN) and Crea +(Cr), patients in the Q1 group had increased levels of cystatin C, a biomarker of early +renal injury. In addition, there were decreased levels of serum magnesium in patients +with lower levels of vitamin D. However, there was no significant difference in male +proportion, BMI, basic vital signs, and other biochemical tests (Table 2). +The rates of comorbidities were high in this study. However, there was no +significant difference in comorbidities among patients with different levels of vitamin +D (Table 2). + +Table 2 Comparison of clinical and biochemical characteristics and comorbidities +among patients with different levels of vitamin D + +Q1 group +(n=162) +Q2 group +(n=164) +Q3 group +(n=164) +Q4 group +(n=165) +Age (years) +86.0 (70.0, 89.0) +75.6 (63.75, 87.0)a +72.5 (63.75, 81.25)ab +73.0 (64.0, 81.0)a +Sex (M/F) +63/99 +71/93 +78/86 +82/83 +TVRC (days) +14 (8, 19) +10 (6, 15)a +10 (7.25, 15)a +11 (7, 13)a +CVD, n (%) +44 (27.16) +37 (22.56) +13 (7.93) +28 (16.97) +HT, n (%) +98 (60.49) +84 (51.22) +93 (56.71) +89 (53.94) +T2DM, n (%) +36 (22.22) +36 (21.95) +48 (29.27) +50 (30.30) + +Tumor, n (%) +17 (10.49) +15 (9.15) +15 (9.15) +14 (8.48) +BMI (Kg/m2) +22.41 ± 3.62 +22.63 ± 3.57 +23.68 ± 36.66ab +23.29 ± 3.60 +CRP (mg/L) +15.46 (5.32, 42.08) +9.45 (3.87, 26.82) a +6.77 (2.68, 19.31) ab +6.29 (2.44, 18.33)a +Temp (℃) +36.67 ± 0.46 +36.66 ± 0.45 +36.74 ± 0.49 +36.74 ± 0.49 +SBP (mmHg) +139.61±22.03 +140.87±22.43 +141.07±19.76 +142.95±19.03 +DBP (mmHg) +76.68 ± 12.55 +81.26 ± 12.43a +80.95 ± 10.80a +80.53 ± 10.90a +HR (bpm) +85.41 ± 15.33 +86.81 ± 17.01 +89.55 ± 14.5 +87.24 ± 14.1 +RR (bpm) +19.47 ± 1.63 +19.66 ± 1.40 +19.50 ± 1.11 +19.45 ± 1.15 +SaO2 (%) +96.69 ± 3.40 +97.46 ± 1.78a +97.52 ± 1.65a +97.62 ± 1.13a +FiO2 (%) +29 (29, 33) +29 (21, 33) a +21 (21, 33) ab +21 (21, 29) ab +WBC (10^9/L) 7.14 ± 3.83 +6.12 ± 2.49a +6.12 ± 2.69a +5.95 ± 3.30a +Monocyte % +7.45 ± 2.90 +8.29 ± 2.70a +8.68 ± 3.31a +8.02 ± 2.96 +Lymphocyte % 21.36 ± 12.13 +25.60 ± 11.72a +26.58 ± 11.46a +28.76 ± 12.81ab +Neutrophil % +69.48 ± 13.30 +63.84 ± 12.84a +62.37 ± 12.48a +61.10 ± 13.25a +PLT (10^9/L) +208.73 ± 90.71 +211.61 ± 87.55 +206.95 ± 72.97 +189.82 ± 68.62a +RBC +(10^12/L) +3.76 ± 0.76 +4.10 ± 0.63a +4.19 ± 0.68a +4.20 ± 0.61a +Hct (%) +34.19 ± 6.97 +38.04 ± 5.87a +39.03 ± 5.62a +39.27 ± 5.29a +Hb (g/L) +116.36 ± 24.47 +123.06 ± 19.16a +126.55 ± 17.76a +127.58 ± 17.93ab +T-Bil (umol/L) +12.72 ± 7.38 +13.29 ± 6.60 +13.38 ± 5.95 +13.24 ± 10.94 +ALT (U/L) +16.47 (12.77, 28.15) +22.34 (13.34, 33.24) +20.14 (13.95, 28.42) +20.00 (14.57, 32.89) +AST (U/L) +24.83 (19.60, 37.41) +24.38 (19.23, 32.94) +23.5 (18.97, 31.12) +24.59 (19.34, 34.13) +AKP (U/L) +83.41 (67.57, 102.71) +79.69 (67.88, 99.37) +78.94 (70.42, 100.54) +76.97 (61.96, +95.47)a +T-Pro (g/L) +58.17 ± 6.46 +61.74 ± 5.68a +63.20 ± 5.40ab +63.18 ± 5.78ab +Alb (g/L) +35.71 ± 4.81 +39.10 ± 4.33a +40.64 ± 4.16ab +41.02 ± 4.26ab +Pre-Alb (g/L) +149.02 (102.26, +203.19) +179.43 (136.11, +227.65) a +194.49 (162.13, 232.86) +ab +190.31 (161, +233.61)a +BUN (umol/L) +6.35 (4.68, 9.10) +5.66 (4.48, 7.16) a +5.77 (4.69, 7.82) +5.64 (4.57, 7.49) +Cr (umol/L) +56.50 (45.85, 82.90) +56.3 (48.7, 74.1) +60.85 (49.35, 72.23) +59.95 (49.3, 73.55) +UA (umol/L) +251.43 (193.66, +349.31) +278.97 (239.34, +373.91) +325.84 (235.18, 372.76) +a +295.65 (257.76, +349.96)a +Cystatin C +(mg/mL) +1.26 (0.98, 1.71) +1.09 (0.93, 1.38) a +1.05 (0.88, 1.37) a +1.03 (0.89, 1.33)a +Lactate +(mmol/L) +2.10 ± 0.93 +1.95 ± 0.76 +2.04 ± 0.80 +2.23 ± 1.06 +FBG (mmol/L) 6.63 ± 3.28 +5.93 ± 2.66a +6.65 ± 3.24b +5.88 ± 2.24ac +LDH (U/L) +232.76 ± 84.93 +209.88 ± 76.38a +203.97 ± 56.08a +201.54 ± 54.01a +K+ (mmol/L) +3.90 ± 0.69 +3.76 ± 0.59a +3.82 ± 0.50 +3.80 ± 0.50 +Na+ (mmol/L) +141.61 ± 5.77 +141.94 ± 5.23 +141.92 ± 3.97 +142.44 ± 3.55 +Cl- (mmol/L) +104.11 ± 5.86 +104.61± 4.83 +104.36 ± 3.77 +104.53 ± 3.61 +Ca2+ (mmol/L) +1.77 ± 0.46 +1.87 ± 0.48 +1.99 ± 0.43 ab +2.05 ± 0.40 ab +Mg2+ +(mmol/L) +0.84 ± 0.11 +0.87 ± 0.09a +0.88 ± 0.09a +0.87 ± 0.10a + +Phosphate +(mmol/L) +1.08 ± 0.59 +1.09 ± 0.37 +1.14 ± 0.23 +1.15 ± 0.28 +Abbreviations: CVD: cardiovascular disease; HT: hormone therapy; T2DM: diabetes mellitus type +2; PLT: platelet count; Hct: red blood cell specific volume; AST: glutamic oxaloacetic transaminase; +AKP: alkaline phosphatase; LDH: lactate dehydrogenase. + a, p < 0.05 compared with Q1 group; b, p<0.05 compared with Q2 group; c, p < 0.05 compared +with Q3 group. + +3. Comparison of inflammatory factors among patients with different levels of +vitamin D +The increased levels of WBC and CRP in patients from Q1 group implicated that +patients with lower levels of serum vitamin D might had high inflammatory state. +Therefore, we measured the serum levels of inflammatory factors in patients from +different groups. However, there was no significant difference of inflammatory factors +among these groups except for lower levels of interferon-γ (IFN-γ) and TNF-α in +patients with lower levels vitamin D (Figure 1, Table S1). + + +Figure 1 Comparison of inflammatory factors among patients with different levels of vitamin D.**, +p < 0.01; ****, p < 0.0001. Abbreviations: IL: interleukins; IFN: interferon; TNF: tumor necrosis +factor. + +4. Association between serum vitamin D level and the severity of COVID-19 + +To further assess the association between serum vitamin D level and the severity +of COVID-19, patients were first divided into 4 groups according to the quartile of +TVRC, which was used as an indicator for the severity of COVID-19. Patients in the +longest TVRC group (TVRC-Q4) had significantly lower serum vitamin D levels +(23.19 [IQR, 14.46-33.77] nmol/L) compared with patients in shorter TVRC groups +(26.74 [IQR, 20.76-38.97] nmol/L in TVRC-Q1, p = 0.0075; 31.02 [IQR, 22.87-41.03] +nmol/L in TVRC-Q2, p < 0.0001; 26.19 [IQR, 18.08, 41.44] nmol/L in TVRC-Q3, p = +0.0461) (Figure 2A). Patients were also grouped into mild, moderate, severe and +critical groups based on the severity classification of COVID-19 according to the +guideline for management of patients with COVID-19 (9th version). There were +significantly lower levels of serum vitamin D in patients with severe (19.53 [IQR, +12.71-27.01] nmol/L) and critical (15.54 [IQR, 8.51-20.68] nmol/L) groups compared +with patients in the mild (31.10 [IQR, 22.73-42.01] nmol/L) and moderate (26.31 [IQR, +17.98-36.51] nmol/L) groups (Figure 2B). Furthermore, patients were divided into 3 +groups based on the prognosis of the disease according to the progression of the disease +changes of the severity classification of COVID-19 when the virus RNA was cleared, +and the relation between serum vitamin D levels and the prognosis was investigated. +Patients with good prognosis had significantly higher levels of serum vitamin D levels +(28.21 [IQR, 20.46-40.22] nmol/L) compared with patients with poor prognosis +(Prognosis-Q1, 19.53 [IQR, 12.11-27.44] nmol/L in Prognosis-Q2, p < 0.0001; 18.03 +[IQR, 10.96-21.56] nmol/L in Prognosis-Q3, p = 0.016) (Figure 2C). + + +Figure 2 Association of vitamin D level with TVRC, classification and prognosis of COVID-19. (A) +Vitamin D levels in each group stratified by TVRC quartile 11 (IQR, 7-16). (B) Vitamin D levels +in each group divided by severity classification of COVID-19. (C) Vitamin D levels in patients with +different progression. + +A +B +** +C +**** +200 +200- +*** +** +200- +**** +150 +25(OH)D3 (nmol/L) +150 +25(OH)D3 ( +100 +25(OH)D3 ( +100 +100 +50- +50 +50 +TVRC-Q4 +Mild +Moderate +Severe +Critica +ROC curve showed the serum vitamin D level could predict the severity +classification and prognosis of COVID-19 significantly (the area under the curve [AUC] += 0.695, 95% CI [0.627-0.764], p < 0.001, for severe and critical of COVID-19, Figure +3A; AUC=0.728, 95% CI [0.585-0.872], p = 0.009, for the aggravation of COVID-19, +Figure 3B). + +Figure 3 ROC curve to investigate the serum vitamin D level in predicting the severity classification +(A) and prognosis (B) of COVID-19. Abbreviations: AUC: the area under the curve; ROC: receiver +operating characteristics. + +5. Association between serum vitamin D levels and clinical parameters +In univariate analyses, serum vitamin D level was negatively associated with +TVRC, age, FiO2, prognosis, IL-10, cystatin C, alkaline phosphatase (AKP), LDH, +direct bilirubin (D-Bil), and CRP. However, BMI, SaO2, DBP, Alb, IL-4, TNF-α, +serum calcium (Ca) levels, indirect bilirubin (I-Bil), serum magnesium (Mg) level, +serum sodium (Na) level, uric acid (UA), pre-albumin (pre-Alb), LDH, Hb, red blood +cell specific volume (Hct) and T-Pro were positively associated with serum vitamin D +level (Table 3). + +Table 3 Correlation between serum vitamin D and other variables +Parameter +r +p-Value +Age (years) +-0.239 +< 0.001 +TVRC (days) +-0.135 +0.001 +BMI (Kg/m2) +0.091 +0.048 +CRP (mg/L) +-0.196 +< 0.001 +DBP (mmHg) +0.096 +0.014 +SaO2 (%) +0.095 +0.016 + +A +B +1.0 +1.0 +0.8 +8'0 +Sensivity +0.6 +Sensivity +0.6 +AUC = 0.695 +AUC = 0.728 +p ≤ 0.001 +p = 0.009 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +0'0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1-Specifity +1-SpecifityFiO2 (%) +-0.227 +< 0.001 +WBC (10^9/L) +-0.116 +0.003 +Monocyte % +0.078 +0.047 +Lymphocyte % +0.226 +< 0.001 +Neutrophil % +-0.23 +< 0.001 +Hct (%) +0.294 +< 0.001 +Hb (g/L) +0.298 +< 0.001 +D-Bil (mmol/L) +-0.112 +0.009 +I-Bil (umol/L) +0.102 +0.017 +AKP (U/L) +-0.104 +0.035 +T-Pro (g/L) +0.3 +< 0.001 +Alb (g/L) +0.405 +< 0.001 +Pre-Alb (g/L) +0.24 +< 0.001 +UA (umol/L) +0.144 +0.001 +Cystatin C (umol/L) +-0.191 +< 0.001 +LDH (U/L) +-0.151 +< 0.001 +Na+ (mmol/L) +0.087 +0.027 +Ca2+ (mmol/L) +0.343 +< 0.001 +Mg2+ (mmol/L) +0.106 +0.015 +Phosphate (mmol/L) +0.211 +< 0.001 +IL-10 +-0.109 +0.007 +IL-4 +0.067 +0.01 +TNF-α +0.202 +< 0.001 +DSS +-0.242 +< 0.001 +Prognosis +-0.194 +< 0.001 +Abbreviations: D-Bil: direct Bilirubin; I-Bil: indirect bilirubin; DSS: disease severity score. + +6. Association between TVRC and clinical parameters +Spearman correlation coefficients were used to evaluate correlations between +TVRC and clinical parameters. The results showed that serum vitamin D level, BMI, +HR, ALB, TNF-α, serum calcium level, serum sodium level, serum phosphorus level, +serum chlorine level, uric acid, pre-Alb, T-Pro, Hb, hematocrit (Hct) and RBC were +negatively associated with TVRC. In addition, age, prognosis, IL-10, Il-12, IL-17, IL- +2, WBC, CRP, Alb, LDH, ALT, glutamic oxaloacetic transaminase (AST), alkaline +phosphatase (AKP), BUN, cystatin C, creatinine, and serum potassium level were +positively associated with TVRC (Table 4). Multiple regression analyses revealed that +only serum vitamin D level was negatively associated with TVRC independently (Table +5). + + +Table 4. Correlation between TVRC and other variables +Variables +Beta coefficient +p-Value +Age (years) +0.253 +< 0.001 +25(OH)D3 (nmol/L) +-0.135 +0.001 +BMI (Kg/m2) +-0.164 +< 0.001 +CRP (mg/L) +0.157 +< 0.001 +HR (bpm) +-0.074 +0.047 +FiO2 (%) +0.242 +< 0.001 +RBC (10^12/L) +-0.185 +< 0.001 +WBC (10^9/L) +0.09 +0.016 +Lymphocyte % +-0.159 +< 0.001 +Neutrophil % +0.158 +< 0.001 +Hct (%) +-0.167 +< 0.001 +Hb (g/L) +-0.186 +< 0.001 +ALT (U/L) +0.076 +0.048 +AST (U/L) +0.081 +0.03 +AKP (U/L) +0.148 +0.001 +r-GT (U/L) +0.074 +0.05 +T-Pro (g/L) +-0.167 +< 0.001 +Alb (g/L) +-0.287 +< 0.001 +Pre-Alb (g/L) +-0.165 +0.001 +BUN (umol/L) +0.244 +< 0.001 +UA (umol/L) +-0.096 +0.026 +Cystatin C (mg/mL) +0.191 +< 0.001 +LDH (U/L) +0.127 +0.002 +Cr (umol/L) +0.116 +0.002 +K+ (mmol/L) +0.207 +< 0.001 +Na+ (mmol/L) +-0.204 +< 0.001 +Cl- (mmol/L) +-0.109 +0.004 +Ca2+ (mmol/L) +-0.119 +0.003 +Phosphate (mmol/L) +-0.229 +< 0.001 +IL-10 +0.087 +0.025 +IL-12 +0.08 +0.04 +IL-17 +0.14 +< 0.001 +IL-1 +0.076 +0.05 +IL-2 +0.124 +0.001 +TNF-α +-0.095 +0.014 +DSS +0.235 +< 0.001 +Prognosis +0.178 +< 0.001 +Abbreviations: r-GT: γ-glutamyl transpeptidase. + +Table 5. Multivariate regression analyses of predictors of TVRC in patients with +COVID-19 + +Variables +Beta coefficient +p-Value +95% CI +25(OH)D3 (nmol/L) +-0.230 +0.016 +-0.168 to -0.018 + +Discussion +As a kind of steroid hormone, vitamin D is tightly linked to a number of different +metabolic processes and immune regulation in the human body. Vitamin D activates +the innate immune system by binding with VDR in immune cells to defend the invasion +of foreign pathogenic microorganisms. For example, 1,25 dihydroxyvitamin D3 (1,25- +(OH)2-D3) could induce the generation of antimicrobial peptides in monocytes to clean +the Mycobacterium tuberculosis[23, 24]. 1,25-(OH)2-D3 also could tune the cellular +and humoral immunity by regulating the differentiation and proliferation of T and B +lymphocytes and the secretion of Th1/Th2 cytokines. In addition, 1,25-(OH)2-D3 could +inhibit the exaggerated inflammatory response via inducing the differentiation of +regulatory T cells (Treg), and have protective effects in inflammatory responses and +autoimmune diseases. +Considering that 25(OH)D3 is the main form of vitamin D in the body and its +stable concentration in circulation, serum 25(OH)D3 was used as an indicator to +evaluate the nutritional status of vitamin D. Presently, vitamin D deficiency, +insufficiency, normal, and sufficiency are defined as <25, 25 to 50, 51 to 75, and > +75nmol/L, respectively[25]. Vitamin D deficiency was defined when the serum level +of 25(OH)D3 was less than 50nmol/L. An epidemiological study in East China showed +that the serum levels of 25(OH)D3 were 40.5 ± 12.5 nmol/L in the normal population, +and 80.3% of the population were vitamin D deficiency[26], which was significantly +higher than that in western countries[27, 28]. In addition, a study in elderly inpatients +showed that the vitamin D levels were 34.6 ± 16.2 nmol/L in the population, of which +17.5% were severely deficient, 73.0% were mildly deficient, 7.5% were insufficient, +and only 2.0% were sufficient[29]. These data suggest that vitamin D deficiency may +be common in the Han population, especially in the elderly and bedridden patients. +This study enrolled 719 patients with COVID-19 and assessed the levels of serum +vitamin D, cytokines and other clinical indicators to investigate the relationship + +between vitamin D levels and TVRC, the classification and prognosis of the disease. +Higher levels of vitamin D were associated with the higher levels of T-Pro, Alb, pre- +Alb, hemoglobin and BMI, which indicated that higher vitamin D levels were +associated with better protein synthesis ability of liver and better nutritional status of +patients. Conversely, the lower levels of vitamin D were associated with the longer +TVRC, higher levels of WBC and CRP, as well as worse oxygenation capacity of the +lung, suggesting that lower vitamin D was associated with severe conditions in these +patients. Meanwhile, lower levels of vitamin D were related with the biomarkers of +early hepatic and renal function impairments, such as lower levels of pre-Alb and higher +levels of cystatin C. In addition, there was positive relationship between vitamin D +levels and serum calcium and phosphorus concentrations. All these results +demonstrated that vitamin D had benefit effects on the clearance of the virus and +alleviating the condition in patients with COVID-19. Further investigation validated +that lower vitamin D levels were associated with longer TVRC, more severe disease +and worse prognosis. Therefore, serum vitamin D level is a predictor of the severity of +disease and prognosis in patients with COVID-19. +Previous studies have shown that the risks of severe infection and mortality were +increased in vulnerable groups (with comorbidities such as diabetes, hypertension, +coronary artery disease and tumors) in patients with COVID-19[29-32]. In this study, +we compared the comorbidities in patients with different vitamin D levels, and found +no significant difference in comorbidities among different groups. The results indicated +the association between vitamin D levels and the prognosis of the disease was less +affected by these chronic comorbidities. Further investigation showed that serum +vitamin D level was correlated with TVRC negatively, and serum vitamin D level was +an independent predictor of TVRC in patients with COVID-19, which further validated +the close relationship between vitamin D and the severity and prognosis of COVID-19. +Vitamin D deficiency is a common phenomenon in Chinese, especially in the +elderly. For its detrimental effects on the immune system, vitamin D deficiency would +impair the clearance of invasive pathogens. This concern is more obvious under the +current situation of the panic of COVID-19 and consistent virus variants. Therefore, it + +is of great significance to investigate how to protect patients with high risk from +infection and improve the prognosis of these patients. Supplement with vitamin D +routinely in patients with COVID-19 is still in debate presently[33-35]. However, the +results of this study demonstrated that early supplement with vitamin D in patients with +COVID-19 and vitamin D deficiency could improve the ability of defensing the +infection of SARS-CoV-2, promoting the clearance of virus and improving the +prognosis in these high-risk patients. However, for the limitation of the observed study, +further prospective randomized controlled trails were needed to investigate the benefits +of supplement of vitamin D in these patients. +Limitations +There are several limitations of our study. Although our study implied that early +supplemented with vitamin D in patients with COVID-19 and vitamin D deficiency +might improve the prognosis of these patients. However, we did not give the therapy +with vitamin D in this population in this retrospective study. In addition, this +retrospective study has some disadvantages compared with prospective studies. +Therefore, further prospective studies are needed to validate the clinical value of serum +vitamin D levels in risk stratifications of patients with COVID-19. +Conclusion +This study demonstrated that serum 25(OH)D3 level was independently associated +with the severity of COVID-19 in elderly, and it could be used as a predictor of the +severity of COVID-19. In addition, supplementation with vitamin D might provide +beneficial effects in old patients with COVID-19. +Sources of Funding +This work was supported by Shanghai Committee of Science and Technology, +China (grant No. 22dz1202304 to Jiajing Cheng). +Author Contributions +Conceptualization, Ruyi Qu, Yuxin Yang and Jinlong Qin; Data curation, Ruyi +Qu, Qiuji Yang and Yingying Bi; Formal analysis, Ruyi Qu and Jinlong Qin; Funding +acquisition, Jiajing Cheng; Inves-tigation, Jiajing Cheng, Mengna He, Xin Wei and +Yiqi Yuan; Methodology, Ruyi Qu, Qiuji Yang, Yingying Bi, Jiajing Cheng, Yuxin + +Yang and Jinlong Qin; Project administration, Jinlong Qin; Re-sources, Jiajing Cheng; +Software, Yingying Bi and Xin Wei; Supervision, Yuxin Yang and Jinlong Qin; +Validation, Qiuji Yang and Yingying Bi; Visualization, Yingying Bi, Mengna He and +Yiqi Yuan; Writing – original draft, Ruyi Qu, Qiuji Yang and Yuxin Yang; Writing – +review & editing, Yuxin Yang and Jinlong Qin. +Acknowledgments +The authors are grateful to all the participants in this study. +Conflicts of Interest +The authors declare no conflict of interest. + + +Reference +1. +WHO Coronavirus (COVID-19) Dashboard. 30 Nov 2022; Available from: +https://covid19.who.int/. +2. +Singh, S.P., et al., Microstructure, pathophysiology, and potential therapeutics of COVID- +19: A comprehensive review. 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Science, 2006. 311(5768): p. 4. +25. +Holick, M.F., Vitamin D and sunlight: strategies for cancer prevention and other health +benefits. Clin J Am Soc Nephrol, 2008. 3(5): p. 1548-54. +26. +Zhen, C., et al., Serum vitamin D levels of the natural population in eastern China. Chinese +Journal of Endocrinology and Metabolism, 2017. 33(9): p. 4. +27. +Looker, A.C., et al., Serum 25-hydroxyvitamin D status of adolescents and adults in two +seasonal subpopulations from NHANES III. Bone, 2002. 30(5): p. 7. +28. +Wielen, R.P.v.d., et al., Serum vitamin D concentrations among elderly people in Europe. +Lancet, 1995. 346(8969): p. 4. +29. +Zhu, L., et al., Association of Blood Glucose Control and Outcomes in Patients with +COVID-19 and Pre-existing Type 2 Diabetes. Cell Metab, 2020. 31(6): p. 1068-1077 e3. +30. +Pranata, R., et al., Hypertension is associated with increased mortality and severity of +disease in COVID-19 pneumonia: A systematic review, meta-analysis and meta-regression. +J Renin Angiotensin Aldosterone Syst, 2020. 21(2): p. 1470320320926899. + +31. +Zhou, F., et al., Clinical course and risk factors for mortality of adult inpatients with +COVID-19 in Wuhan, China: a retrospective cohort study. The Lancet, 2020. 395(10229): +p. 1054-1062. +32. +Liang, W., et al., Cancer patients in SARS-CoV-2 infection: a nationwide analysis in China. +The Lancet Oncology, 2020. 21(3): p. 335-337. +33. +Gibbons, J.B., et al., Association between vitamin D supplementation and COVID-19 +infection and mortality. Sci Rep, 2022. 12(1): p. 19397. +34. +Jolliffe, D.A., et al., Effect of a test-and-treat approach to vitamin D supplementation on +risk of all cause acute respiratory tract infection and covid-19: phase 3 randomised +controlled trial (CORONAVIT). BMJ, 2022. 378: p. e071230. +35. +Bychinin, M.V., et al., Effect of vitamin D3 supplementation on cellular immunity and +inflammatory markers in COVID-19 patients admitted to the ICU. Sci Rep, 2022. 12(1): p. +18604. + +Supplemental data + +Table 1. Comparison of inflammatory factors among patients with different levels of +vitamin D + +Q1 group +(n=150) +Q2 group +(n=153) +Q3 group +(n=157) +Q4 group +(n=149) +IL-1 +0.80(0.16,1.69) +0.96(0.21,2.03) +0.89(0.37,2.07) +0.95(0.29,2.07) +IL-2 +0.08(0.05,0.82) +0.08(0.04,0.88) +0.08(0.04,0.77) +0.08(0.05,0.88) +IL-4 +0.44(0.07,1.90) +0.78(0.07,2.37) +0.98(0.08,5.47) +0.6(0.08,3.77) +IL-5 +0.07(0.03,0.14) +0.07(0.04,0.14) +0.07(0.03,0.21) +0.07(0.04,0.1) +IL-6 +80.44(21.22,204.04) +67.63(24.65,228.57) +107.85(26.4,312.4) +70.84(19.37,288.68) +IL-8 +101.77(32.25,229.92) +108.2(23.96,255.95) +122.3(44.53,242.43) +108.06(32.42,244.3) +IL-10 +4.88(2.79,9.32) +4.28(2.51,7.21) +4.26(2.53,8.01) +3.85(2.09,6.45)a +IL-12 +0.07(0.04,0.18) +0.07(0.04,0.62) +0.07(0.04,0.5) +0.08(0.04,0.4) +IL-17 +1.11(0.36,2.93) +0.91(0.26,2.37) +1.05(0.33,2.67) +1.05(0.32,2.31) +IFN-α +0.07(0.04,0.10) +0.07(0.03,0.09) +0.07(0.04,0.1) +0.06(0.04,0.1) +IFN-γ +1.27(0.38,3.79) +1.93(0.56,5)a +2.35(0.4,7.37)a +1.72(0.42,5.34) +TNF-α +4.32(1.82,9.12) +6.29(2.91,11.04)a +7.38(3.52,12.74)a +8.78(3.39,17.19)ab + + +Table 2. Comparison of clinical and biochemical characteristics among patients with +different severity rating + +Q1 group +(n=330) +Q2 group +(n=309) +Q3 group +(n=71) +Q4 group +(n=9) +Age (years) +68(59,78) +81(71,88) a +86(77,90) a +87(79.5,91) a +Sex (M/F) +149/181 +130/179 +34/37 +4/5 +TVRC (days) +9(6,13) +12(8,17) a +13(9,20.25) a +12(3.25,14.75) +25(OH)D3 +31.10(22.73,42.01) +26.31(17.98,36.51) a +19.53(12.71,27.01) ab +15.54(8.51,20.68) ab +BMI (Kg/m2) +23.42±3.59 +22.74±3.61 +21.73±3.18 +NA +CRP (mg/L) +5.46(2.33,14.47) +10.27(4.38,27.91) a +36.65(19.13,76.31) ab +99.08(56.11,155.47) ab + +Temp (℃) +36.74±0.48 +36.68±0.46 +36.70±0.50 +37.00±0.41 +SBP (mmHg) +141.06±19.57 +140.53±21.84 +141.38±21.72 +140.00±21.47 +DBP (mmHg) +81.42±11.06 +78.75±12.02 +77.87±13.31 +76.11±15.60 +HR (bpm) +88.33±15.10 +86.01±14.55 +85.58±17.10 +89.22±17.14 +RR (bpm) +19.41±1.24 +19.58±1.23 +19.69±2.14 +20.78±2.99 +SaO2 (%) +97.72±1.21 +97.38±1.49 +95.57±3.22 ab +91.38±11.88 abc +Fio2 (%) +21(21,29) +29(21,33) a +33(33,41) ab +61(29,141) ab +RBC +(10^12/L) +4.25±0.57 +3.99±0.72 a +3.53±0.74 ab +3.20±0.71 abc +WBC (10^9/L) 5.75±2.58 +6.45±3.03 a +8.41±3.93 ab +10.49±5.51 abc +Monocyte % +8.27±2.94 +8.15±2.92 +6.54±2.75 +7.97±6.25 +Lymphocyte +% +29.23±11.37 +24.32±11.64 a +13.59±8.56 ab +8.78±5.00 abc +Neutrophil % +60.23±11.98 +65.46±12.49 a +78.03±10.61 ab +82.88±9.49 abc +PLT (10^9/L) +196.96±65.99 +210.42±86.97 +221.68±94.84 +252.33±159.25 +Hct (%) +39.35±5.35 +36.91±6.27 a +32.64±6.81 ab +29.43±6.56 abc +Hb (g/L) +127.60±18.05 +118.93±20.43 a +107.37±26.52 ab +106.56±21.07 ab +T-Bil (umol/L) +12.73±5.94 +12.89±6.22 +14.98±16.83 +17.20±9.87 +ALT (U/L) +19.93(13.52,30.35) +20.20(13.91,31.77) +20.35(13.17,30.83) +19.02(14.79,30.27) +AST (U/L) +22.71(18.48,29.55) +24.98(19.32,34.96) a +30.38(21.24,42.76) a +45.24(34.48,50.38) a +AKP (U/L) +78.10(63.65,95.12) +83.77(69.13,99.65) +80.64(67.78,102.90) +84.03(67.08,112.51) +T-Pro (g/L) +62.91±5.54 +61.30±5.92 +57.54±6.56 ab +52.45±7.15 abc +Alb (g/L) +41.01±4.28 +38.29±4.43 a +34.38±4.45 ab +31.75±4.24 abc +Pre-Alb (g/L) +196.58(160.86,238.85) +181.60(127.98,291.55) +a +98.48(80.37,148.75) ab +98.73(65.29,151.81) a +BUN (umol/L) +5.43(4.39,6.79) +6.13(4.80,8.32) a +6.33(4.92,11.15) a +13.85(7.56,20.02) a +Cr (umol/L) +57.50(48.80,71.28) +58.40(47.80,78.70) +54.20(37.70,80.60) +89.50(38.10,169.30) +UA (umol/L) +299.76(243.44,359.89) +291.43(224.63,376.58) +242.34(142.73,319.98) ab +252.97(148.49,377.04) +a +Cystatin C +(mg/mL) +0.98(0.86,1.21) +1.16(0.95,1.56) a +1.39(1.06,1.88) ab +1.60(1.16,2.55) a +Lactate +(mmol/L) +1.94±0.71 +2.00±0.78 +2.21±1.04 +2.77±1.23 +FBG (mmol/L) 5.81±2.72 +6.35±2.88 a +7.69±3.00 ab +8.07±1.25 abc +LDH (U/L) +194.28±48.28 +215.18±70.80 a +244.00±90.07 ab +358.20±107.77 abc +K (mmol/L) +3.75±0.51 +3.87±0.63 +3.89±0.54 +4.20±0.66 +Na(mmol/L) +142.95±3.54 +141.19±5.27 +140.75±6.28 +139.56±7.09 +Cl (mmol/L) +105.00±3.52 +103.94±5.25 +103.75±6.22 +102.33±7.35 ab +Ca (mmol/L) +2.05±0.37 +1.84±0.49 a +1.53±0.49 ab +1.68±0.44 ab +Mg (mmol/L) +0.88±0.08 +0.86±0.10 +0.82±0.11 a +0.81±0.10 a +P (mmol/L) +1.19±0.34 +1.12±0.43 +0.88±0.36 ab +0.72±0.37 abc + + + +Table 3. Comparison of clinical and biochemical characteristics among patients with +different prognosis + +P1 group +(n=638) +P2 group +(n=70) +P3 group +(n=11) +Age (years) +74.5(64,86) +85.5(78.75,90.25) a +81(66,89) +Sex (M/F) +277/361 +34/36 +6/5 +TVRC (days) +10(7,15) +14(9,23) a +18.5(14.75,22) a +BMI (Kg/m2) +23.17±3.60 +21.79±2.70 +19.92±4.98 +25(OH)D3 +28.21(20.46,40.22) +19.53(12.11,27.44) a +18.03(10.96,21.56) a +CRP (mg/L) +7.22(2.96,20.25) +45.98(22.03,93.56) a +69.08(17.99,105.47) a +Temp (℃) +36.71±0.47 +36.71±0.49 +36.78±0.41 +SBP (mmHg) +140.69±20.61 +142.04±22.37 +142.55±20.86 +DBP (mmHg) +79.84±11.47 +80.24±14.89 +77.91±12.76 +HR (bpm) +87.47±14.69 +83.21±17.82 +88.45±18.97 +RR (bpm) +19.48±1.22 +19.99±2.36 a +19.64±1.86 +SaO2 (%) +97.49±1.52 +95.68±4.87 a +96.00±3.55 +Fio2 (%) +29(21,33) +33(29,41) a +41(37,53) a +RBC (10^12/L) +4.12±0.65 +3.47±0.80 a +3.72±0.64 +WBC (10^9/L) +6.09±2.84 +8.35±3.54 a +10.12±6.57 a +Monocyte % +8.21±2.94 +6.95±2.83 a +5.48±5.14 a +Lymphocyte % +26.83±11.81 +13.69±7.76 a +11.76±11.01 a +Neutrophil % +62.76±12.58 +77.69±9.44 a +82.25±13.39 a +PLT (10^9/L) +205.06±78.08 +211.16±93.12 +219.82±133.25 +Hct (%) +38.16±5.90 +32.22±7.32 a +33.75±6.11 +Hb (g/L) +123.41±19.53 +104.79±23.48 a +125.00±39.87 b +T-Bil (umol/L) +12.80±6.08 +15.83±17.02 a +11.66±6.80 +ALT (U/L) +20.08(13.79,30.75) +18.68(13.30,32.71) a +21.80(10.16,32.22) a +AST (U/L) +23.58(18.86,32.43) +31.92(20.11,47.30) a +35.85(23.21,47.03) +AKP (U/L) +79.99(65.62,95.93) +88.07(69.46,113.97) +80.64(65.92,98.15) +T-Pro (g/L) +62.21±5.75 +56.77±6.47 a +54.37±6.76 a +Alb (g/L) +39.72±4.53 +34.15±4.47 a +32.74±4.76 a +Pre-Alb (g/L) +190.22(147.24,232.20) +100.75(81.27,146.45) a +129.31(82.19,197.08) +BUN (umol/L) +5.65(4.51,7.43) +7.32(5.28,12.94) a +10.41(6.20,18.20) a +Cr (umol/L) +57.80(48.20,72.48) +62.15(41.40,90.65) +67.20(32.10,129.40) +UA (umol/L) +291.73(230.62,365.03) +275.16(171.31,363.57) +148.57(79.31,240.75) ab +Cystatin C (mg/mL) +1.05(0.89,1.38) +1.47(1.23,2.15) a +1.20(1.01,2.35) +Lactate (mmol/L) +1.96±0.74 +2.24±1.07 +2.89±1.09 a +FBG (mmol/L) +6.00±2.67 +8.07±3.41 a +9.64±3.43 a +LDH (U/L) +203.50±58.94 +251.11±97.54 a +334.63±115.47 ab +K (mmol/L) +3.79±0.55 +4.01±0.71 a +4.17±0.62 +Na(mmol/L) +142.20±4.30 +139.64±6.25 a +140.82±12.58 +Cl (mmol/L) +104.46±4.39 +103.93±5.77 +102.91±11.60 +Ca (mmol/L) +1.93±0.45 +1.67±0.50 a +1.29±0.39 ab +Mg (mmol/L) +0.87±0.09 +0.83±0.11 a +0.81±0.10 + +P (mmol/L) +1.15±0.39 +0.95±0.37 a +0.63±0.19 ab + + diff --git a/DdE0T4oBgHgl3EQfygJ9/content/tmp_files/load_file.txt b/DdE0T4oBgHgl3EQfygJ9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf5c39bc5ac631ca9eb8c0620621ac02391be76a --- /dev/null +++ b/DdE0T4oBgHgl3EQfygJ9/content/tmp_files/load_file.txt @@ -0,0 +1,1977 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf,len=1976 +page_content='Title: Decreased serum vitamin D level as 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Tongji University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Shanghai 200434,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' China 4 Department of Radiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=" Shanghai Fourth People's Hospital," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Tongji University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Shanghai 200434,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' China 5 Clinical Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=" Shanghai Fourth People's Hospital," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Tongji University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Shanghai 200434,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' China 6 Department of Obstetrics and Gynecology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Shanghai Fourth People’s Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' School of Life and Sciences and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Tongji University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Shanghai 200434,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' China Correspondence: 22310050@tongji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='cn (Yuxin Yang);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 2180129@tongji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='cn (Jinlong Qin) # These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Abstract Background: The corona virus disease 2019 (COVID-19) pandemic, which is caused by severe acute respiratory syndrome coronavirus 2, is still localized outbreak and has resulted in a high rate of infection and severe disease in older patients with comorbidities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The vitamin D status of the population has been found to be an important factor that could influence outcome of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' However, whether vitamin D can lessen the symptoms or severity of COVID-19 still remains controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Methods: A total of 719 patients with confirmed COVID-19 were enrolled retrospectively in this study from April 13 to June 6, 2022 at Shanghai Forth People’s Hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The circulating levels of 25(OH)D3, inflammatory factors, and clinical parameters were assayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Time to viral RNA clearance (TVRC), classification and prognosis of COVID-19 were used to evaluate the severity of COVID-19 infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Results: The median age was 76 years (interquartile range, IQR, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5-84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6), 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1% of patients were male, and the TVRC was 11 days (IQR, 7-16) in this population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The median level of 25(OH)D3 was 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15 (IQR, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89) nmol/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Patients with lower serum 25(OH)D3 had prolonged time to viral clearance, more obvious inflammatory response, more severe respiratory symptoms and higher risks of impaired hepatic and renal function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Multiple regression analyses revealed that serum 25(OH)D3 level was negatively associated with TVRC independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' ROC curve showed the serum vitamin D level could predict the severity classification and prognosis of COVID-19 significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Conclusions: Serum 25(OH)D3 level is independently associated with the severity of COVID-19 in elderly, and it could be used as a predictor of the severity of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, supplementation with vitamin D might provide beneficial effects in old patients with COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Keywords: COVID-19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' vitamin D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' time to viral RNA clearance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' inflammatory response Introduction As of 30 November 2022, the corona virus disease 2019 (COVID-19) pandemic has resulted in more than 639 million confirmed cases with more than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6 million deaths[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' China has also experienced several waves of COVID-19 pandemic and has explored various strategies to protect the susceptible people from infection, especially the elderly and patients with comorbidities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Therefore, it is of great significance to analyze the risk factors of COVID-19 and investigate inventions to reduce the risks of infection or serious illness for the prevention and treatment of COVID-19 in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition to the injury of alveolar epithelial cells mediated by angiotensin- converting enzyme 2 (ACE2), severe acute respiratory syndrome coronavirus 2 (SARS- CoV-2) could activate macrophages via ACE2 receptors [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The interleukins 1 (IL- 1), IL-6 and tumor necrosis factor (TNF-α) released by the activated macrophage could further stimulate the neutrophils and T lymphocytes, followed by the release of a large number of inflammatory factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' This inflammatory cascade and inflammation storm are considered to be the main pathogenesis of acute respiratory distress syndrome in COVID-19 [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition to regulating calcium and phosphorus metabolism, vitamin D is also closely related to immune regulation, cardiovascular diseases, metabolic syndrome, obesity, diabetes, hypertension, cancer, infection and other diseases [6-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The multiple effects of vitamin D are related to the wide distribution of the vitamin D receptor (VDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' After binding with VDR, the activated vitamin D acts on the cis-acting elements in the promoter of target genes and thus regulating the transcription of the target genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Most immune cells, including dendritic cells, T lymphocytes, and B lymphocytes, have high levels of VDR that could modulate the cellular response to viruses as binding with vitamin D [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, there are expressions of VDR in the lung tissue, which are related with the severity of lung injury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Previous studies showed that mice with VDR knockout had more serious lung injury induced by LPS compared with WT mice [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The histological study showed increased alveolar permeability, pulmonary vascular exudation, neutrophil infiltration and inflammatory factors in the lungs of VDR knockout mice [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Although vitamin D has multiple beneficial effects, the nutritional status of vitamin D in the population is unsatisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' An epidemiological study in more than 40 countries conducted by Lips P et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' found that vitamin D deficiency was present in more than 50% of the population, especially among nursing home residents (mainly elderly) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In Europe, approximately 40% of the population is vitamin D deficient (< 20 nmol/L), as well as in USA (24%) and Canada (37%) [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Previous study showed there was a close relationship between the risk of respiratory tract infection and vitamin D [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Patients with daily or weekly vitamin D supplementation, especially those with 25(OH)D3 < 10 nmol/L, were found to have a reduced risk of respiratory tract infections [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Studies in patients with COVID-19 also demonstrated that vitamin D levels were related with the infectious risk and severity of COVID-19 [18-20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' A retrospective study by Angelidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' found that patients with low serum 25(OH)D3 levels had increased mortality and risk of invasive mechanical ventilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The median 25(OH)D3 level in this population was 28 nmol/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The mortality in patients with 25(OH)D3 < 30 nmol/L was 25%, compared with the 9% mortality in patients with > 30 nmol/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The results were similar when the cutoff value was adjusted as 20 nmol/L [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Although treatment with vitamin D failed to improve the survival in critically ill patients with COVID-19 [21], epidemiological studies showed that patients with high vitamin D levels had low risks of infection with COVID-19 [22], which demonstrated that vitamin D could prevent people from COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Therefore, vitamin D levels have the potential to be a predictor of the risk of infection and the severity of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, it is unclear whether some subgroups of patients would benefit from treatment with vitamin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In this study, we aimed to assess vitamin D levels in patients with COVID-19 infection, and to investigate the relationship between vitamin D levels and time to clearance of virus, the classification and progression of the disease, which might provide evidence for identifying of high-risk patients for COVID-19, and protecting these vulnerable patients from infection and critical illness of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=" Patients and Methods Study Population This is a retrospective cohort study of 719 patients aged 22 to 92 years with confirmed COVID-19 pneumonia hospitalized at the Shanghai Fourth People's Hospital, School of Medicine, Tongji University, Shanghai, China." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' All patients were diagnosed with COVID-19 pneumonia according to World Health Organization interim guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' According to hospital data, patients were admitted from April 13 to June 6, 2022, the final date of follow-up was June 20, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The study was approved by the ethics committee of Shanghai Fourth People’s Hospital (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 2020012) and individual consent for this retrospective analysis was waived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Data Collection The epidemiological, clinical evaluation and outcomes data of all participants during hospitalization were collected from electronic medical records by a trained team of physicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The individual components of clinical outcomes were reviewed independently and recorded into the computer data base by 2 authors (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The clinical outcomes (including the time to viral RNA clearance (TVRC), the classification and progression of COVID-19) were monitored up to June 20, 2022, the final date of follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The Viral Nucleic Acid Kit (Health) was used to extract nucleic acids from clinical throat swab samples obtained from all patients at admission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' A 2019- nCoV detection kit (Bioperfectus) was used to detect the ORF1ab gene (nCovORF1ab) and the Ngene (nCoV-NP) according to the manufacturer’s instructions using real-time reverse transcriptase–polymerase chain reaction (qPCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' COVID-19 infection was considered laboratory-confirmed if both the nCovORF1ab and nCoV-NP tests showed positive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Liver and kidney function, lipids and electrolytes were measured by CH930, Atellica Solution, Siemens, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Cytokines were measured by Deflex, Beckman flow cytometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Blood Routine and C-reactive protein (CRP) were measured by BC7500, Mindray, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Serum 25 hydroxyvitamin D was determined by cobas 8000, Roche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Statistical Analysis All statistical analyses were performed using IBM SPSS Statistics (Version 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0, 147 SPSS, IBM Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=', Armonk, New York, USA), GraphPad Prism 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='2 (GraphPad Software, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=', San Diego CA, USA) and R (version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1, R Foundation for Statistical Computing, Vienna, Austria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Continuous variables were presented as mean ± Standard Deviation (SD) or median (quartile), and categorical variables were summarized as counts (frequency percentages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' χ2 or Fisher exact test (for small cell counts) was applied to compare categorical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' For continuous variables, normal distribution was evaluated with Kolmogorov-Smirnov test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Then One-way ANOVA (if homogeneity of variances was assumed) or Wilcoxon-Mann-Whitney U test (if homogeneity of variances was not met) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Furthermore, receiver operating characteristics (ROC) curves were performed to investigate the value of serum vitamin D level in predicting the severity classification and prognosis of COVID-19 in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' All reported values were two-sided and P < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 was considered as statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Clinical baseline characteristics of enrolled patients A total of 719 patients with confirmed COVID-19 were enrolled retrospectively in this study from April 13 to June 6, 2022 at Shanghai Forth People’s Hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In these patients, the median age was 76 years (interquartile range, IQR, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5-84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6), 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1% of patients were male, and the TVRC was 11 days (IQR, 7-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The median level of 25(OH)D3 was 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15 (IQR, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89) nmol/L in these patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The body mass index (BMI) was 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='59 Kg/m2 in these patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' There were slightly increased levels of mean systolic blood pressure (SBP, 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='85 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 mmHg) and respiratory rate (RR, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39 bpm), but normal levels of mean diastolic blood pressure (DBP, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='85 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84 mmHg), heart rate (HR, 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11 bpm), temperature (Temp, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47℃) and oxygen saturation (SaO2, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The fraction of inspiration O2 (FiO2) was 29 (21- 33) %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The CRP levels were 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39 - 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='9) mg/L, but almost normal levels of white blood cells (WBC, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1×10^9/L), red blood cell (RBC, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='70×10^9/L) and hemoglobin (Hb, 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07g/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, the levels of serum bilirubin (Bil), albumin (Alb), alanine aminotransferase (ALT)，fasting glucose (FBG) and renal function were in normal ranges (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Table 1 Clinical baseline characteristics of enrolled patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Parameter Value Age (years) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0 (64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5, 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6) Sex (M/F) 317/402 TVRC (days) 11 (7, 16) 25(OH)D3 (nmol/L) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31, 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89) BMI (Kg/m2) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='59 CRP (mg/L) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='9) Temp (℃) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47 SBP (mmHg) 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='85 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 DBP (mmHg) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='85 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84 HR (bpm) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11 RR (bpm) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39 SaO2 (%) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54 FiO2 (%) 29 (21, 33) WBC (10^9/L) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1 RBC (10^12/L) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='70 FBG (mmol/L) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='28 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57 Hb (g/L) 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07 T-Bil (mmol/L) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89 ALT (U/L) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87) T-Pro (g/L) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11 Alb (g/L) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64) Pre-Alb (g/L) 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='86 (136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10, 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='50) BUN (mmol/L) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81) Cr (umol/L) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='9 (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1, 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='7) UA (umol/L) 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16 (225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48, 363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44) Cystatin C (mg/mL) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='91, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44) Abbreviations: M: male;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' F: female;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' TVRC: time to viral RNA clearance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' BMI: body mass index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' CRP: C-reaction protein;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Temp: temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' SBP: systolic blood pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' DBP: diastolic blood pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' HR: heart rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' RR: respiration rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' SaO2: oxygen saturation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' FiO2: fraction of inspiration O2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' WBC: white blood corpuscle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' RBC: red blood corpuscle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' FBG: fasting blood glucose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Hb: hemoglobin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' T-Bil: total bilirubin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' ALT: alanine aminotransferase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' T-Pro: total protein;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Alb: albumin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Pre-Alb: prealbumin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' BUN: blood urea nitrogen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Cr: crea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' UA: uric acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Comparison of clinical baseline characteristics and comorbidities among patients with different levels of vitamin D The levels of serum vitamin D were measured in 609 patients with COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Then patients were divided into 4 groups according to the quartile values of serum vitamin D levels: Q1 < 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='59, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56) nmol/L, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='59, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56) nmol/L < Q2 < 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='13) nmol/L, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='13) nmol/L < Q3 < 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='42 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='9, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35) nmol/L, and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='42 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='9, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35) nmol/L < Q4 < 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21, 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29) nmol/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Table 2 showed the clinical baseline characteristics among the four groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Compared with patients with higher levels of serum vitamin D, patients in Q1 group were older, and had more severe illness, which manifested as longer TVRC, lower oxygen saturation, and FiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Patients in the Q1 group also had higher levels of inflammation, which included higher levels of CRP and WBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' There was increased percentage of neutrophil and decreased percentage of monocyte and lymphocyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Patients in Q1 group also had decreased levels of total protein (T-Pro), Alb, and increased levels of lactate dehydrogenase (LDH), which indicated that patients with low vitamin D levels had impaired hepatic synthetical function and nutritional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Although there was no significant difference in the blood urea nitrogen (BUN) and Crea (Cr), patients in the Q1 group had increased levels of cystatin C, a biomarker of early renal injury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, there were decreased levels of serum magnesium in patients with lower levels of vitamin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' However, there was no significant difference in male proportion, BMI, basic vital signs, and other biochemical tests (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The rates of comorbidities were high in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' However, there was no significant difference in comorbidities among patients with different levels of vitamin D (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Table 2 Comparison of clinical and biochemical characteristics and comorbidities among patients with different levels of vitamin D Q1 group (n=162) Q2 group (n=164) Q3 group (n=164) Q4 group (n=165) Age (years) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0 (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0, 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75, 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0)a 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75, 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25)ab 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0 (64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0, 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0)a Sex (M/F) 63/99 71/93 78/86 82/83 TVRC (days) 14 (8, 19) 10 (6, 15)a 10 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25, 15)a 11 (7, 13)a CVD, n (%) 44 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16) 37 (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56) 13 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93) 28 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97) HT, n (%) 98 (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49) 84 (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22) 93 (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71) 89 (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='94) T2DM, n (%) 36 (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22) 36 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95) 48 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='27) 50 (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='30) Tumor, n (%) 17 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49) 15 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15) 15 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15) 14 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48) BMI (Kg/m2) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='41 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='62 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='63 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68 ± 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='66ab 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='60 CRP (mg/L) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='32, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82) a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31) ab 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='33)a Temp (℃) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49 SBP (mmHg) 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03 DBP (mmHg) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='26 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43a 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80a 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='90a HR (bpm) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='41 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='33 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1 RR (bpm) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='63 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='40 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='50 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15 SaO2 (%) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='40 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='78a 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='52 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65a 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='62 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='13a FiO2 (%) 29 (29, 33) 29 (21, 33) a 21 (21, 33) ab 21 (21, 29) ab WBC (10^9/L) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='83 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='12 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='12 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='30a Monocyte % 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='90 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='70a 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31a 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='02 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96 Lymphocyte % 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='36 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='13 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='60 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72a 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='58 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46a 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81ab Neutrophil % 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='30 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84a 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48a 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25a PLT (10^9/L) 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='73 ± 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61 ± 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95 ± 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82 ± 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='62a RBC (10^12/L) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='63a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61a Hct (%) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='19 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87a 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='62a 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='27 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29a Hb (g/L) 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='36 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='06 ± 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16a 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76a 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='58 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93ab T-Bil (umol/L) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='60 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} 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+page_content='40ab 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='18 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='78ab Alb (g/L) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='33a 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16ab 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='02 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='26ab Pre-Alb (g/L) 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='02 (102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='26, 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='19) 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43 (136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11, 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65) a 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49 (162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='13, 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='86) ab 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31 (161, 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61)a BUN (umol/L) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='66 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16) a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49) Cr (umol/L) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='50 (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='85, 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='90) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='3 (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='7, 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='85 (49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95 (49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='3, 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55) UA (umol/L) 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43 (193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='66, 349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31) 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97 (239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='34, 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='91) 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84 (235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='18, 372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76) a 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65 (257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76, 349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96)a Cystatin C (mg/mL) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='26 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38) a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37) a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='33)a Lactate (mmol/L) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='06 FBG (mmol/L) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='63 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='66a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24b 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24ac LDH (U/L) 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 ± 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88 ± 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38a 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97 ± 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08a 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54 ± 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01a K+ (mmol/L) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='59a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='50 Na+ (mmol/L) 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='94 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='92 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55 Cl- (mmol/L) 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='86 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='83 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='36 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61 Ca2+ (mmol/L) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43 ab 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='40 ab Mg2+ (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10a Phosphate (mmol/L) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='28 Abbreviations: CVD: cardiovascular disease;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' HT: hormone therapy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' T2DM: diabetes mellitus type 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' PLT: platelet count;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Hct: red blood cell specific volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' AST: glutamic oxaloacetic transaminase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' AKP: alkaline phosphatase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' LDH: lactate dehydrogenase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' a, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 compared with Q1 group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' b, p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 compared with Q2 group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' c, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 compared with Q3 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Comparison of inflammatory factors among patients with different levels of vitamin D The increased levels of WBC and CRP in patients from Q1 group implicated that patients with lower levels of serum vitamin D might had high inflammatory state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Therefore, we measured the serum levels of inflammatory factors in patients from different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' However, there was no significant difference of inflammatory factors among these groups except for lower levels of interferon-γ (IFN-γ) and TNF-α in patients with lower levels vitamin D (Figure 1, Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Figure 1 Comparison of inflammatory factors among patients with different levels of vitamin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='**, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' ****, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Abbreviations: IL: interleukins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' IFN: interferon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' TNF: tumor necrosis factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Association between serum vitamin D level and the severity of COVID-19 To further assess the association between serum vitamin D level and the severity of COVID-19, patients were first divided into 4 groups according to the quartile of TVRC, which was used as an indicator for the severity of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Patients in the longest TVRC group (TVRC-Q4) had significantly lower serum vitamin D levels (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='19 [IQR, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46-33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77] nmol/L) compared with patients in shorter TVRC groups (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='74 [IQR, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97] nmol/L in TVRC-Q1, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0075;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='02 [IQR, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87-41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03] nmol/L in TVRC-Q2, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='19 [IQR, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44] nmol/L in TVRC-Q3, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0461) (Figure 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Patients were also grouped into mild, moderate, severe and critical groups based on the severity classification of COVID-19 according to the guideline for management of patients with COVID-19 (9th version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' There were significantly lower levels of serum vitamin D in patients with severe (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53 [IQR, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01] nmol/L) and critical (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54 [IQR, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='51-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68] nmol/L) groups compared with patients in the mild (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 [IQR, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='73-42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01] nmol/L) and moderate (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31 [IQR, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98-36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='51] nmol/L) groups (Figure 2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Furthermore, patients were divided into 3 groups based on the prognosis of the disease according to the progression of the disease changes of the severity classification of COVID-19 when the virus RNA was cleared, and the relation between serum vitamin D levels and the prognosis was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Patients with good prognosis had significantly higher levels of serum vitamin D levels (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21 [IQR, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46-40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22] nmol/L) compared with patients with poor prognosis (Prognosis-Q1, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53 [IQR, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44] nmol/L in Prognosis-Q2, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03 [IQR, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56] nmol/L in Prognosis-Q3, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='016) (Figure 2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Figure 2 Association of vitamin D level with TVRC, classification and prognosis of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' (A) Vitamin D levels in each group stratified by TVRC quartile 11 (IQR, 7-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' (B) Vitamin D levels in each group divided by severity classification of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' (C) Vitamin D levels in patients with different progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' A B ** C **** 200 200- *** ** 200- **** 150 25(OH)D3 (nmol/L) 150 25(OH)D3 ( 100 25(OH)D3 ( 100 100 50- 50 50 TVRC-Q4 Mild Moderate Severe Critica ROC curve showed the serum vitamin D level could predict the severity classification and prognosis of COVID-19 significantly (the area under the curve [AUC] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='695, 95% CI [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='627-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='764], p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001, for severe and critical of COVID-19, Figure 3A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='728, 95% CI [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='585-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='872], p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='009, for the aggravation of COVID-19, Figure 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Figure 3 ROC curve to investigate the serum vitamin D level in predicting the severity classification (A) and prognosis (B) of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Abbreviations: AUC: the area under the curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' ROC: receiver operating characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Association between serum vitamin D levels and clinical parameters In univariate analyses, serum vitamin D level was negatively associated with TVRC, age, FiO2, prognosis, IL-10, cystatin C, alkaline phosphatase (AKP), LDH, direct bilirubin (D-Bil), and CRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' However, BMI, SaO2, DBP, Alb, IL-4, TNF-α, serum calcium (Ca) levels, indirect bilirubin (I-Bil), serum magnesium (Mg) level, serum sodium (Na) level, uric acid (UA), pre-albumin (pre-Alb), LDH, Hb, red blood cell specific volume (Hct) and T-Pro were positively associated with serum vitamin D level (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Table 3 Correlation between serum vitamin D and other variables Parameter r p-Value Age (years) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='239 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 TVRC (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 BMI (Kg/m2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='048 CRP (mg/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='196 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 DBP (mmHg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='014 SaO2 (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='016 A B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content="8 8'0 Sensivity 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6 Sensivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6 AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='695 AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='728 p ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content="0 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0 1-Specifity 1-SpecifityFiO2 (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='227 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 WBC (10^9/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='003 Monocyte % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='047 Lymphocyte % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='226 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Neutrophil % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Hct (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='294 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Hb (g/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='298 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 D-Bil (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='009 I-Bil (umol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='017 AKP (U/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='035 T-Pro (g/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Alb (g/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='405 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Pre-Alb (g/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 UA (umol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Cystatin C (umol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='191 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 LDH (U/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='151 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Na+ (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='027 Ca2+ (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='343 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Mg2+ (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='015 Phosphate (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='211 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 IL-10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='007 IL-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01 TNF-α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='202 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 DSS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='242 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Prognosis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='194 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Abbreviations: D-Bil: direct Bilirubin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' I-Bil: indirect bilirubin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' DSS: disease severity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Association between TVRC and clinical parameters Spearman correlation coefficients were used to evaluate correlations between TVRC and clinical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The results showed that serum vitamin D level, BMI, HR, ALB, TNF-α, serum calcium level, serum sodium level, serum phosphorus level, serum chlorine level, uric acid, pre-Alb, T-Pro, Hb, hematocrit (Hct) and RBC were negatively associated with TVRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, age, prognosis, IL-10, Il-12, IL-17, IL- 2, WBC, CRP, Alb, LDH, ALT, glutamic oxaloacetic transaminase (AST), alkaline phosphatase (AKP), BUN, cystatin C, creatinine, and serum potassium level were positively associated with TVRC (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Multiple regression analyses revealed that only serum vitamin D level was negatively associated with TVRC independently (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Correlation between TVRC and other variables Variables Beta coefficient p-Value Age (years) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='253 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 25(OH)D3 (nmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 BMI (Kg/m2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='164 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 CRP (mg/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='157 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 HR (bpm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='047 FiO2 (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='242 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 RBC (10^12/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='185 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 WBC (10^9/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='016 Lymphocyte % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='159 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Neutrophil % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='158 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Hct (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='167 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Hb (g/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='186 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 ALT (U/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='048 AST (U/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03 AKP (U/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 r-GT (U/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 T-Pro (g/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='167 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Alb (g/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='287 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Pre-Alb (g/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 BUN (umol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='244 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 UA (umol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='026 Cystatin C (mg/mL) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='191 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 LDH (U/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='002 Cr (umol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='002 K+ (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='207 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Na+ (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='204 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Cl- (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='004 Ca2+ (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='003 Phosphate (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='229 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 IL-10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='025 IL-12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04 IL-17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 IL-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 IL-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 TNF-α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='014 DSS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='235 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Prognosis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='178 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='001 Abbreviations: r-GT: γ-glutamyl transpeptidase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Multivariate regression analyses of predictors of TVRC in patients with COVID-19 Variables Beta coefficient p-Value 95% CI 25(OH)D3 (nmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='168 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='018 Discussion As a kind of steroid hormone, vitamin D is tightly linked to a number of different metabolic processes and immune regulation in the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Vitamin D activates the innate immune system by binding with VDR in immune cells to defend the invasion of foreign pathogenic microorganisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' For example, 1,25 dihydroxyvitamin D3 (1,25- (OH)2-D3) could induce the generation of antimicrobial peptides in monocytes to clean the Mycobacterium tuberculosis[23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 1,25-(OH)2-D3 also could tune the cellular and humoral immunity by regulating the differentiation and proliferation of T and B lymphocytes and the secretion of Th1/Th2 cytokines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, 1,25-(OH)2-D3 could inhibit the exaggerated inflammatory response via inducing the differentiation of regulatory T cells (Treg), and have protective effects in inflammatory responses and autoimmune diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Considering that 25(OH)D3 is the main form of vitamin D in the body and its stable concentration in circulation, serum 25(OH)D3 was used as an indicator to evaluate the nutritional status of vitamin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Presently, vitamin D deficiency, insufficiency, normal, and sufficiency are defined as <25, 25 to 50, 51 to 75, and > 75nmol/L, respectively[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Vitamin D deficiency was defined when the serum level of 25(OH)D3 was less than 50nmol/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' An epidemiological study in East China showed that the serum levels of 25(OH)D3 were 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5 nmol/L in the normal population, and 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='3% of the population were vitamin D deficiency[26], which was significantly higher than that in western countries[27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, a study in elderly inpatients showed that the vitamin D levels were 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='2 nmol/L in the population, of which 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5% were severely deficient, 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0% were mildly deficient, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5% were insufficient, and only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='0% were sufficient[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' These data suggest that vitamin D deficiency may be common in the Han population, especially in the elderly and bedridden patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' This study enrolled 719 patients with COVID-19 and assessed the levels of serum vitamin D, cytokines and other clinical indicators to investigate the relationship between vitamin D levels and TVRC, the classification and prognosis of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Higher levels of vitamin D were associated with the higher levels of T-Pro, Alb, pre- Alb, hemoglobin and BMI, which indicated that higher vitamin D levels were associated with better protein synthesis ability of liver and better nutritional status of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Conversely, the lower levels of vitamin D were associated with the longer TVRC, higher levels of WBC and CRP, as well as worse oxygenation capacity of the lung, suggesting that lower vitamin D was associated with severe conditions in these patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Meanwhile, lower levels of vitamin D were related with the biomarkers of early hepatic and renal function impairments, such as lower levels of pre-Alb and higher levels of cystatin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, there was positive relationship between vitamin D levels and serum calcium and phosphorus concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' All these results demonstrated that vitamin D had benefit effects on the clearance of the virus and alleviating the condition in patients with COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Further investigation validated that lower vitamin D levels were associated with longer TVRC, more severe disease and worse prognosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Therefore, serum vitamin D level is a predictor of the severity of disease and prognosis in patients with COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Previous studies have shown that the risks of severe infection and mortality were increased in vulnerable groups (with comorbidities such as diabetes, hypertension, coronary artery disease and tumors) in patients with COVID-19[29-32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In this study, we compared the comorbidities in patients with different vitamin D levels, and found no significant difference in comorbidities among different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' The results indicated the association between vitamin D levels and the prognosis of the disease was less affected by these chronic comorbidities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Further investigation showed that serum vitamin D level was correlated with TVRC negatively, and serum vitamin D level was an independent predictor of TVRC in patients with COVID-19, which further validated the close relationship between vitamin D and the severity and prognosis of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Vitamin D deficiency is a common phenomenon in Chinese, especially in the elderly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' For its detrimental effects on the immune system, vitamin D deficiency would impair the clearance of invasive pathogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' This concern is more obvious under the current situation of the panic of COVID-19 and consistent virus variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Therefore, it is of great significance to investigate how to protect patients with high risk from infection and improve the prognosis of these patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Supplement with vitamin D routinely in patients with COVID-19 is still in debate presently[33-35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' However, the results of this study demonstrated that early supplement with vitamin D in patients with COVID-19 and vitamin D deficiency could improve the ability of defensing the infection of SARS-CoV-2, promoting the clearance of virus and improving the prognosis in these high-risk patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' However, for the limitation of the observed study, further prospective randomized controlled trails were needed to investigate the benefits of supplement of vitamin D in these patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Limitations There are several limitations of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Although our study implied that early supplemented with vitamin D in patients with COVID-19 and vitamin D deficiency might improve the prognosis of these patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' However, we did not give the therapy with vitamin D in this population in this retrospective study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, this retrospective study has some disadvantages compared with prospective studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Therefore, further prospective studies are needed to validate the clinical value of serum vitamin D levels in risk stratifications of patients with COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Conclusion This study demonstrated that serum 25(OH)D3 level was independently associated with the severity of COVID-19 in elderly, and it could be used as a predictor of the severity of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' In addition, supplementation with vitamin D might provide beneficial effects in old patients with COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Sources of Funding This work was supported by Shanghai Committee of Science and Technology, China (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 22dz1202304 to Jiajing Cheng).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Author Contributions Conceptualization, Ruyi Qu, Yuxin Yang and Jinlong Qin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Data curation, Ruyi Qu, Qiuji Yang and Yingying Bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Formal analysis, Ruyi Qu and Jinlong Qin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Funding acquisition, Jiajing Cheng;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Inves-tigation, Jiajing Cheng, Mengna He, Xin Wei and Yiqi Yuan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Methodology, Ruyi Qu, Qiuji Yang, Yingying Bi, Jiajing Cheng, Yuxin Yang and Jinlong Qin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Project administration, Jinlong Qin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Re-sources, Jiajing Cheng;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Software, Yingying Bi and Xin Wei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Supervision, Yuxin Yang and Jinlong Qin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Validation, Qiuji Yang and Yingying Bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Visualization, Yingying Bi, Mengna He and Yiqi Yuan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Writing – original draft, Ruyi Qu, Qiuji Yang and Yuxin Yang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Writing – review & editing, Yuxin Yang and Jinlong Qin.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Cell Metab, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 31(6): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 1068-1077 e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Pranata, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Gibbons, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=', Association between vitamin D supplementation and COVID-19 infection and mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Sci Rep, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 12(1): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 19397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Jolliffe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=', Effect of a test-and-treat approach to vitamin D supplementation on risk of all cause acute respiratory tract infection and covid-19: phase 3 randomised controlled trial (CORONAVIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' BMJ, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 378: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' e071230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Bychinin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=', Effect of vitamin D3 supplementation on cellular immunity and inflammatory markers in COVID-19 patients admitted to the ICU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Sci Rep, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 12(1): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' 18604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Supplemental data Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Comparison of inflammatory factors among patients with different levels of vitamin D Q1 group (n=150) Q2 group (n=153) Q3 group (n=157) Q4 group (n=149) IL-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07) IL-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88) IL-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='90) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='78(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='6(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77) IL-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1) IL-6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44(21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22,204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='63(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65,228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57) 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='85(26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='4,312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='4) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84(19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37,288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68) IL-8 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77(32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25,229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='92) 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='2(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96,255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95) 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='3(44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53,242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43) 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='06(32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='42,244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='3) IL-10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='79,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='32) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='28(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='51,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='26(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='85(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45)a IL-12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='62) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='4) IL-17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='36,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='91(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='26,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='33,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='67) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='32,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31) IFN-α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='1) IFN-γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='27(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='79) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56,5)a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='4,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37)a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='42,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='34) TNF-α 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='32(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='12) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='91,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04)a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='52,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='74)a 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='78(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='19)ab Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Comparison of clinical and biochemical characteristics among patients with different severity rating Q1 group (n=330) Q2 group (n=309) Q3 group (n=71) Q4 group (n=9) Age (years) 68(59,78) 81(71,88) a 86(77,90) a 87(79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5,91) a Sex (M/F) 149/181 130/179 34/37 4/5 TVRC (days) 9(6,13) 12(8,17) a 13(9,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25) a 12(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75) 25(OH)D3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='73,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31(17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98,36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='51) a 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53(12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01) ab 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54(8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='51,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68) ab BMI (Kg/m2) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='42±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='59 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='74±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='73±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='18 NA CRP (mg/L) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='33,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='27(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='91) a 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65(19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='13,76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31) ab 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08(56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11,155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47) ab Temp (℃) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='50 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='41 SBP (mmHg) 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='06±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53±21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38±21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00±21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47 DBP (mmHg) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='42±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='06 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='02 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='60 HR (bpm) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='33±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='58±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14 RR (bpm) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='41±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='58±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='78±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='99 SaO2 (%) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22 ab 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88 abc Fio2 (%) 21(21,29) 29(21,33) a 33(33,41) ab 61(29,141) ab RBC (10^12/L) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='99±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72 a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='74 ab 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71 abc WBC (10^9/L) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='58 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03 a 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='41±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93 ab 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='51 abc Monocyte % 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='27±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='94 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='92 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25 Lymphocyte % 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='32±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64 a 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='59±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56 ab 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='78±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00 abc Neutrophil % 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49 a 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61 ab 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49 abc PLT (10^9/L) 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96±65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='99 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='42±86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68±94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='33±159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25 Hct (%) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='91±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='27 a 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81 ab 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56 abc Hb (g/L) 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='60±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43 a 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37±26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='52 ab 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56±21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07 ab T-Bil (umol/L) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='73±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='94 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98±16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='83 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87 ALT (U/L) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='52,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='91,31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='17,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='83) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='02(14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='79,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='27) AST (U/L) 22.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76) a 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24(34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38) a AKP (U/L) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10(63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65,95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='12) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77(69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='13,99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64(67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='78,102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='90) 84.' 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+page_content='73,319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98) ab 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97(148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49,377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04) a Cystatin C (mg/mL) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='86,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56) a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='06,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88) ab 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='60(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55) a Lactate (mmol/L) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23 FBG (mmol/L) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88 a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00 ab 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25 abc LDH (U/L) 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='28±48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='28 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='18±70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80 a 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00±90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07 ab 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20±107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77 abc K (mmol/L) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='66 Na(mmol/L) 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='19±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='27 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='28 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09 Cl (mmol/L) 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='52 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='94±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='33±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35 ab Ca (mmol/L) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49 ab 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44 ab Mg (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 a P (mmol/L) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='36 ab 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37 abc Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content=' Comparison of clinical and biochemical characteristics among patients with different prognosis P1 group (n=638) P2 group (n=70) P3 group (n=11) Age (years) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5(64,86) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5(78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75,90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25) a 81(66,89) Sex (M/F) 277/361 34/36 6/5 TVRC (days) 10(7,15) 14(9,23) a 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='5(14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75,22) a BMI (Kg/m2) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='17±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='60 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='79±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='70 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='92±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98 25(OH)D3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21(20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53(12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44) a 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96,21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56) a CRP (mg/L) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='98(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03,93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='56) a 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08(17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='99,105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47) a Temp (℃) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='41 SBP (mmHg) 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='61 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='04±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='86 DBP (mmHg) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='91±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 HR (bpm) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97 RR (bpm) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='99±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='36 a 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='86 SaO2 (%) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='49±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='52 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87 a 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55 Fio2 (%) 29(21,33) 33(29,41) a 41(37,53) a RBC (10^12/L) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80 a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64 WBC (10^9/L) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='84 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54 a 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='12±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57 a Monocyte % 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='94 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='83 a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='14 a Lymphocyte % 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='83±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 a 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01 a Neutrophil % 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='58 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='69±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='44 a 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39 a PLT (10^9/L) 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='06±78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16±93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='12 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82±133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25 Hct (%) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='90 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='32 a 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11 Hb (g/L) 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='41±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='79±23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48 a 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00±39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87 b T-Bil (umol/L) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='83±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='02 a 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='66±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80 ALT (U/L) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='79,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='68(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='30,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71) a 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22) a AST (U/L) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='58(18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='86,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='92(20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='30) a 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='85(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03) AKP (U/L) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='99(65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='62,95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07(69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46,113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='97) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64(65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='92,98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15) T-Pro (g/L) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='21±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47 a 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 a Alb (g/L) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='72±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='53 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47 a 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='74±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='76 a Pre-Alb (g/L) 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='22(147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24,232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20) 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75(81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='27,146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45) a 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31(82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='19,197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='08) BUN (umol/L) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='51,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='32(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='28,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='94) a 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='41(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20) a Cr (umol/L) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='80(48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20,72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='48) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15(41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='40,90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='65) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20(32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10,129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='40) UA (umol/L) 291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='73(230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='62,365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='03) 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='16(171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31,363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57) 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='57(79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='31,240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='75) ab Cystatin C (mg/mL) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='05(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='38) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='23,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15) a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='35) Lactate (mmol/L) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='24±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='89±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09 a FBG (mmol/L) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='00±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='67 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='07±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='41 a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='43 a LDH (U/L) 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='50±58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='94 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11±97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='54 a 334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='63±115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='47 ab K (mmol/L) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='71 a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='62 Na(mmol/L) 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='20±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='30 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='64±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='25 a 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='82±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='58 Cl (mmol/L) 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='46±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='77 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='91±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='60 Ca (mmol/L) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='50 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39 ab Mg (mmol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='11 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='10 P (mmol/L) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='37 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfygJ9/content/2301.02660v1.pdf'} +page_content='63±0.' 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stabilization +CLÉMENT GOÏCOECHÉA,1,2 THOMAS BILLOTTE,2 MATTHIEU CHAFER,1,2 +MARTIN MAUREL,1,2 JENNY JOUIN,3 PHILIPPE THOMAS,3 DEVANG NAIK,2 +FRÉDÉRIC GÉRÔME,1,2 BENOÎT DEBORD,1,2 AND FETAH BENABID1,2,* +1GLOphotonics SAS, 123 avenue Albert Thomas, 87060 Limoges, France +2GPPMM group, Xlim Research Institute, CNRS UMR 7252, University of Limoges, Limoges, France +3IRCER UMR CNRS 7315, Centre Européen de la Céramique, 12 rue Atlantis, 87068 Limoges, France +*f.benabid@xlim.fr +Abstract: We report on the development of all-fiber stand-alone Iodine-filled Photonic +Microcells demonstrating record absorption contrast at room temperature. The microcell’s fiber +is made of inhibited coupling guiding hollow-core photonic crystal fibers. The fiber-core +loading with Iodine was undertaken at 10-1-10-2mbar vapor pressure using a novel gas-manifold +based on metallic vacuum parts with ceramic coated inner surfaces for corrosion resistance. +The fiber is then sealed on the tips and mounted on FC/APC connectors for better integration +with standard fiber components. The stand-alone microcells display Doppler lines with +contrasts up to 73% in the 633nm wavelength range, and an insertion loss between 3 to 4dB. +Sub-Doppler spectroscopy based on saturable absorption has been carried out to resolve the +hyperfine structure of the P(33)6-3 lines at room temperature with a full-width at half maximum +of 24MHz on the b4 component with the help of lock-in amplification. Also, we demonstrate +distinguishable hyperfine components on the R(39)6-3 line at room temperature without any +recourse to signal-to-noise ratio amplification techniques. + +1. Introduction +In the last 20 years, efforts have been made towards the miniaturization of frequency standards +with the emergence of new technological devices, such as clocks based on +microelectromechanical systems (MEMS) [1], planar devices mounted on silicon chip, +using hollow-core anti-resonant reflecting optical waveguides (ARROW) [2], or compact +engineered circular multi-pass cells [3]. Amongst these devices, the hollow-core photonic +crystal fiber (HCPCF) technology appeared to be an excellent and promising alternative to +bulky cells for portable, small footprint applications by filling and sealing atomic or molecular +vapor inside its core [4], culminating in the form of the photonic microcell (PMC) an atom- +photonics component that can be integrated with small insertion loss while using existing fiber +connectors into any optical set-up [5]. By confining the atoms/molecules alongside light over +modal areas of as small as a few μm2, whilst keeping them in interaction over length scales a +million times longer than the Rayleigh range, the resulting enhanced atom-laser interaction +efficiency leads to strong absorption despite low gas density and low light-level, resulting in +an increased signal to noise ratio compared to other technologies. The unique optical properties +of HCPCF offer a large range of core sizes and lengths coupled with low loss, alongside +reduced power consumption and micro-structured cladding providing versatile modes +composition allowing transverse light structuring [6]. + +HCPCFs light guiding performances are continuously improving on a broad spectral range with +loss figures competing with standard solid-core fibers on telecom spectral ranges with loss +down to 0.174dB/km in the C-band [7]. Low-loss figures are also accessible in the visible range +down to 0.9dB/km at 558nm [8]. +Since the advent of the PMC as a photonic component, a plethora of work has been carried out +on the types of enclosed gas and the sealing techniques. The evolution of the different +fabrication techniques of PMC has been mainly dictated by the type of fiber as gas-container +and by the nature of the gas. In fact, compatibility of the solid-core single mode fiber (SMF) +splicing technique with photonic bandgap (PBG) fibers [9,10] with its associated core diameter +range of 5-20µm is no longer viable with inhibited-coupling guiding HCPCF (IC-HCPCF) +because of their larger core sizes (typically between 20 and 100 µm), and thus implies a strong +mode mismatch induced loss. Therefore, mode-field adapters have been introduced to render +IC guiding fiber technology compatible with SMF, either by tapering HCPCF as reported by +Wheeler et al. [11] or by implementing graded-index fibers [12,13]. Another configuration has +been also reported based on glass cells glued on the tips of PBG HCPCF to manage gas filling +and proceed to gas-enclosing by collapsing a section of the glass cell [14]. All the different +techniques mentioned above suffer from drawbacks linked to the use of glue or exposition to +ambient air leading to gas contamination and degradation of PMC spectroscopic performances +on the long term. Recently, Billotte et al. introduced a novel PMC assembly process that no +longer requires helium buffer gas or gluing stage [5]. Based on a glass sleeve collapse, a 1.5dB +insertion loss microcell has been achieved with a 7m long IC-HCPCF filled with acetylene at +80µbar pressure. Doppler-free spectroscopy showed constant linewidth and contrast over more +than 3 months, thus highlighting the quality/purity of the sealed-gas medium and the impact of +this contaminant-free technique for the creation of next-generation PMCs. +The above work on PMC assembly was chiefly motivated for frequency standards, which thus +far was limited to molecular gases, such as Acetylene (C2H2, 1550nm region) [4,5,11] or +Carbon Dioxide (CO2) [15–17]. The actual state-of-the-art for a sealed PMC is set by Triches +et al. and Billotte et al., each exhibiting an instability around 2.10-11 @1000s [14,18]. Both are +working with acetylene around the telecom wavelength band. +Extending PMC technology to atomic or molecular vapors, such as alkali vapors or molecular +halogen gas (such as iodine (I2)), have been very challenging because of their physio-chemical +reactivity and the required vacuum environment. This in turn limited the impact of PMC based +optical frequency references to a restricted number of spectral ranges. For example, in the green +and red spectral range, I2 is known for displaying a very dense Doppler-lines spectrum useful +for the development of visible broadband frequency references [19,20]. This is illustrated by +the fact that several I2 ro-vibrational absorption lines are among Bureau International des Poids +et Mesures (BIPM) recommended frequency references for the realization of the meter [21]. +Also, I2 based frequency stabilization [22–26] proved to be valuable for several technological +applications [27–30]. Consequently, the advent of iodine filled PMC (I2-PMC) would be +extremely useful for applications like guiding star lasers [31], high resolution LIDAR [32], or +laser frequency stabilization [26], which requires these performances to be delivered in a +compact and mobile physical package. However, encapsulating iodine vapor into glass cells +carries specific challenges because of I2 physical and chemical properties. In particular, its high + +reactivity and corrosion with metals [33,34] requires the use of complex glass-manifold filling +system. Consequently, the development of I2-PMC represents a technical challenge both in +vapor handling, loading and in-HCPCF sealing. Indeed, the use of common metallic vacuum +parts is inadequate for the I2-PMC assembly process, and the commonly used glass manifold +alternative for I2 cell manufacturing is too complex and expensive for scalable I2-PMC +fabrication. Within this context, we note the previous work on I2-filled HCPCF [24,35,36]. +Lurie et al. have shown that the hollow-core fiber for I2-microcell development and saturated +absorption spectroscopy applications demonstrated a strong efficiency thanks to strong overlap +of pump and probe beams [35], and laser stabilization with fractional frequency stability of +2.3×10-12 at 1s for a HCPCF mounted on a glass vacuum manifold was achieved [24]. Impact +of residual gases in the vacuum system alongside I2 pressure have been raised by the authors +as obstacles to reach transit limited sub-Doppler linewidths. This seminal work within the I2- +filled HCPCF framework has been followed up in 2015 by the first demonstration of a +hermetically sealed I2 Kagome HCPCF [36]. However, despite a demonstration of laser +stabilization with fractional frequency stabilities of 3.10-11 at 100s, the sealed HCPCF has been +marred by a strong 21.5dB insertion loss and cannot be coined PMC because the gas sealing +was achieved by fusion-collapsing a section of the fiber, thus eliminating its optical guidance +features at the collapsed section. +We report in this work on an I2-PMC development based on a scalable, corrosion resistant and +contamination free fabrication process [5]. For example, a 2.5 and 4 meter long patch-cord like +iodine PMC based on tubular IC-HCPCF [37] and an overall transmission efficiency as high +as 40% have been fabricated. These PMCs exhibit, at room temperature, high performances in +term of absorption contrast reaching respectively 60% and 53% on the P(33)6-3 transition - at +632.991nm [38]. Furthermore, we demonstrate room temperature resolution of Iodine +hyperfine spectrum observation, over 10GHz spectral range around 633nm, of several sub- +Doppler spectral transparencies from different broadened lines, thanks to saturable absorption. +The exceptional lifetime of these microcells is demonstrated through the unaffected P(33)6-3 +absorption contrast and Doppler linewidth of 4 year-old I2-PMC. +2. Experimental set-up for I2-PMC fabrication +An IC-HCPCF based on tubular lattice cladding has been designed and fabricated for optimal +guidance on several hundred thousand hyperfine transitions of iodine in the green-to-red +spectral range (Fig. 1(a)) with loss below 30dB/km level between 530nm and 668nm. The fiber +(Fig. 1(a)) with an outer diameter of 200µm exhibits a core diameter of 30µm surrounded by 8 +isolated tubes cladding. This fiber presents an excellent modal behavior with quasi single-mode +guidance as illustrated by the near-field intensity profile at 633nm (see Fig. 1(a)) measured at +the output of a 4m long fiber. + + +Fig. 1. (a) Measured loss spectrum of the experimental IC tubular fiber related to the +developed PMCs (see Fig. 2). On the right : micrograph picture of the fiber cross section and +near field intensity distribution at 633nm. (b) Overview of the experimental set-up for I2-filling +and fiber-sealing. The vacuum system is represented in gray with valves and the gauge +represented by yellow circles. The fiber is represented in dark blue. PBS: polarizing beam +splitter. Lock-in system is represented in blue color. AOM : acousto-optic modulator. (c) +Schematic of set-up for PMC characterization. +Two PMCs, PMC#1 and PMC#2, have been fabricated 3 years apart in 2017 and 2020 +respectively from similar fibers using an in-house gas-vacuum manifold, represented +schematically in Fig. 1(b) and purposely designed for I2 loading into HCPCF and for I2-PMC +assembly. +The manifold is composed of three main compartments separated by vacuum valves. The +central part holds the HCPCF and acts as the fiber loading section. On the right side of the fiber +loading section, a turbo-molecular vacuum-pump is connected via a cryogenic trap to prevent +any contamination to the vacuum-pump during I2 releasing. The left side corresponds to the I2 +dispenser. Here, iodine chips are placed in glass test-tube, which is hermetically connected to +the manifold via a metallic fitting. This section is under pressure and/or temperature regulation +for I2 sublimation and release into the fiber loading section. The fiber loading goes through the +following sequence. First, the fiber loading section is evacuated to a vacuum pressure of less +than 10-6mbar. Similarly, the iodine dispenser section is evacuated while ensuring the I2 +remains solid by regulating the iodine chip temperature. The above ensures the leakproofness +of the manifold and high vacuum quality. Once this process is achieved, the I2 is sublimated by +increasing the temperature until a pressure of around 10-1 mbar is reached. Second, the I2 vapor +is released in the fiber loading section by opening the valve between the two sections and + +W +Pump +WWclosing the valve to the vacuum pump section. During this loading process, we continuously +monitor the fiber transmission spectrum derived from a tunable external cavity diode laser +(TOPTICA DL-PRO, 631-635nm range) tuned to a frequency range corresponding to one of +the iodine rovibrational lines. A second beam from the same laser is sent to an Iodine +macroscopic gas cell and serves as a reference. The spectroscopic signature of the Iodine +absorption lines (spanning over a set of 3 lines around the P(33)6-3 transition) can be observed +after 10 minutes of loading. The loading is then kept on until the desired contrast is reached. +It is noteworthy, that the metallic parts of the whole manifold have been post-processed against +corrosion and chemical reaction with iodine by applying a ceramic coating on the inner metallic +surfaces. This allows an outstanding ease-of-use with several HCPCFs being loaded and +assembled over several years. +Before mounting and splicing the HCPCF (described in Fig. 1(a)), it was flushed with Helium +or Argon gas and heated for several hours in the oven at ~100°C to reduce any residual gas +inside the fiber. The fiber is then end-capped on one extremity by collapsing a borosilicate +capillary with an inner diameter fitting the outer diameter of the fiber, following the process +mentioned in [5]. The sealed and polished extremity is then mounted on a FC/APC optical +connector with a measured coupling loss in the range of 1 to 1.5dB at 633nm, 20dB lower than +the splicing loss obtained by collapsing the fiber on itself in [36]. +The second extremity of the HCPCF is connected by borosilicate sleeve fusion splicing to a +30cm piece of the same HCPCF. The second tip of 30cm long HCPCF is hermetically attached +to the loading compartment of the manifold via a home-made fiber-feedthrough (Fig. 1(a) of +[5]). The end-capped fiber is then evacuated by pumping the valve-controlled middle chamber +of the vacuum system down to the range of 10-6 mbar. Once the desired contrast is reached +(here around 60%), we hermetically seal the fiber by end-capping with a splicing machine +based on sleeve collapse around the tip of the HCPCF, as described in [5]. The tips of the +resulted PMC are then polished and mounted on FC/APC fiber connectors. Figure 2(a) shows +the photography of a typical FC/APC connectorized patch-cord like PMC in its final form. +Figure 2(b) shows the reconstructed near-field intensity profile of the transmitted light from +the developed I2-PMC. The measured transmission was in the range of 40-50%, corresponding +to an insertion loss of less than 4 dB, which is 17.5dB lower than the one measured in [36]. +Figure 2(d) represents the normalized transmission spectra at the output of PMC#1 (red curve), +PMC#2 (orange curve) and the commercial macroscopic cell (black curve) measured +consecutively with the same laser configurations and room temperature conditions. The bottom +axis and the top axis of the graph give the frequency and the relative frequency from that of the +R(39)6-3 transition, respectively. The two PMCs exhibit contrast between 53% to 60% for the +P(33)6-3 line and 61% to 73% for the R(39)6-3. This is respectfully 5.9 to 6.7 and 5.1 to 6.1 +times larger than the ones obtained with the 10cm long commercial I2 macroscopic gas cell +(resp. 9% & 12%) corresponding to 1.3.10-1 – 1.6.101 mbar of I2 vapor pressure specifications +given by manufacturer. These contrasts are 2.2 to 2.9 times larger than the one obtained at room +temperature in previously reported work [36], which is to our knowledge the only reported +work on low-loss I2-loaded sealed HCPCF. + + +Fig. 2. (a) Photography of I2 PMC#1 mounted on FC/APC connectors. (b) Measured near field +intensity profile at 633nm at the output of PMC#1. (c) Contrast evolution of the PMC#2 over 2 +years. (d) Normalized transmission spectra through the fabricated PMCs (PMC#1 in red color, +PMC#2 in orange) and through a macroscopic commercial gas cell (black). +Finally, comparison of the shown transmission spectra with those recorded at the time of PMC +sealing and more recently on October 2022 (see Fig.2(c)) shows comparable contrasts. In fact, +evolution of the contrast of P(33)6-3 line through PMC#2 has been studied along 2 years since +its encapsulation at t0. The different measurements are summarized on table from Fig. 2(c). The +measured contrast of the P(33)6-3 line of I2 was found to remain constant within a range of +8,5% around the extrema average value of 61%. Observed fluctuations have been attributed +to the different temperature conditions and laser diode ampereage setpoint, and corroborated +by an additional study using another PMC (based on the same fiber and fabrication process). +The result has shown that by considering the extreme experimental values of room temperature +(i.e. from 19 to 22.5°C) and laser diode ampereage, these two major contributions of contrast +change can lead to a variation of 9%. +The stability of the absorption contrast highlights the leak proofness of the PMCs, and the +reliability and repeatability of the developed process. +3. Sub-Doppler spectroscopy with I2 PMC +The fabricated PMCs have shown their potential for sub-Doppler spectroscopy through the +resolution of the hyperfine structure of the P(33)6-3 line. To do so, Saturated Absorption + +PMC#2ContrastevolutionofP(33)6-3 +Deviation from +Acquisitiontime +Contrast +average (%) +to +0.691 +10.6 +to+5months +0.527 +15.7 +to+10 months +0.658 +5.3Spectroscopy (SAS) measurements have been done following the set-up shown in Fig. 1(c). +The laser beam is separated into counter-propagating pump and probe beams with 4mW and +40µW output power, respectively. The pump beam is obtained from the first order diffracted +beam off an acousto-optic modulator (AOM) operating at 64MHz frequency. This was +motivated so to avoid interference between the probe and back-reflected pump during the +propagation in the fiber. The spectroscopic transparency signal is obtained by redirecting the +PMC-transmitted probe beam on to a photodiode with the help of a polarizing cube. Half and +quarter waveplates are used to improve both the optical PMC transmission and the intensity of +the redirected probe beam on the photodetector. + + +Fig. 3. (a) Example of Iodine ro-vibrational hyperfine energy levels. This structure is usually +not observable through a simple macroscopic cell at room temperature. (b) Measured R(39)6- +3 Doppler line at room temperature through PMC#1. (c) Hyperfine components structure of +the P(33)6-3 Doppler line obtained with a lock-in amplifier detection scheme. Measured peaks +(red) are compared with tabulated values components for P(33)6-3 & R(127)11-15 +(respectively in blue and green). Data have been fitted with lorentzian multi-peak fit function. +Figure 3(b) shows the probe signal when the laser frequency is tuned in the vicinity of R(39)6- +3 line and recorded directly by the photodetector at room temperature. In addition of the +Doppler-broadened absorption line, the trace shows the 21 hyperfine b-lines [39]. To our +knowledge, this is the first time that such Iodine transparencies are observed on a cell without +any means of signal-to-noise ratio (SNR) post-acquisition amplification such as lock-in + +[(b) +0.24 +Signal input (a.u.) +0.26 +0.28 +-0.30 +-25-20-15-10 +-5 +0 +5 +10 +5 +20 +25 +Acquisitiontime(ms) +PMC#1 Lock-inamplifier output (a.u.) +6 +PMC#1Lock-inamplifieroutput +Lorentzianmulti-fitpeak +P(33)6-3bcomponents(BIPMdatabase) +CumulativeFitPeak +R(127)11-15acomponents(BIPMdatabase) +D +4 +2 +-1000 +-900 +-800 +-700 +-600 +-500 +-400 +-300 +-200 +-100 +0 +100 +RelativefrequencyfromP(33)6-3b21component(MHz)detection. In order to improve the hyperfine structure resolution we used a lock-in amplification +detection scheme with a squared-modulated pump beam of 1MHz (amplitude modulation). The +spectrum obtained as output of the lock-in amplifier is shown in Fig. 3(c). The 21 hyperfine b- +components of the P(33)6-3 line (“b” energy level scheme shown in Fig.3(a)) have been +identified and are in good agreement with the optical frequencies tabulated by the BIPM in blue +[38]. One can notice some shift and/or additional peak, such as between b11 and b18, that could +be explained by the overlap between P(33)6-3 and R(127)11-5 absorption lines of I2. Hence, +additional weaker peaks could come from the SAS of R(127)11-5 line. +Figure 4 shows the b4 hyperfine component of the P(33)6-3 line, previously displayed in Fig. +3(c), on which a Lorentzian fitting shows a FWHM of 21MHz for 4mW pump beam. +Contribution of broadening sources such as wall collisions and the natural linewidth can be +directly calculated [4] leading to 2.95MHz and 3.23MHz respectively (lifetime about 310ns +[40,41]). The laser linewidth provided by the manufacturer is about 0.20MHz. Considering the +pressure of I2 inside the PMC of around 10-2mbar, we can estimate a few 40kHz [42] +intermolecular collision broadening. Therefore, the minimum linewidth obtainable using this +setup is 6.42MHz. The larger measured linewidth of 21MHz is explained by power broadening +coming from the pump beam intensity of 10MW/m², 357 times bigger than saturation intensity +of 28kW/m² [43]. + +Fig. 4. Zoom-in on b4 component of the hyperfine structure displayed in Fig. 3(c). Lock-in +signal is displayed in red line. A 21MHz FWHM Lorentzian curve has been fitted in black +line. + + + + +3.0 +PMC#1Lock-inamplifieroutput +Lorentzianmulti-fitpeak +PMC#1 Lock-in amplifier output (a.u.) +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +-710 +-700 +-690 +-680 +-670 +-660 +-650 +Relative frequency from P(33)6-3 b21 component (MHz)4. Conclusion +As a summary, we reported on the first fabrication of meter-long low-optical loss pure I2 PMCs +based on a new process for creating all-fibered stand-alone PMCs. Absorption contrasts up to +73% have been measured at room temperature with PMC insertion loss of 4dB, 5.4 times lower +than the state-of-the-art. The good sealing quality is demonstrated by a PMC being still +functional after 4 years and the stable absorption measured throughout both PMCs, as well as +the performance of Doppler-free signal measurements for the first PMC on the P(33)6-3 line +of I2 at room temperature. The b4 component of this line shows a FWHM of 24MHz at 4mW +output pump beam power. By calculating the different broadening sources, we identify the +dominant one as power broadening. An optimization of I2 pressure, PMC length and pump +power should allow us to reduce this FWHM while keeping the same SA contrast. These results +are very promising for many compact sensing applications and laser stabilization. +Funding: Région Nouvelle-Aquitaine. +Disclosures: The authors declare no conflicts of interest. +References +1. +S. Knappe, V. Shah, P. D. D. Schwindt, L. 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Meas. 46(2), 186–190 (1997). + + diff --git a/EtE4T4oBgHgl3EQfGgz2/content/tmp_files/load_file.txt b/EtE4T4oBgHgl3EQfGgz2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b148127be65cf420d10b8974145641cc44d34c99 --- /dev/null +++ b/EtE4T4oBgHgl3EQfGgz2/content/tmp_files/load_file.txt @@ -0,0 +1,746 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf,len=745 +page_content='Fabrication and characterization of iodine photonic microcell for sub-Doppler spectroscopy and laser stabilization CLÉMENT GOÏCOECHÉA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2 THOMAS BILLOTTE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2 MATTHIEU CHAFER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2 MARTIN MAUREL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2 JENNY JOUIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='3 PHILIPPE THOMAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='3 DEVANG NAIK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2 FRÉDÉRIC GÉRÔME,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2 BENOÎT DEBORD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2 AND FETAH BENABID1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='* 1GLOphotonics SAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 123 avenue Albert Thomas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 87060 Limoges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' France 2GPPMM group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Xlim Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' CNRS UMR 7252,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' University of Limoges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Limoges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' France 3IRCER UMR CNRS 7315,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Centre Européen de la Céramique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 12 rue Atlantis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 87068 Limoges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' France f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='benabid@xlim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='fr Abstract: We report on the development of all-fiber stand-alone Iodine-filled Photonic Microcells demonstrating record absorption contrast at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The microcell’s fiber is made of inhibited coupling guiding hollow-core photonic crystal fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The fiber-core loading with Iodine was undertaken at 10-1-10-2mbar vapor pressure using a novel gas-manifold based on metallic vacuum parts with ceramic coated inner surfaces for corrosion resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The fiber is then sealed on the tips and mounted on FC/APC connectors for better integration with standard fiber components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The stand-alone microcells display Doppler lines with contrasts up to 73% in the 633nm wavelength range, and an insertion loss between 3 to 4dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Sub-Doppler spectroscopy based on saturable absorption has been carried out to resolve the hyperfine structure of the P(33)6-3 lines at room temperature with a full-width at half maximum of 24MHz on the b4 component with the help of lock-in amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Also, we demonstrate distinguishable hyperfine components on the R(39)6-3 line at room temperature without any recourse to signal-to-noise ratio amplification techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Introduction In the last 20 years, efforts have been made towards the miniaturization of frequency standards with the emergence of new technological devices, such as clocks based on microelectromechanical systems (MEMS) [1], planar devices mounted on silicon chip, using hollow-core anti-resonant reflecting optical waveguides (ARROW) [2], or compact engineered circular multi-pass cells [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Amongst these devices, the hollow-core photonic crystal fiber (HCPCF) technology appeared to be an excellent and promising alternative to bulky cells for portable, small footprint applications by filling and sealing atomic or molecular vapor inside its core [4], culminating in the form of the photonic microcell (PMC) an atom- photonics component that can be integrated with small insertion loss while using existing fiber connectors into any optical set-up [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' By confining the atoms/molecules alongside light over modal areas of as small as a few μm2, whilst keeping them in interaction over length scales a million times longer than the Rayleigh range, the resulting enhanced atom-laser interaction efficiency leads to strong absorption despite low gas density and low light-level, resulting in an increased signal to noise ratio compared to other technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The unique optical properties of HCPCF offer a large range of core sizes and lengths coupled with low loss, alongside reduced power consumption and micro-structured cladding providing versatile modes composition allowing transverse light structuring [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' HCPCFs light guiding performances are continuously improving on a broad spectral range with loss figures competing with standard solid-core fibers on telecom spectral ranges with loss down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='174dB/km in the C-band [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Low-loss figures are also accessible in the visible range down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='9dB/km at 558nm [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Since the advent of the PMC as a photonic component, a plethora of work has been carried out on the types of enclosed gas and the sealing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The evolution of the different fabrication techniques of PMC has been mainly dictated by the type of fiber as gas-container and by the nature of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' In fact, compatibility of the solid-core single mode fiber (SMF) splicing technique with photonic bandgap (PBG) fibers [9,10] with its associated core diameter range of 5-20µm is no longer viable with inhibited-coupling guiding HCPCF (IC-HCPCF) because of their larger core sizes (typically between 20 and 100 µm), and thus implies a strong mode mismatch induced loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Therefore, mode-field adapters have been introduced to render IC guiding fiber technology compatible with SMF, either by tapering HCPCF as reported by Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' [11] or by implementing graded-index fibers [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Another configuration has been also reported based on glass cells glued on the tips of PBG HCPCF to manage gas filling and proceed to gas-enclosing by collapsing a section of the glass cell [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' All the different techniques mentioned above suffer from drawbacks linked to the use of glue or exposition to ambient air leading to gas contamination and degradation of PMC spectroscopic performances on the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Recently, Billotte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' introduced a novel PMC assembly process that no longer requires helium buffer gas or gluing stage [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Based on a glass sleeve collapse, a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='5dB insertion loss microcell has been achieved with a 7m long IC-HCPCF filled with acetylene at 80µbar pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Doppler-free spectroscopy showed constant linewidth and contrast over more than 3 months, thus highlighting the quality/purity of the sealed-gas medium and the impact of this contaminant-free technique for the creation of next-generation PMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The above work on PMC assembly was chiefly motivated for frequency standards, which thus far was limited to molecular gases, such as Acetylene (C2H2, 1550nm region) [4,5,11] or Carbon Dioxide (CO2) [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The actual state-of-the-art for a sealed PMC is set by Triches et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' and Billotte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=', each exhibiting an instability around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='10-11 @1000s [14,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Both are working with acetylene around the telecom wavelength band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Extending PMC technology to atomic or molecular vapors, such as alkali vapors or molecular halogen gas (such as iodine (I2)), have been very challenging because of their physio-chemical reactivity and the required vacuum environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' This in turn limited the impact of PMC based optical frequency references to a restricted number of spectral ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' For example, in the green and red spectral range, I2 is known for displaying a very dense Doppler-lines spectrum useful for the development of visible broadband frequency references [19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' This is illustrated by the fact that several I2 ro-vibrational absorption lines are among Bureau International des Poids et Mesures (BIPM) recommended frequency references for the realization of the meter [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Also, I2 based frequency stabilization [22–26] proved to be valuable for several technological applications [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Consequently, the advent of iodine filled PMC (I2-PMC) would be extremely useful for applications like guiding star lasers [31], high resolution LIDAR [32], or laser frequency stabilization [26], which requires these performances to be delivered in a compact and mobile physical package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' However, encapsulating iodine vapor into glass cells carries specific challenges because of I2 physical and chemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' In particular, its high reactivity and corrosion with metals [33,34] requires the use of complex glass-manifold filling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Consequently, the development of I2-PMC represents a technical challenge both in vapor handling, loading and in-HCPCF sealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Indeed, the use of common metallic vacuum parts is inadequate for the I2-PMC assembly process, and the commonly used glass manifold alternative for I2 cell manufacturing is too complex and expensive for scalable I2-PMC fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Within this context, we note the previous work on I2-filled HCPCF [24,35,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Lurie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' have shown that the hollow-core fiber for I2-microcell development and saturated absorption spectroscopy applications demonstrated a strong efficiency thanks to strong overlap of pump and probe beams [35], and laser stabilization with fractional frequency stability of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='3×10-12 at 1s for a HCPCF mounted on a glass vacuum manifold was achieved [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Impact of residual gases in the vacuum system alongside I2 pressure have been raised by the authors as obstacles to reach transit limited sub-Doppler linewidths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' This seminal work within the I2- filled HCPCF framework has been followed up in 2015 by the first demonstration of a hermetically sealed I2 Kagome HCPCF [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' However, despite a demonstration of laser stabilization with fractional frequency stabilities of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='10-11 at 100s, the sealed HCPCF has been marred by a strong 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='5dB insertion loss and cannot be coined PMC because the gas sealing was achieved by fusion-collapsing a section of the fiber, thus eliminating its optical guidance features at the collapsed section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' We report in this work on an I2-PMC development based on a scalable, corrosion resistant and contamination free fabrication process [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' For example, a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='5 and 4 meter long patch-cord like iodine PMC based on tubular IC-HCPCF [37] and an overall transmission efficiency as high as 40% have been fabricated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' These PMCs exhibit, at room temperature, high performances in term of absorption contrast reaching respectively 60% and 53% on the P(33)6-3 transition - at 632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='991nm [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Furthermore, we demonstrate room temperature resolution of Iodine hyperfine spectrum observation, over 10GHz spectral range around 633nm, of several sub- Doppler spectral transparencies from different broadened lines, thanks to saturable absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The exceptional lifetime of these microcells is demonstrated through the unaffected P(33)6-3 absorption contrast and Doppler linewidth of 4 year-old I2-PMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Experimental set-up for I2-PMC fabrication An IC-HCPCF based on tubular lattice cladding has been designed and fabricated for optimal guidance on several hundred thousand hyperfine transitions of iodine in the green-to-red spectral range (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 1(a)) with loss below 30dB/km level between 530nm and 668nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The fiber (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 1(a)) with an outer diameter of 200µm exhibits a core diameter of 30µm surrounded by 8 isolated tubes cladding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' This fiber presents an excellent modal behavior with quasi single-mode guidance as illustrated by the near-field intensity profile at 633nm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 1(a)) measured at the output of a 4m long fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (a) Measured loss spectrum of the experimental IC tubular fiber related to the developed PMCs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' On the right : micrograph picture of the fiber cross section and near field intensity distribution at 633nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (b) Overview of the experimental set-up for I2-filling and fiber-sealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The vacuum system is represented in gray with valves and the gauge represented by yellow circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The fiber is represented in dark blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' PBS: polarizing beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Lock-in system is represented in blue color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' AOM : acousto-optic modulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (c) Schematic of set-up for PMC characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Two PMCs, PMC#1 and PMC#2, have been fabricated 3 years apart in 2017 and 2020 respectively from similar fibers using an in-house gas-vacuum manifold, represented schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 1(b) and purposely designed for I2 loading into HCPCF and for I2-PMC assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The manifold is composed of three main compartments separated by vacuum valves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The central part holds the HCPCF and acts as the fiber loading section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' On the right side of the fiber loading section, a turbo-molecular vacuum-pump is connected via a cryogenic trap to prevent any contamination to the vacuum-pump during I2 releasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The left side corresponds to the I2 dispenser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Here, iodine chips are placed in glass test-tube, which is hermetically connected to the manifold via a metallic fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' This section is under pressure and/or temperature regulation for I2 sublimation and release into the fiber loading section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The fiber loading goes through the following sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' First, the fiber loading section is evacuated to a vacuum pressure of less than 10-6mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Similarly, the iodine dispenser section is evacuated while ensuring the I2 remains solid by regulating the iodine chip temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The above ensures the leakproofness of the manifold and high vacuum quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Once this process is achieved, the I2 is sublimated by increasing the temperature until a pressure of around 10-1 mbar is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Second, the I2 vapor is released in the fiber loading section by opening the valve between the two sections and W Pump WWclosing the valve to the vacuum pump section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' During this loading process, we continuously monitor the fiber transmission spectrum derived from a tunable external cavity diode laser (TOPTICA DL-PRO, 631-635nm range) tuned to a frequency range corresponding to one of the iodine rovibrational lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' A second beam from the same laser is sent to an Iodine macroscopic gas cell and serves as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The spectroscopic signature of the Iodine absorption lines (spanning over a set of 3 lines around the P(33)6-3 transition) can be observed after 10 minutes of loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The loading is then kept on until the desired contrast is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' It is noteworthy, that the metallic parts of the whole manifold have been post-processed against corrosion and chemical reaction with iodine by applying a ceramic coating on the inner metallic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' This allows an outstanding ease-of-use with several HCPCFs being loaded and assembled over several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Before mounting and splicing the HCPCF (described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 1(a)), it was flushed with Helium or Argon gas and heated for several hours in the oven at ~100°C to reduce any residual gas inside the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The fiber is then end-capped on one extremity by collapsing a borosilicate capillary with an inner diameter fitting the outer diameter of the fiber, following the process mentioned in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The sealed and polished extremity is then mounted on a FC/APC optical connector with a measured coupling loss in the range of 1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='5dB at 633nm, 20dB lower than the splicing loss obtained by collapsing the fiber on itself in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The second extremity of the HCPCF is connected by borosilicate sleeve fusion splicing to a 30cm piece of the same HCPCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The second tip of 30cm long HCPCF is hermetically attached to the loading compartment of the manifold via a home-made fiber-feedthrough (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 1(a) of [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The end-capped fiber is then evacuated by pumping the valve-controlled middle chamber of the vacuum system down to the range of 10-6 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Once the desired contrast is reached (here around 60%), we hermetically seal the fiber by end-capping with a splicing machine based on sleeve collapse around the tip of the HCPCF, as described in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The tips of the resulted PMC are then polished and mounted on FC/APC fiber connectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Figure 2(a) shows the photography of a typical FC/APC connectorized patch-cord like PMC in its final form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Figure 2(b) shows the reconstructed near-field intensity profile of the transmitted light from the developed I2-PMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The measured transmission was in the range of 40-50%, corresponding to an insertion loss of less than 4 dB, which is 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='5dB lower than the one measured in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Figure 2(d) represents the normalized transmission spectra at the output of PMC#1 (red curve), PMC#2 (orange curve) and the commercial macroscopic cell (black curve) measured consecutively with the same laser configurations and room temperature conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The bottom axis and the top axis of the graph give the frequency and the relative frequency from that of the R(39)6-3 transition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The two PMCs exhibit contrast between 53% to 60% for the P(33)6-3 line and 61% to 73% for the R(39)6-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' This is respectfully 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='9 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='7 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='1 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='1 times larger than the ones obtained with the 10cm long commercial I2 macroscopic gas cell (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 9% & 12%) corresponding to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='10-1 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='101 mbar of I2 vapor pressure specifications given by manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' These contrasts are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='9 times larger than the one obtained at room temperature in previously reported work [36], which is to our knowledge the only reported work on low-loss I2-loaded sealed HCPCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (a) Photography of I2 PMC#1 mounted on FC/APC connectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (b) Measured near field intensity profile at 633nm at the output of PMC#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (c) Contrast evolution of the PMC#2 over 2 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (d) Normalized transmission spectra through the fabricated PMCs (PMC#1 in red color, PMC#2 in orange) and through a macroscopic commercial gas cell (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Finally, comparison of the shown transmission spectra with those recorded at the time of PMC sealing and more recently on October 2022 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='2(c)) shows comparable contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' In fact, evolution of the contrast of P(33)6-3 line through PMC#2 has been studied along 2 years since its encapsulation at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The different measurements are summarized on table from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The measured contrast of the P(33)6-3 line of I2 was found to remain constant within a range of \uf0b18,5% around the extrema average value of 61%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Observed fluctuations have been attributed to the different temperature conditions and laser diode ampereage setpoint, and corroborated by an additional study using another PMC (based on the same fiber and fabrication process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The result has shown that by considering the extreme experimental values of room temperature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' from 19 to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='5°C) and laser diode ampereage, these two major contributions of contrast change can lead to a variation of \uf0b19%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The stability of the absorption contrast highlights the leak proofness of the PMCs, and the reliability and repeatability of the developed process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Sub-Doppler spectroscopy with I2 PMC The fabricated PMCs have shown their potential for sub-Doppler spectroscopy through the resolution of the hyperfine structure of the P(33)6-3 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' To do so, Saturated Absorption PMC#2ContrastevolutionofP(33)6-3 Deviation from Acquisitiontime Contrast average (%) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='691 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='6 to+5months 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='527 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='7 to+10 months 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='658 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='3Spectroscopy (SAS) measurements have been done following the set-up shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The laser beam is separated into counter-propagating pump and probe beams with 4mW and 40µW output power, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The pump beam is obtained from the first order diffracted beam off an acousto-optic modulator (AOM) operating at 64MHz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' This was motivated so to avoid interference between the probe and back-reflected pump during the propagation in the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The spectroscopic transparency signal is obtained by redirecting the PMC-transmitted probe beam on to a photodiode with the help of a polarizing cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Half and quarter waveplates are used to improve both the optical PMC transmission and the intensity of the redirected probe beam on the photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (a) Example of Iodine ro-vibrational hyperfine energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' This structure is usually not observable through a simple macroscopic cell at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (b) Measured R(39)6- 3 Doppler line at room temperature through PMC#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' (c) Hyperfine components structure of the P(33)6-3 Doppler line obtained with a lock-in amplifier detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Measured peaks (red) are compared with tabulated values components for P(33)6-3 & R(127)11-15 (respectively in blue and green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Data have been fitted with lorentzian multi-peak fit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Figure 3(b) shows the probe signal when the laser frequency is tuned in the vicinity of R(39)6- 3 line and recorded directly by the photodetector at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' In addition of the Doppler-broadened absorption line, the trace shows the 21 hyperfine b-lines [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' To our knowledge, this is the first time that such Iodine transparencies are observed on a cell without any means of signal-to-noise ratio (SNR) post-acquisition amplification such as lock-in [(b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='24 Signal input (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='30 25-20-15-10 5 0 5 10 5 20 25 Acquisitiontime(ms) PMC#1 Lock-inamplifier output (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=') 6 PMC#1Lock-inamplifieroutput Lorentzianmulti-fitpeak P(33)6-3bcomponents(BIPMdatabase) CumulativeFitPeak R(127)11-15acomponents(BIPMdatabase) D 4 2 1000 900 800 700 600 500 400 300 200 100 0 100 RelativefrequencyfromP(33)6-3b21component(MHz)detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' In order to improve the hyperfine structure resolution we used a lock-in amplification detection scheme with a squared-modulated pump beam of 1MHz (amplitude modulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The spectrum obtained as output of the lock-in amplifier is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The 21 hyperfine b- components of the P(33)6-3 line (“b” energy level scheme shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='3(a)) have been identified and are in good agreement with the optical frequencies tabulated by the BIPM in blue [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' One can notice some shift and/or additional peak, such as between b11 and b18, that could be explained by the overlap between P(33)6-3 and R(127)11-5 absorption lines of I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Hence, additional weaker peaks could come from the SAS of R(127)11-5 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Figure 4 shows the b4 hyperfine component of the P(33)6-3 line, previously displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 3(c), on which a Lorentzian fitting shows a FWHM of 21MHz for 4mW pump beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Contribution of broadening sources such as wall collisions and the natural linewidth can be directly calculated [4] leading to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='95MHz and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='23MHz respectively (lifetime about 310ns [40,41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The laser linewidth provided by the manufacturer is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='20MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Considering the pressure of I2 inside the PMC of around 10-2mbar, we can estimate a few 40kHz [42] intermolecular collision broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Therefore, the minimum linewidth obtainable using this setup is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='42MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The larger measured linewidth of 21MHz is explained by power broadening coming from the pump beam intensity of 10MW/m², 357 times bigger than saturation intensity of 28kW/m² [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Zoom-in on b4 component of the hyperfine structure displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Lock-in signal is displayed in red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' A 21MHz FWHM Lorentzian curve has been fitted in black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='0 PMC#1Lock-inamplifieroutput Lorentzianmulti-fitpeak PMC#1 Lock-in amplifier output (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='0 710 700 690 680 670 660 650 Relative frequency from P(33)6-3 b21 component (MHz)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Conclusion As a summary, we reported on the first fabrication of meter-long low-optical loss pure I2 PMCs based on a new process for creating all-fibered stand-alone PMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Absorption contrasts up to 73% have been measured at room temperature with PMC insertion loss of 4dB, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content='4 times lower than the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The good sealing quality is demonstrated by a PMC being still functional after 4 years and the stable absorption measured throughout both PMCs, as well as the performance of Doppler-free signal measurements for the first PMC on the P(33)6-3 line of I2 at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' The b4 component of this line shows a FWHM of 24MHz at 4mW output pump beam power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' By calculating the different broadening sources, we identify the dominant one as power broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' An optimization of I2 pressure, PMC length and pump power should allow us to reduce this FWHM while keeping the same SA contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' These results are very promising for many compact sensing applications and laser stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Funding: Région Nouvelle-Aquitaine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Disclosures: The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Knappe, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' Shah, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE4T4oBgHgl3EQfGgz2/content/2301.04896v1.pdf'} 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BURITY, CLETO B. MIRANDA-NETO, AND ZAQUEU RAMOS +Abstract. Free divisors form a celebrated class of hypersurfaces which has been extensively studied +in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors +and investigate central aspects of the blowup algebras of their Jacobian ideals. +For instance, for +all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay – a desirable +feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the +Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property +with tools ranging from cohomology to asymptotic depth. In addition, as an application, we give an +ideal-theoretic homological criterion for homaloidal divisors, i.e., hypersurfaces whose polar maps are +birational. +Dedicated with gratitude to the memory of Professor Wolmer V. Vasconcelos, +mentor of generations of commutative algebraists. +Introduction +The well-studied theory of free divisors – or free hypersurfaces – has its roots in the seminal work +of K. Saito [35], and in subsequent papers of H. Terao [43, 44, 45] mostly concerned with the case +of hyperplane arrangements. The original environment was the complex analytic setting, and the +motivation was the computation of Gauss-Manin connections for the universal unfolding of an isolated +singularity; for instance, it was proved that the discriminant in the parameter space of the universal +unfolding is a free divisor. Over time, different approaches, viewpoints, and interests have emerged, +including algebraic (and algebro-geometric) adaptations and even generalizations that have drawn +the attention of an increasing number of researchers over the last fifteen years. The list of references +is huge; see, e.g., Abe [1], Abe, Terao and Yoshinaga [2], Buchweitz and Conca [8], Buchweitz and +Mond [9], Calder´on-Moreno and Narv´aez-Macarro [12], Damon [15], Dimca [17, 18], Dimca and +Sticlaru [20], Miranda-Neto [31, 32], Schenck [37], Schenck, Terao and Yoshinaga [38], Schenck and +Tohˇaneanu [39], Simis and Tohˇaneanu [41], Tohˇaneanu [47], and Yoshinaga [51]. In particular, nice +references containing interesting open problems on the subject (including the celebrated Terao’s +Conjecture) are Dimca’s book [17] and Schenck’s survey [37]. +In the present paper, the general goal is to present progress on the algebraic side of the theme, +by means of various techniques. +First, we explicitly describe four new families of homogeneous +free divisors in standard graded polynomial rings over a field k with char k = 0. +Second, and +in the same graded setup, we turn our angle to investigating blowup algebras of Jacobian ideals of +polynomials. More precisely, we prove that for our families the Rees algebra is Cohen-Macaulay – i.e., +from a geometric point of view, blowing-up their singular loci yields arithmetically Cohen-Macaulay +schemes. Furthermore we characterize in various ways, via tools varying from (local) cohomology to +asymptotic depth, the maximality of the dimension of the special fiber ring for polynomials which +are no longer required to be free. The relevance of the latter lies in connections to the important +2010 Mathematics Subject Classification. Primary: 14J70, 32S05, 13A30, 14E05, 14M05; Secondary: 13C15, 13H10, +14E07, 32S22, 32S25. +Key words and phrases. Free divisor, Jacobian ideal, blowup algebra, Rees algebra, analytic spread, homaloidal +divisor. +Corresponding author: Cleto B. Miranda-Neto (cleto@mat.ufpb.br). +1 + +2 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +theory of homaloidal divisors, i.e., homogeneous polynomials f ∈ k[x1, . . . , xn] (where, typically, the +field k is assumed to be algebraically closed) for which the associated polar map +Pf = +� ∂f +∂x1 +: · · · : +∂f +∂xn +� +: Pn−1 ��� Pn−1 +is birational – i.e., Pf is a Cremona transformation. In fact we provide, as an application, an ideal- +theoretic (also homological) homaloidness criterion. It is worth mentioning that the modern theory +about such polynomials began in Ein and Shepherd-Barron [23], where it was proved for instance +that the relative invariant of a regular prehomogeneous complex vector space is homaloidal. Another +classical reference is Dolgachev [21]. +In order to introduce the other main concepts of interest to this paper, let k denote a field of +characteristic zero and, given n ≥ 3, let R = k[x1, . . . , xn] be a standard graded polynomial ring +over k. +Let R+ = (x1, . . . , xn) be the irrelevant ideal. +Given a non-zero reduced homogeneous +polynomial f ∈ R2 ++ (whose partial derivatives will be assumed, to avoid pathologies, to be k-linearly +independent), it is well-known that the property of f being a free divisor can be translated into saying +that the corresponding Jacobian ideal Jf = (∂f/∂x1, . . . , ∂f/∂x1) ⊂ R is perfect of codimension 2 +– in particular, a free f must be highly singular in the sense that codim Sing V (f) = 2 regardless +of n. +Therefore, the intervention of Jf in the theory is (naturally) crucial, and as a bonus this +allows for an interesting link to the study of blowup algebras, particularly the traditional problem +of describing ideals for which such rings are Cohen-Macaulay. Here, we are especially interested in +the Rees algebra +R(Jf) = R +� ∂f +∂x1 +t, . . . , ∂f +∂xn +t +� +⊂ R[t] +and its special fiber ring F(Jf) = R(Jf) ⊗R k, which, as is well-known from blowup theory, encode +relevant geometric information. Recall that the analytic spread of Jf, denoted ℓ(Jf), is the Krull +dimension of F(Jf), which is bounded above by n. Saying that Jf has maximal analytic spread +means ℓ(Jf) = n. +Next we briefly describe the contents of each section of the paper. +Section 1 invokes the definitions that are central to this paper, such as the notions of free divisor +and blowup algebras of ideals, as well as a few auxiliary facts which will be used in some parts of +the paper. Also, some conventions are established. +In Section 2 we present our first family of free divisors in R, with n ≥ 4. They are reducible and, +in fact, linear in the sense that in addition the Jacobian ideal Jf is linearly presented, i.e., the entries +of the corresponding Hilbert-Burch matrix are (possibly zero) linear forms. We also determine the +defining equations of F(Jf) and compute the analytic spread as well as the reduction number of Jf. +Moreover, we prove that R(Jf) and F(Jf) are Cohen-Macaulay. +Section 3 describes our second family of free divisors, in an even number of at least 4 variables, and +again reducible and linear in the above sense. We exhibit a well-structured minimal set of generators +for the module of syzygies of Jf. In addition, as in the previous family – but via different methods +– we show that R(Jf) and F(Jf) are Cohen-Macaulay (the latter is in fact shown to be a generic +determinantal ring) and determine the analytic spread and the reduction number of Jf. +In Section 4, our third family is presented as a two-parameter family of (no longer linear) free +divisors f = fα,β in 3 variables and of degree αβ, where α, β ≥ 2. For the one-parameter family with +β = 2 (also for (α, β) = (2, 3)), we show that R(Jf) is Cohen-Macaulay and derive that F(Jf) is +isomorphic to a polynomial ring over k (so that Jf has reduction number zero). Also we prove that +f is reducible if k = C, and in case k ⊆ R we verify that f is reducible if β is odd and irreducible +otherwise. In addition, if β ≥ 3 is odd, we show how to derive yet another two-parameter family of +free divisors g = gα,β (of degree αβ − α) from fα,β; for the one-parameter family with β = 3, we +deduce that R(Jg) is Cohen-Macaulay. + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +3 +In Section 5 we introduce our fourth family of free divisors, in 3 variables. Such (reducible) divisors +have the linearity property – as in the two first families – and are constructed as the determinant of +the Jacobian matrix of a set of quadrics which we associate to 3 given suitable linear forms. This +family is in fact partially new, because if k = C then its members are recovered by the well-known +classification of linear free divisors in at most 4 variables, whereas on the other hand our (permanent) +assumption on k is that char k = 0. Furthermore, we show that Jf is of linear type – i.e., the canonical +epimorphism from the symmetric algebra of Jf onto R(Jf) is an isomorphism – and we derive that +R(Jf) is a complete intersection ring. +For f ∈ R belonging to any of the four (or five) families, we use a result from Miranda-Neto [31] +to easily determine the Castelnuovo-Mumford regularity of the graded module Derk(R/(f)) formed +by the k-derivations of the ring R/(f). Needless to say, the regularity is an important invariant +which controls the complexity of a module (being related to bounds on the degrees of syzygies), +whereas the derivation module is a classical object as it collects the tangent vector fields defined on +the hypersurface V (f) ⊂ Pn−1 +k +. +We close the paper with Section 6, where we address the question as to when, for a (not necessarily +free) polynomial f ∈ R, the ideal Jf has maximal analytic spread – the relevance of this task is the +already mentioned connection to the theory of homaloidal divisors. We provide a number of charac- +terizations of when such maximality holds, including cohomological conditions on a suitable auxiliary +module as well as the asymptotic depth associated to both adic and integral closure filtrations of Jf. +We also point out that the main problems we raise in this paper appear in this section. This includes +a conjecture predicting that if f is linear free divisor satisfying ℓ(Jf) = n then Jf is of linear type +(the case of interest is n ≥ 5), as well as the question of whether the reduced Hessian determinant of +a homaloidal polynomial must necessarily be a (linear) free divisor. For all such problems we were +motivated and guided by several examples, and the computations were performed with the aid of +the program Macaulay of Bayer and Stillman [5]. +1. Preliminaries: Free divisors and blowup algebras +We begin by invoking some definitions and auxiliary facts. +First we establish the convention +that, throughout the entire paper, k denotes a field of characteristic zero. A few other conventions +(including notations) will be made in this section. +Let R = k[x1, . . . , xn] be a standard graded +polynomial ring in n ≥ 3 indeterminates x1, . . . , xn over k, and let R+ = (x1, . . . , xn) denote the +homogeneous maximal ideal of R. +1.1. Free divisors. Fix a non-zero homogeneous polynomial f ∈ R2 ++. A logarithmic derivation of +f is an operator θ = �n +i=1 gi∂/∂xi, for homogeneous polynomials g1, . . . , gn ∈ R satisfying +θ(f) = +n +� +i=1 +gi +∂f +∂xi +∈ (f). +Geometrically, θ can be interpreted as a vector field defined on Pn−1 +k +that is tangent along the +(smooth part of the) hypersurface V (f). From now on we suppose f is reduced in the usual sense +that fred = f, that is, f is (at most) a product of distinct irreducible factors. In addition, we assume +throughout – with no further mention – that the partial derivatives of f are k-linearly independent +so as to prevent f from being a cone (recall that a polynomial g ∈ R is a cone if, after some linear +change of coordinates, g depends on at most n − 1 variables). Denote by TR/k(f) the R-module +formed by the logarithmic derivations of f, which is also called tangential idealizer (or Saito-Terao +module) of f, and commonly denoted Derlog(−V (f)). It is easy to see that TR/k(f) has (generic) +rank n as an R-module. +Definition 1.1. f is a free divisor if the R-module TR/k(f) is free. + +4 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +This concept, which originated in [35], has been shown to be of great significance to a variety of +branches in mathematics. We recall yet another classical object. +Definition 1.2. If fxi := ∂f/∂xi, i = 1, . . . , n, then the Jacobian ideal of f (also called gradient +ideal of f) is given by +Jf = (fx1, . . . , fxn) ⊂ R. +Note that, because f is not a cone, the ideal Jf is minimally generated by the n partial derivatives +of f. Also recall that the Euler derivation +εn := +n +� +i=1 +xi +∂ +∂xi +is logarithmic for the homogeneous polynomial f by virtue of the well-known Euler’s identity +�n +i=1 xifxi = (deg f)f. +Now we remark that, since TR/k(f) decomposes into the direct sum of +the module of syzygies of Jf and the cyclic module Rεn (see [31, Lemma 2.2]), a free basis of TR/k(f) +if f is a free divisor consists of the derivations corresponding to the columns of a minimal syzygy +matrix of fx1, . . . , fxn together with εn. +Next we recall a useful characterization which is even adopted as the definition of free divisor by +some authors, and moreover highlights the central role that commutative algebra plays in the theory. +Lemma 1.3. ([31, Lemma 4.1]) f ∈ R is a free divisor if and only if Jf is a codimension 2 perfect +ideal (equivalently, the ideal Jf has projective dimension 1). +In other words, f is a free divisor if and only if R/Jf is a Cohen-Macaulay ring and ht Jf = 2, +where, here and in the entire paper, ht I stands for the height of an ideal I ⊂ R. It follows that the +classical Hilbert-Burch theorem plays a major role in the algebraic side of free divisor theory. It is +also worth mentioning that this fruitful interplay holds in a more general setting (see [32]). +Below we invoke a well-known and very useful criterion of freeness detected by Saito himself in +case k = C, but which is known to hold over any field of characteristic zero (see [8, Theorem 2.4]). +Lemma 1.4. ([35, Theorem 1.8(ii)], also [17, Theorem 8.1]) f ∈ R is a free divisor if and only if +there exist n vector fields θ1, . . . , θn ∈ TR/k(f) such that +det [θj(xi)]i,j=1,...,n = λf +for some non-zero λ ∈ k. In this case, the set {θ1, . . . , θn} is a free basis of TR/k(f). +As already pointed out, up to elementary operations in the columns of an n × (n − 1) syzygy +matrix ϕ of Jf, the derivations θj’s of the free basis above correspond to the columns of ϕ along with +the Euler vector field εn. +There is also the following important subclass introduced in [9]. +Definition 1.5. f is a linear free divisor if f is a free divisor and the ideal Jf is linearly presented. +Stated differently, f is a linear free divisor if and only if Jf admits a minimal graded R-free +resolution of the form +0 −→ R(−n)n−1 −→ R(−n + 1)n −→ Jf −→ 0. +In particular, the degree of a linear free divisor is necessarily equal to n (and thus has minimal +degree, since any free divisor is seen to have degree at least n). +Now let us provisionally consider a more general setup. Let S be any Noetherian commutative +ring containing k. A k-derivation of S is defined as an additive map ϑ: S → S which vanishes on +k and satisfies Leibniz’ rule: ϑ(uv) = uϑ(v) + vϑ(u), for all u, v ∈ S. Such objects are collected +in an S-module, denoted Derk(S). In particular, if again R = k[x1, . . . , xn], we get the R-module +Derk(R), which is free on the ∂/∂xi’s. +Now if f ∈ R2 ++ is as above, we can also consider the + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +5 +derivation module Derk(R/(f)), which can be graded as follows. +First, assume that Derk(R) is +given the grading inherited from the natural Z-grading of the Weyl algebra of R, so that each ∂/∂xi +has degree −1. We endow TR/k(f) with the induced grading from Derk(R), that is, a logarithmic +derivation �n +i=1 gi∂/∂xi ∈ TR/k(f) has degree δ if g1, . . . , gn ∈ R have degree δ + 1. For example, +εn ∈ [TR/k(f)]0. Finally recall that there is an identification (see, e.g., [31, Lemma 2.1]) +Derk(R/(f)) = TR/k(f)/fDerk(R). +Then we let Derk(R/(f)) be graded with the grading induced from TR/k(f) by means of this quotient. +The next auxiliary lemma is concerned with the graded module Derk(R/(f)). Let us first recall +the concept of Castelnuovo-Mumford regularity of a finitely generated graded module E over the +graded polynomial ring R. Let 0 → Fp → . . . → F0 → E → 0 be a minimal graded R-free resolution +of E, where Fi := �bi +j=1 R(−ai,j), i = 0, . . . , p. Note that p is the projective dimension of E. +Definition 1.6. If mi := max{ai,j | 1 ≤ j ≤ bi}, i = 0, . . . , p, then the Castelnuovo-Mumford +regularity of E is defined as reg E = max{mi − i | 0 ≤ i ≤ p}. +This gives in some sense a numerical measure of the complexity of the module. There are more +general definitions given in terms of sheaf and local cohomologies (which in turn are related), but +the one given above suffices for our purposes in this paper. We refer, e.g., to [4] and [7, Chapter 15]. +Lemma 1.7. ([31, Corollary 2.5(i)]) If f ∈ R is a free divisor of degree d, then reg Derk(R/(f)) = +d − 2. In particular, if f ∈ R is a linear free divisor then reg Derk(R/(f)) = n − 2. +It is worth mentioning that some authors have investigated the Castelnuovo-Mumford regularity +of other objects that are also “differentially related” to f, such as the Milnor algebra R/Jf (see [11]) +and the module TR/k(f) itself (see [16, Theorem 5.5] and [36, Section 3]). +1.2. Blowup algebras. We close the section with a brief review on blowup algebras and a few +closely related notions. We fix a homogeneous proper ideal I of R. +Definition 1.8. The Rees algebra of I is the graded ring +R(I) = +� +i≥0 +Iiti ⊂ R[t], +where t is an indeterminate over R. This R-algebra defines the blowup along the subscheme corre- +sponding to I. The special fiber ring of I, sometimes dubbed fiber cone of I, is the special fiber of +R(I), i.e., the (standard) graded k-algebra +F(I) = R(I) ⊗R k ∼= +� +i≥0 +Ii/R+Ii. +The analytic spread of I is ℓ(I) = dim F(I). There are bounds ht I ≤ ℓ(I) ≤ n. +Alternatively, R(I) can be realized as the quotient of the symmetric algebra SymRI (a basic +construct in algebra) by its R-torsion submodule, which is in fact an ideal. Thus there is a natural +R-algebra epimorphism SymRI → R(I). If this map is an isomorphism, I is said to be of linear type. +Since R is in particular a domain, this is tantamount to saying that SymRI is a domain as well. For +instance, any ideal generated by a regular sequence is of linear type. +Next we provide a useful formula for the computation of the analytic spread by means of a +Jacobian matrix (in characteristic zero, as we have permanently assumed). To this end we consider +an even more concrete description of the Rees algebra (hence of its special fiber), to wit, if we fix +generators I = (f1, . . . , fν) ⊂ R = k[x1, . . . , xn], then R(I) is just the R-subalgebra generated by +f1t, . . . , fνt ∈ R[t]. In the particular case where the fi’s are all homogeneous of the same degree +– e.g., the partial derivatives of a homogeneous polynomial – we can write the special fiber as +F(I) ∼= k[f1, . . . , fν] as a k-subalgebra of R. + +6 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +Lemma 1.9. ([40, Proposition 1.1]) Write I = (f1, . . . , fν) and suppose all the fi’s are homogeneous +of the same degree. Set Θ := +� +∂fi +∂xj +� +, 1 ≤ i ≤ ν, 1 ≤ j ≤ n. Then ℓ(I) = rank Θ. +Finally recall that a subideal K ⊂ I is a reduction of I if the induced extension of Rees algebras +R(K) ⊂ R(I) is integral; equivalently, there exists r ≥ 0 such that Ir+1 = KIr. The minimal such +r is denoted rK(I). The reduction K is minimal if it is minimal with respect to inclusion. Now the +reduction number of I is defined as +r(I) = min{rK(I) | K is a minimal reduction of I}. +For instance, it is a standard fact (as k is infinite) that r(I) = 0 if and only if I can be generated by +ℓ(I) elements, which occurs if for example I is of linear type. More generally, the following basic result +gives a way to compute this number in the presence of a suitable condition on the standard graded +k-algebra F(I), which can be also regarded (for the purpose of reading the Castelnuovo-Mumford +regularity off a minimal graded free resolution) as a cyclic graded module over a polynomial ring +k[t1, . . . , tν] whenever I can be generated by ν forms in R. +Lemma 1.10. ([26, Proposition 1.2]) If F(I) is Cohen-Macaulay, then r(I) = reg F(I). +2. First family: linear free divisors in Pn−1 +Before presenting our first family of free divisors as well as properties of related blowup algebras, +let us record a couple of basic calculations which will be used without further mention in the proof +of Theorem 2.2 below. +Remark 2.1. Let S = k[w, u] be a standard graded polynomial ring in 2 variables w, u, and consider +the ideal n = (w, u). Given an integer r ≥ 2, the following facts are well-known and easy to see. +(a) The ideal nr = (wr, wr−1u, . . . , wur−1, ur) is a perfect ideal of codimension 2, having the +following (r + 1) × r syzygy matrix: +(1) +ϕr = + + +−w +0 +. . . +0 +u +−w +. . . +0 +0 +u +. . . +0 +... +... +... +... +0 +0 +. . . +−w +0 +0 +. . . +u + + +; +(b) The presentation ideal of the Rees algebra R(nr), that is, the kernel of the surjective map of +S-algebras +S[y1, . . . , yr+1] ։ R(nr), +yi �→ wr−i+1ui−1, +is equal to Q = (I1(y · ϕr), I2(B)), where y = +� +y1 +· · · +yr+1 +� +and B = +� +y1 +· · · +yr +y2 +· · · +yr+1 +� +. +Theorem 2.2. Consider the standard graded polynomial ring R = k[x1, . . . , xn], where n ≥ 4. +Denote xn−1 = w and xn = u. Let +f = 2wn−1u + +n−2 +� +i=1 +xiwi−1un−i. +Then f is a linear free divisor. +Proof. First notice that +(2) +fxi = wi−1un−i +for each +1 ≤ i ≤ n − 2, + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +7 +fw = 2(n − 1)wn−2u + +n−2 +� +i=2 +(i − 1)xiwi−2un−i +and +fu = 2wn−1 + +n−2 +� +i=1 +(n − i)xiwi−1un−(i+1). +In particular, the subideal (fx1, . . . , fxn−2) of Jf is equal to the ideal u2(w, u)n−3. Thus, if ϕn−3 is the +(n − 2) × (n − 3) syzygy matrix of the ideal (w, u)n−3 (see (1)), then the columns of the n × (n − 3) +matrix +η = +� ϕn−3 +0 +� +are syzygies of the gradient ideal Jf. We also have the following equalities +(3) +ufw = 2(n − 1)wn−2u2 + +n−2 +� +i=2 +(i − 1)xiwi−2un−(i−1) = 2(n − 1)wfxn−2 + +n−2 +� +i=2 +(i − 1)xifxi−1 +and +(4) (n − 1)ufu = 2(n − 1)wn−1u + +n−2 +� +i=1 +(n − 1)(n − i)xiwi−1un−i = wfw + +n−1 +� +i=1 +(n(n − i − 1) + 1)xifxi. +Now note that (3) and (4) yield two new (linear) syzygies of Jf, to wit, +δ1 = +� α2x2 +α3x3 +· · · +αn−2xn−2 +2(n − 1)w +−u +0 �t +and +δ2 = +� +β1x1 +β2x2 +· · · +βn−2xn−2 +w +−(n − 1)u +�t +where αi = i − 1 if 2 ≤ i ≤ n − 2, and βi = n(n − i − 1) + 1 whenever 1 ≤ i ≤ n − 2. +Claim 1. The minimal graded free resolution of Jf is +(5) +0 → R(−n)n−1 +ψ +−→ R(−n + 1)n → Jf → 0 +where ψ = +� +η +δ1 +δ2 +� +. +From the discussion above, we already know that the sequence (5) is a complex. To prove that it +is in fact a minimal graded free resolution of Jf, it suffices to verify that ht In−1(ψ) ≥ 2. Note we +can write ψ in the form +ψ = +� ϕn−3 +∗ +0 +Φ +� +where Φ = +� +−u +w +0 +−(n − 1)u +� +. Thus, det Φ · In−3(ϕn−3) = u2 · (w, u)n−3 ⊂ In−1(ψ). In particular, +un−1 ∈ In−1(ψ). On the other hand, if we specialize the entries of ψ via the k-algebra endomorphism +of R that fixes the variables w, u and maps the remaining ones to 0, we obtain the matrix +ψ = + + +−w +0 +. . . +0 +0 +0 +u +−w +. . . +0 +0 +0 +0 +u +. . . +0 +0 +0 +... +... +... +... +... +... +0 +0 +. . . +−w +0 +0 +0 +0 +. . . +u +2(n − 1)w +0 +0 +0 +. . . +0 +−u +w +0 +0 +. . . +0 +0 +−(n − 1)u + + +. +The (n − 1)-minor of ψ obtained by omitting the last row is cwn−1 for a certain non-zero c ∈ k. +Therefore, the (n − 1)-minor of ψ obtained by omitting the last row has the shape cwn−1 + G, for a +suitable G ∈ (x1, . . . , xn−2). Hence, (un−1, cwn−1 + G) ⊂ In−1(ψ). Hence, ht In−1(ψ) ≥ 2 as desired. + +8 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +A computation shows that, in the first case covered by the theorem (i.e., n = 4), the Jacobian +ideal Jf is of linear type; in particular, ℓ(Jf) = 4 and r(Jf) = 0. The case n ≥ 5 is treated in +Proposition 2.3 below. As ingredients we consider the polynomial rings +T := k[y1, . . . , yn−2, s, t] +and +T ′ := k[y1, . . . , yn−2] +as well as the k-algebras +A := k[fx1, . . . , fxn−2, fw, fu] +and +A′ := k[fx1, . . . , fxn−2]. +By factoring out u2 from the polynomials fx1, . . . , fxn−2 (see (2)), we see that +A′ ∼= A′′ := k[wn−3, wn−4u, . . . , un−3]. +All these rings are related via the following commutative diagram of k-algebras: +(6) +T +� � A +T ′�� +� +� � A′ +∼ += +� +�� +� +A′′ +Proposition 2.3. Maintain the above notations, and let f ∈ R be as in Theorem 2.2, with n ≥ 5. +The following assertions hold: +(i) F(Jf) ∼= T/I2 +� +y1 +· · · +yn−3 +y2 +· · · +yn−2 +� +as k-algebras. In particular, F(Jf) is Cohen-Macaulay; +(ii) ℓ(Jf) = 4; +(iii) r(Jf) = 1; +(iv) reg Derk(R/(f)) = n − 2 (also for n = 4). +Proof. (i) Since Jf is a homogeneous ideal generated in the same degree, there is an isomorphism +of graded k-algebras F(Jf) ∼= A, so that F(Jf) ∼= T/Q where Q := ker (T ։ A). By the diagram +(6), we get Q′T ⊂ Q, where Q′ := ker (T ′ ։ A′). From Remark 2.1(b) we have +Q′T = I2 +� +y1 +· · · +yn−3 +y2 +· · · +yn−2 +� +. +Hence, ht Q ≥ ht Q′T = n−4. Thus, in order to prove that Q = I2 +� +y1 +· · · +yn−3 +y2 +· · · +yn−2 +� +, we must show +ht Q ≤ n − 4, or equivalently, dim A ≥ 4. Now, on the other hand, Lemma 1.9 gives dim A = rank Θ, +where Θ is the Hessian matrix of f. Notice that the (Jacobian) matrix Θ can be written in blocks as +Θ = +� +0 +Θw,u +Θt +w,u +∗ +� +where Θw,u is the (n − 2) × 2 Jacobian matrix of fx1, . . . , fxn−2 with respect to w, u. +Clearly, +I2(Θw,u) ̸= 0. In particular, I4(Θ) ̸= 0 because I2(Θw,u)2 ⊂ I4(Θ). Hence rank Θ ≥ 4, so that +dim A = rank Θ ≥ 4, as needed. The Cohen-Macaulayness of F(Jf) will be confirmed below, in the +proof of item (iii). +(ii) Since ℓ(Jf) = dim F(Jf), the statement follows directly from the proof of (i). +(iii) It is well-known that I2 +� +y1 +· · · +yn−3 +y2 +· · · +yn−2 +� +is a perfect ideal with linear resolution (in fact, this +ideal is resolved by the Eagon-Northcott complex). In particular, the ring F(Jf) ∼= T/Q is Cohen- +Macaulay and its Castelnuovo-Mumford regularity is 1. Thus, by Lemma 1.10, r(Jf) = reg F(Jf) = +1. +(iv) By Theorem 2.2, f is a linear free divisor. Now the assertion follows from Lemma 1.7. + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +9 +Our next goal is to prove that, for f as above, R(Jf) is Cohen-Macaulay. +First, we need an +auxiliary lemma. +Lemma 2.4. Let k[z, v] = k[z1, . . . , zm, v1, . . . , vm] be a polynomial ring, with m ≥ 2. Consider +F = a1v1z1 + · · · + amvmzm, +where a1, . . . , an are non-zero elements of k. Let +M = +� +z1 +z2 +. . . +zm−1 +z2 +z3 +. . . +zm +� +. +Then, k[z, v]/(I2(M), F) is a Cohen-Macaulay domain of codimension m − 1. +Proof. By [24, Section 4], we have that k[z, v]/I2(M) is a Cohen-Macaulay domain of codimension +m−2. In particular, since F /∈ I2(M), the ring k[z, v]/(I2(M), F) is Cohen-Macaulay of codimension +m − 1, and so it remains to show that it is a domain. Clearly, we can assume m ≥ 3. +Claim. z1 is a (k[z, v]/(I2(M), F))-regular element. +Suppose that z1 is not (k[z, v]/(I2(M), F))-regular. Then, z1 ∈ p for some associated prime p of +k[z, v]/(I2(M), F), and hence in particular (z1, I2(M), F) ⊂ p. Now let M1 be the matrix obtained +from M by deletion of its first column. Then it is easy to see that (z1, z2, I2(M1), F) ⊂ p. If M2 is the +matrix obtained by deletion of the first column of M1, then (z1, z2, z3, I2(M2), F) ⊂ p. Proceeding +in this way, we get +(z1, z2, . . . , zm−1, F) ⊂ p. +Since ht (z1, z2, . . . , zm−1, F) = m, it follows that ht p ≥ m. But, this is a contradiction because p +is a associated prime of the Cohen-Macaulay (hence unmixed) ring k[z, v]/(I2(M), F), which has +codimension m − 1. This proves the Claim. +Finally, localizing in z1 we deduce the isomorphism +k[z, v][z−1 +1 ] +(I2(M), F)k[z, v][z−1 +1 ] +∼= +k[z, v2, . . . , vm][z−1 +1 ] +I2(M)k[z, v2, . . . , vm][z−1 +1 ]. +But the ring on the right side of the isomorphism is a domain. +By the claim, it follows that +k[z, v]/(I2(M), F) is a domain. +Theorem 2.5. Let f ∈ R be as in Theorem 2.2. Then, R(Jf) is Cohen-Macaulay. +Proof. Consider the natural epimorphism +C := k[x1, . . . , xn−2, w, u, y1, . . . , yn−2, s, t] ։ R(Jf), +whose kernel we denote J . From the previous considerations (and notations), it follows that +K := (I1(γ · ψ), Q) ⊂ J +where γ = +� y1 +· · · +yn−2 +s +t � +. Note we can rewrite the ideal K as +K = I2 +� +u +y1 +· · · +yn−3 +w +y2 +· · · +yn−2 +� ++ (G, H), +G = γ · δ1 = −su + 2(n − 1)yn−2w + +n−2 +� +i=2 +αiyi−1xi +and +H = γ · δ2 = −(n − 1)tu + sw + +n−2 +� +i=1 +βiyixi. +Claim 1. Let K1 := I2 +� +u +y1 +· · · +yn−3 +w +y2 +· · · +yn−2 +� ++(H) ⊂ C. Then, C/K1 is a Cohen-Macaulay domain. +Denote K0 := I2 +� +u +y1 +· · · +yn−3 +w +y2 +· · · +yn−2 +� +. It is well-known that C/K0 is a Cohen-Macaulay integral +domain of dimension n + 3. Moreover, since H /∈ K0, this polynomial must be C/K0-regular. Hence, + +10 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +the ring C/K1 = C/(K0, H) is Cohen-Macaulay of dimension n + 2. In particular, C/K1 satisfies +Serre’s condition S2 (see the definition in Subsection 6.1). We claim that, even more, the ring C/K1 +is normal, from which its integrality will follow. In order to show that C/K1 is normal, it remains +to verify that it is locally regular in codimension 1. Note ht K1 = n − 2. By the classical Jacobian +criterion, it suffices to prove that +ht (K1, In−2(Θ)) ≥ n, +where Θ denotes the Jacobian matrix of K1. Notice that the matrix Θ, after a reordering of its +columns (which obviously does not affect ideals of minors), can be written in the format +Θ = +� Θ′ +0 +∗ +Θ′′ +� +. +Precisely, Θ′ is the Jacobian matrix of K0 with respect the variables w, u, y1, . . . , yn−2 and Θ′′ is the +(row) Jacobian matrix of H with respect to the variables x1, . . . , xn−2, s, t. In particular, +(K1, I1(Θ′′) · In−3(Θ′)) ⊂ (K1, In−2(Θ)). +Now pick a minimal prime q of (K1, In−2(Θ)). In particular, I1(Θ′′) · In−3(Θ′) ⊂ q, which yields +I1(Θ′′) ⊂ q or In−3(Θ′) ⊂ q. If I1(Θ′′) ⊂ q then ht q ≥ n because I1(Θ′′) = (y1, . . . , yn−2, w, u). On +the other hand, if In−3(Θ′) ⊂ q then (K0, In−3(Θ′)) ⊂ q. But it is well-known that +ht (K0, In−3(Θ′)) = n. +Therefore, ht q ≥ n in any case, and we get ht (K1, In−2(Θ)) ≥ n, as desired. +Claim 2. C/K is a Cohen-Macaulay domain of dimension n + 1. +By Claim 1 and its proof, C/K1 is a Cohen-Macaulay domain of dimension n + 2. Thus, since +G /∈ K1, the ring C/K = C/(K1, G) is Cohen-Macaulay of dimension n + 1. It remains to prove that +C/K is a domain. First we claim that u is C/K-regular. Suppose otherwise. Then u ∈ p for some +associated prime p of C/K, which gives +(u, wy1, . . . , wyn−3, I2(N), G, H) ⊂ p, +where +N := +� +y1 +· · · +yn−3 +y2 +· · · +yn−2 +� +. +In particular, +(7) +Q1 := (u, w, I2(N), G, H) ⊂ p +or +Q2 := (u, y1, . . . , yn−3, G, H) ⊂ p. +We have +C/(u, w, I2(N), H) ∼= (k[x1, . . . , xn−2, y1, . . . , yn−2]/(I2(N), β1y1x1 + · · · + βn−2yn−2xn−2))[s, t]. +From this isomorphism and Lemma 2.4, the ring C/(u, w, I2(N), H) is a Cohen-Macaulay domain +of dimension (n − 1) + 2 = n + 1. +Thus, since G /∈ (u, w, I2(N), H), we obtain that C/Q1 = +C/(u, w, I2(N), G, H) is a Cohen-Macaulay ring of dimension n. In particular, ht Q1 = n. On the +other hand, +C/Q2 ∼= k[x1, . . . , xn−2, w, yn−2, s, t]/(yn−2w, sw + βn−2yn−2xn−2) +is a Cohen-Macaulay ring of dimension n, which yields ht Q2 = n. It follows, by (7), that ht p ≥ n. +This is a contradiction, because p is an associated prime of C/K, which is Cohen-Macaulay of codi- +mension n−1. So, u is C/K-regular. Now, by localizing in u and setting D := k[x1, . . . , xn−2, w, u, y1, s, t], +routine calculations give +(C/K)[u−1] ∼= D[u−1]/(G, H)D[u−1] ∼= k[x1, . . . , xn−2, w, u, y1][u−1], +which is a domain. Hence, C/K is a domain, which proves Claim 2. +To conclude the proof of the theorem, we notice that since K ⊂ J are prime ideals of the same +codimension, then necessarily K = J . In particular, R(Jf) ∼= C/J is Cohen-Macaulay. + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +11 +3. Second family: linear free divisors in P2n−1 +In order to describe our second family of free divisors, consider the standard graded polynomial +ring R = k[x1, . . . , x2n−2, w, u] in 2n ≥ 4 indeterminates over k. Let +f = wuq, +q = (x1u − x2w)(x3u − x4w) · · · (x2(n−1)−1u − x2(n−1)w). +For every 1 ≤ i ≤ n − 1, denote qi = q/(x2i−1u − x2iw) ∈ R. Then, +(8) +fx2i−1 = u2wqi +and +fx2i = −w2uqi +(1 ≤ i ≤ n − 1), +(9) +fw = qu − wu +n−1 +� +i=1 +x2iqi +and +fu = qw + wu +n−1 +� +i=1 +x2i−1qi. +Using (8) and (9) we easily deduce the following relations: +(10) +det +� fx2i−1 +fx2j−1 +fx2i +fx2j +� += 0 +(1 ≤ i < j ≤ n − 1), +(11) +wfx2i−1 + ufx2i = 0 +(1 ≤ i ≤ n − 1), +wfw + ufu = (n + 1)f, +(12) +x2i−1fx2i−1 + x2ifx2i = f +(1 ≤ i ≤ n − 1), +(13) +(n + 1)x2i−1fx2i−1 + (n + 1)x2ifx2i − ufu − wfw = 0 +(1 ≤ i ≤ n − 1). +Set α = a1a2, β = b1b2 and γ = a1b2 + a2b1. In addition to the equalities above, we have +(n + 1) +n−1 +� +i=1 +x2ifx2i += +−(n + 1)w2u +n−1 +� +i=1 +x2iqi += +(n + 1)[w(fw − qu)] += +nwfw − ufu + (ufu + wfw) − (n + 1)f += +nwfw − ufu. +(14) +Now we are in a position to prove the first result of this section. +Theorem 3.1. Maintain the above notations. The following assertions hold: +(i) f is a linear free divisor; +(ii) The 2n × (2n − 1) matrix +(15) +ψn = + + +w (n + 1)x1 +. . . +0 +0 +0 +u +(n + 1)x2 +. . . +0 +0 +(n + 1)x2 +... +... +... +... +... +... +0 +0 +· · · +w (n + 1)x2n−3 +0 +0 +0 +· · · +u +(n + 1)x2n−2 (n + 1)x2n−2 +0 +−w +· · · +0 +−w +−nw +0 +−u +· · · +0 +−u +u + + +is a syzygy matrix of Jf. Thus a free basis of TR/k(f) is {θ1, . . . , θ2n−1, ε2n}, where the θi’s +correspond to the columns of ψn; +(iii) reg Derk(R/(f)) = 2(n − 1). + +12 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +Proof. (i) Consider the 2n × 2n matrix +M = + + +w x1 +. . . +0 +0 +0 +x1 +u +x2 +. . . +0 +0 +(n + 1)x2 +x2 +... +... +... +... +... +... +... +0 +0 +· · · +w x2n−3 +0 +x2n−3 +0 +0 +· · · +u +x2n−2 (n + 1)x2n−2 x2n−2 +0 +0 +· · · +0 +0 +−nw +w +0 +0 +· · · +0 +0 +u +u + + +. +Using (11), (12), (14), and the Euler relation, it is easy to see that +∇f · M = [ 0 +f +· · · +0 +f +0 +2nf ], +so that ∇f · M ≡ 0 mod f. Moreover, (n + 1)f = det M . Thus, by Lemma 1.4 (or, in this case, by +the version of Saito’s criterion stated in [8, Theorem 2.4]), we conclude that f is a linear free divisor. +(ii) For simplicity, write ψn = ψ. By (i) and Lemma 1.3, we know that Jf is a codimension 2 +perfect ideal, so it suffices to prove that ∇f ·ψ = 0 and that ψ has maximal rank. The former follows +by (11), (13) and (14). Now denote by ∆ the (2n − 1)-minor of ψ obtained by omitting the 2n-th +row of ψ. It is easy to see that ∆ modulo w is given by (n + 1)nx1x3 · · · x2n−3un. In particular, ∆ is +non-zero as well. Hence, ψ has maximal rank. +(iii) By part (i), f is a linear free divisor (in 2n variables). Now we apply Lemma 1.7. +For the next results, we consider a set of 2n variables z1, . . . , z2n−2, s, t over R as well as the natural +epimorphism +S := k[x1, . . . , x2n−2, w, u, z1, . . . , z2n−2, s, t] ։ R(Jf) +whose kernel we denote J . By the equalities (10) we have an inclusion +I2 +� +z1 +z3 +. . . +z2n−3 +z2 +z4 +. . . +z2n−2 +� +⊂ J . +Therefore, +K := +� +I1(γ · ψn), I2 +� +z1 +z3 +. . . +z2n−3 +z2 +z4 +. . . +z2n−2 +�� +⊂ J +where γ = +� +z1 +. . . +z2n−2 +s +t +� +. The generators of I1(γ · ψn) are of three types: +(16) +wz2i−1 + uz2i +(1 ≤ i ≤ n − 1), +Fi := (n + 1)(x2i−1z2i−1 + x2iz2i) − ws − ut +(1 ≤ i ≤ n − 1), +G := (n + 1) +n−1 +� +i=1 +x2iz2i − nws + ut. +We can use the generators of type (16) as well as the ideal I2 +� +z1 +z3 +. . . +z2n−3 +z2 +z4 +. . . +z2n−2 +� +to rewrite K as +K = + + + + +I2 +� +z1 +z3 +. . . +z2n−3 +−u +z2 +z4 +. . . +z2n−2 +w +� +� +�� +� +=:K0 +, F1, . . . , Fn−1, G + + + + + +With this, we have +S/K ∼= A[x1, . . . , x2n−2, s, t]/(F1, . . . , Fn−1, G)A[x1, . . . , x2n−2, s, t] + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +13 +where A := k[z1, . . . , z2n−2, w, u]/K0. Now, consider the 2n × n matrix +ζ = + + +(n + 1)z1 +0 +. . . +0 +0 +(n + 1)z2 +0 +. . . +0 +(n + 1)z2 +... +... +... +... +... +0 +0 +. . . (n + 1)z2n−3 +0 +0 +0 +. . . (n + 1)z2n−2 (n + 1)z2n−2 +−w +−w . . . +−w +−nw +−u +−u . . . +−u +u + + +taken as a matrix with entries in the domain A. We denote by M the A-module defined as the +cokernel of ζ. +Proposition 3.2. Maintain the above notations. Then: +(i) M is an A-module of projective dimension 1; +(ii) The symmetric algebra SymAM is a Cohen-Macaulay domain of dimension 2n + 1. +Proof. (i) Consider the complex +(17) +0 −→ An +ζ +−→ A2n −→ M −→ 0. +By the well-known Buchsbaum-Eisenbud acyclicity criterion, in order to show that (17) is exact it +suffices to confirm that rank ζ = n. To this end, consider the following n × n submatrix of ζ: +η := + + +(n + 1)z1 +· · · +0 +0 +... +... +... +... +0 +. . . +(n + 1)z2n−3 +0 +−w +. . . +−w +−nw + + . +We have det η = −n(n + 1)n−1z1 · · · z2n−3w ̸= 0 (modK0). Hence, ζ has rank n. +(ii) In addition to the property given in (i), recall A is a Cohen-Macaulay domain and ht K0 = n−1. +Then, because of [29, Theorem 1.1], it suffices to show that +ht(It(ζ) + K0) ≥ 2n − t + 1 +for every 1 ≤ t ≤ n. For this note first that, by suitably permuting the rows of ζ, we obtain a matrix +N of the form +N = + + +∗z1 · · · +0 +. . . +0 +0 +... +... +... +. . . +... +... +0 +0 +∗z2i−1 . . . +0 +0 +... +... +... +... +... +... +0 +0 +0 +. . . ∗z2n−3 +0 +−u −u +−u +. . . +−u +u +∗z2 · · · +0 +. . . +0 +∗z2 +... +... +... +· · · +... +... +0 +· · · +∗z2i +. . . +0 +∗z2i +... +... +... +... +... +... +0 +0 +0 +. . . ∗z2n−2 ∗z2n−2 +−w −w +−w +. . . +−w +−nw + + +where all coefficients ∗ are equal to n + 1. Let us denote the top and bottom blocks of N by Nodd +and Neven, respectively. Our goal is to prove ht(It(N) + K0) ≥ 2n − t + 1 whenever 1 ≤ t ≤ n, where + +14 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +as before +K0 = I2 +� +z1 +z3 +. . . +z2n−3 +−u +z2 +z4 +. . . +z2n−2 +w +� +. +Let P be a prime ideal containing It(N) + K0 and having the same codimension. From Nodd it is +easy to see that the ideal Ct generated by the t-products of the set {z1, z3, . . . , z2n−3, z2n−1 := u} is +contained in P. But, by [48, Section 2], the minimal primes of Ct are of the form (z2j1−1, . . . , z2jn−t+1−1) +for certain 1 ≤ j1 < · · · < jn−t+1 ≤ n. Hence, we can suppose that (z2j1−1, . . . , z2jn−t+1−1) ⊂ P with +1 ≤ j1 < · · · < jn−t+1 ≤ n. Analogously, from Neven we can write (z2i1, . . . , z2in−t+1) ⊂ P for certain +1 ≤ i1 < · · · < in−t+1 ≤ n (we put z2n := w). +Let us assume that the following condition takes place: +(†) +There exists j ∈ {1, . . . , n} such that {z2j−1, z2j} ∩ P = {z2j−1} or {z2j}. +Suppose {z2j−1, z2j} ∩ P = {z2j−1}. Then, by the relations in K0, all the odd variables belong to +P. Therefore, we have +(n − t + 1) +� +�� +� +even variables ++ +n +���� +odd variables += 2n − t + 1 variables in P, +which gives ht(It(N) + K0) ≥ 2n − t + 1 for 1 ≤ t ≤ n. The argument for the case {z2j−1, z2j} ∩ P = +{z2j} is similar. +Now, suppose that (†) is not true. Without loss of generality, we may assume (j1, . . . , jn−t+1) = +(1, . . . , n − t + 1). It follows that z1, z2, . . . , z2(n−t+1)−1, z2(n−t+1) ∈ P. Consider the following t × t +submatrix of ζ: + + +0 +∗z2(n−t+1)+1 +0 +. . . +0 +0 +0 +0 +∗z2(n−t+2)+1 . . . +0 +0 +... +... +... +... +... +... +0 +0 +0 +. . . ∗z2n−3 +0 +−u +−u +−u +. . . +−u +u +−w +−w +−w +. . . +−w +−nw + + +. +The determinant of this matrix is cz2(n−t+1)+1 · · · z2n−3z2n−1z2n for some c ∈ k; in particular, this +determinant lies in P and hence zs ∈ P for some s with 2(n − t + 1) + 1 ≤ s ≤ 2n. Therefore, +since we are assuming that (†) does not hold, there exist two consecutive indices 2j − 1, 2j with +2(n − t + 1) + 1 ≤ 2j − 1, 2j ≤ 2n satisfying z2j−1, z2j ∈ P. Now we can suppose, without loss of +generality, that 2j − 1 = 2(n − t + 1) + 1. We have +(z1, z2, . . . , z2(n−t+1)−1, z2(n−t+1), z2(n−t+1)+1, z2(n−t+1)+2) + K0 ⊂ P. +Hence, considering the subideal +�K0 := I2 +� z2(n−t+2)+1 +z2(n−t+3)+1 +. . . +z2n−3 +−u +z2(n−t+3) +z2(n−t+4) +. . . +z2n−2 +w +� +⊂ K0, +we observe that all the variables appearing in �K0 are different from the 2(n − t + 2) variables that +already belong to P; since in addition ht �K0 = t − 3, we conclude +ht(It(N) + K0) = ht P ≥ 2(n − t + 2) + (t − 3) = 2n − t + 1 +whenever 1 ≤ t ≤ n, as needed. +So we have shown that SymAM is a Cohen-Macaulay domain. Note that M possesses a rank as +an A-module (equal to n, by (17)). Now recall that the Rees algebra of the A-module M, denoted +RA(M), can be defined as the quotient of SymAM by its A-torsion submodule (see [42] for the +general theory). Consequently, since in this case A and SymAM are both domains, we can identify +SymAM = RA(M); in particular, using [42, Proposition 2.2] (which gives a formula for the dimension + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +15 +of the Rees algebra of a module with rank) and noticing that dim A = 2n − (n − 1) = n + 1, we +finally get +dim SymAM = dim RA(M) = dim A + rankAM = (n + 1) + n = 2n + 1. +Theorem 3.3. Maintain the above notations. Then: +(i) K = J ; +(ii) The Rees algebra R(Jf) is Cohen-Macaulay; +(iii) Let T = k[z1, . . . , z2n−2, s, t], with n ≥ 3. Then, +F(Jf) ∼= T/I2 +� +z1 +z3 +. . . +z2n−3 +z2 +z4 +. . . +z2n−2 +� +as k-algebras. In particular, F(Jf) is Cohen-Macaulay, ℓ(Jf) = n + 2, and r(Jf) = 1. +Proof. We have a natural epimorphism +SymAM ∼= S/K ։ S/J ∼= R(Jf). +By Proposition 3.2, K is a prime ideal of S, and dim SymAM = 2n + 1 = dim R + 1 = dim R(Jf). +So ht K = ht J , and then K = J . +Using Proposition 3.2 once again, we obtain that R(Jf) is +Cohen-Macaulay. This proves (i) and (ii). +In order to prove (iii), let R+ be the homogeneous maximal ideal of R. We have +F(Jf) ∼= R(Jf)/R+R(Jf) ∼= S/(R+S, K) ∼= T/I2 +� +z1 +z3 +. . . +z2n−3 +z2 +z4 +. . . +z2n−2 +� +, +which, as is well-known (being a generic determinantal ring), is Cohen-Macaulay of dimension n + 2 +and moreover has regularity 1. The latter, by Lemma 1.10, gives r(Jf) = 1. +Remark 3.4. (a) The ideal Jf is of linear type if and only if n = 2 (i.e., the case where R is a +polynomial ring in 4 variables). Indeed, by Theorem 3.3(iii), if n ≥ 3 then r(Jf) = 1 ̸= 0, hence Jf +cannot be of linear type. Conversely, let n = 2, so that R = k[x1, x2, w, u]. We can check that the +ideals of minors of ψ2 (see (15)) satisfy +ht Is(ψ2) ≥ 5 − s = (2n − 1) + 2 − s +for +s = 1, 2, 3. +It follows by [29, Theorem 1.1] that Jf is of linear type (in particular, r(Jf) = 0). Note in addition +that SymRJf is a complete intersection, i.e., the polynomials +L1 = 3x1z1 + 3x2z2 − ws − ut, L2 = wz1 + uz2, L3 = x2z1 + x1z2 + us + wt +form an R[z1, z2, s, t]-sequence. +It is also worth mentioning that, in 3 variables, if g ∈ k[x, y, z] defines a rank 3 central hyperplane +arrangement, then it has been recently shown that Jg is of linear type and moreover that the property +of the symmetric algebra of Jg being a complete intersection characterizes the freeness of g (see [10, +Proposition 2.14 and Corollary 2.15]). +(b) Let n = 3, i.e., R = k[x1, x2, x3, x4, w, u]. In this case, a computation shows that the entries of the +product +� +z1 +· · · +z4 +s +t +� +· ψ3 form a regular sequence, i.e., SymRJf is a complete intersection +once again. +Question 3.5. For an arbitrary n, is SymRJf a complete intersection ring? + +16 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +4. Third family: non-linear free plane curves +In this section we furnish our third family of free divisors and some of its properties. In fact, from +such a family we will derive yet another one; see Remark 4.3. Similar examples (also in 3 variables) +can be found, e.g., in [20] and [34]. +Theorem 4.1. Consider the two-parameter family of polynomials +f = fα,β = (xα − yα−1z)β + yαβ ∈ R = k[x, y, z], +for integers α, β ≥ 2. The following assertions hold: +(i) f is a free divisor; +(ii) R(Jf) is Cohen-Macaulay if (α, β) = (2, 3) or if α ≥ 2 and β = 2; +(iii) Jf is not of linear type if α = 2 and β ≥ 3 or if α ≥ 3 and β = 2. In these cases, F(Jf) is +a polynomial ring over k and then r(Jf) = 0; +(iv) f is reducible over k = C. If k ⊆ R then f is reducible if β is odd and irreducible otherwise; +(v) reg Derk(R/(f)) = αβ − 2. +Proof. (i) We have +fx = αβxα−1(xα−yα−1z)β−1, +fy = αβyαβ−1−(α−1)βyα−2z(xα−yα−1z)β−1, fz = −βyα−1(xα−yα−1z)β−1 +Note that we can write fx, fy and fz as +fx = αxα−1G, +fy = xα−1P + yα−1Q, +fz = −yα−1G +for certain G, P, Q ∈ R. Thus, +(18) +Jf = I2 + + +yα−1 +−α−1P +0 +G +αxα−1 +Q + + . +In particular, since Jf has codimension two, it follows by the Hilbert-Burch theorem that Jf is a +perfect ideal. By Lemma 1.3, f is a free divisor. +(ii) In the specific cases (α, β) = (2, 2) and (α, β) = (2, 3), the Cohen-Macaulayness of R(Jf) can +be confirmed by a routine computation. Therefore, we may suppose α ≥ 3 and β = 2. Determining +G, P, Q in this situation, we obtain from (18) that a syzygy matrix of Jf is +ϕ = + + +yα−1 +(α − 1)xyα−2z +0 +α(xα − yα−1z) +αxα−1 +α2yα + α(α − 1)yα−2z2 + + . +Let us denote by Q the (prime) ideal of k[x, y, z, s, t, u] = R[s, t, u] defining R(Jf). Notice that +(19) +I1 +�� s +t +u � +· ϕ +� += (syα−1 + αuxα−1, yα−2H + αxαt) ⊂ Q, +where H := (α − 1)xzs + (α2y2 + α(α − 1)z2)u − αyzt. Clearly, we can rewrite I1([ s +t +u ] · ϕ) as +I1 +�� yα−2 +xα−2 � +· +� +sy +H +αux +αx2t +�� +. +Now it follows from Cramer’s rule that +(20) +det +� +sy +H +αux +αx2t +� += αx2yst − αuxH = αx(xyst − uH) ∈ Q. +From (19) and (20) we deduce an inclusion +P := (syα−1 + αuxα−1, yα−2H + αxαt, xyst − uH) ⊂ Q. +Claim 1. P is a perfect ideal of height 2. + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +17 +Clearly, ht P ≥ 2 and +P = I2 + + +H +−xt +−ys +u +αxα−1 +yα−2 + + +Now, Claim 1 follows by the Hilbert-Burch theorem. +Claim 2. P = Q. +Since P ⊆ Q and ht P = ht Q = 2, it suffices to prove that the ideal P is prime. Note first that x is +regular modulo P. To show this, suppose otherwise. Then we would have x ∈ p for some associated +prime p of R[s, t, u]/P. In particular, (x, syα−1, yα−2H, uH) ⊂ p. Using the explicit format of H +given above, it is easy to see that +ht (x, syα−1, yα−2H, uH) ≥ 3. +In particular, ht p ≥ 3. But this is a contradiction because, by Claim 1, P is perfect of height 2. +Now, by inverting the element x we get +D := +k[x, y, z, s, t, u][x−1] +Pk[x, y, z, s, t, u][x−1] += +k[x, y, z, s, t, u][x−1] +(u + α−1x1−αyα−1s, xt + α−1x1−αyα−2H, xyst − uH) += +k[x, y, z, s, t, u][x−1] +(u + α−1x1−αyα−1s, xt + α−1x1−αyα−2H) +∼= +k[x, y, z, s, t][x−1] +(at + bs) +∼= +k[x, y, z, x−1][s, t] +(at + bs) +where a := x(1 − x−αyα−1z) +and +b := x1−αyα−2[(α − 1)za + x1−αyα+1] are elements in the +coefficient ring k[x, y, z, x−1]. Since a and b are easily seen to be relatively prime in this facto- +rial domain, the element at + bs must be irreducible in k[x, y, z, x−1][s, t], so that the quotient +k[x, y, z, x−1][s, t]/(at+bs) ∼= D is a domain. This means (as x is regular modulo P) that R[s, t, u]/P +is a domain, as needed. +Finally, by Claim 1 and Claim 2, we conclude that R(Jf) is Cohen-Macaulay. +(iii) First, if α = 2, a simple inspection shows that the linear type property of Jf fails in case +β ≥ 3 (and holds if β = 2), by analyzing the saturation of the ideal S of 2 linear forms defining +SymRJf in R[s, t, u] by the ideal Jf. The resulting ideal – which thus defines R(Jf) (see, e.g., [31, +Lemma 2.11]) – turns out to strictly contain S . In addition, it is contained in (x, y, z)R[s, t, u], so +that F(Jf) ∼= k[s, t, u]. As to the case where α ≥ 3 and β = 2, we can use a previous calculation. +Precisely, by the structure of the defining ideal Q = P ⊂ k[x, y, z, s, t, u] of R(Jf) as obtained in +item (ii), we readily get that Jf is not of linear type. Moreover, by looking at the non-linear Rees +equation +xyst − uH ∈ (x, y, z)R[s, t, u] +we conclude that, once again, F(Jf) ∼= k[s, t, u]. In either case, ℓ(Jf) = 3 and (by Lemma 1.10) +r(Jf) = 0. +(iv) Let k = C and assume first that β = 2m, m ≥ 1. We have f = (xα − yα−1z)2m + y2mα and +hence, for i = √−1 and A := xα − yα−1z, +(21) +f = (iAm + ymα)(−iAm + ymα). +Now, assume β ≥ 3 is odd. Write f = (xα − yα−1z + yα − yα)β + yαβ and set B := xα − yα−1z + yα. +Thus we can rewrite +f = (B − yα)β + yαβ = [Bg + (−1)βyαβ] + yαβ = Bg + +18 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +for a suitable g := gα,β ∈ R of degree α(β − 1). Of course, this also shows that if k ⊆ R and β is +odd, then f is reducible over k. +Finally, if k ⊆ R then the irreducibility of f over k for even β follows from the structure of the +factors described in (21) over the unique factorization domain C[x, y, z]. +(v) According to item (i), f is a free divisor. Noticing that its degree is αβ, the assertion follows +by Lemma 1.7. +Remark 4.2. Computations strongly suggest that the cases described in item (ii) are precisely the +ones where the Cohen-Macaulayness of R(Jf) takes place. Concerning the linear type property of +Jf, computations also indicate that Jf is not of linear type if α, β ≥ 3 – cases not covered by part +(iii). The (partially computer-assisted) conclusion is that Jf is of linear type if and only if α = β = 2. +Remark 4.3. Here we want to point out that the form g = gα,β ∈ Rα(β−1) defined in the proof of +item (iv) is also a free divisor provided that β ≥ 3 is odd. First, note that g is defined by means of +Bg = (B − yα)β + yαβ, where B = xα − yα−1z + yα. Explicitly, from +Bg = (B − yα)β + yαβ = +β +� +j=0 +�β +j +� +Bβ−j(−yα)j + yαβ = B · + + +β−1 +� +j=0 +(−1)j +�β +j +� +Bβ−j−1yαj + + +we get g = +β−1 +� +j=0 +(−1)j +�β +j +� +Bβ−j−1yαj. An elementary calculation shows that we can write gx, gy, gz +as gx = αxα−1T, gy = xα−1U + yα−1V , gz = yα−1T, for certain T, U, V ∈ R. Thus, +Jg = I2 + + +yα−1 +−α−1U +0 +T +αxα−1 +V + + . +Since ht Jg = 2, the ideal Jg must be perfect by the Hilbert-Burch theorem. Therefore, g is a free +divisor whenever β ≥ 3 is odd. In particular, by Lemma 1.7 we have reg Derk(R/(g)) = αβ − α − 2. +We also observe that, for every odd β ≥ 3, the form g is reducible over k = C. Indeed, let +Φ = +β−1 +� +j=0 +(−1)j +�β +j +� +Zβ−j−1W j ∈ C[Z, W], +which then factors as a product of linear forms in Z and W. Now let σ: C[Z, W] → C[x, y, z] be the +homomorphism given by Z �→ B and W �→ yα. Then, the form σ(Φ) = g is reducible in C[x, y, z]. +Finally, let us study the algebra R(Jgα,β) in the cases β = 3 and β = 5. +If β = 3 then first as a matter of illustration we explicitly have +gα,3 = B2 − 3Byα + 3y2α += +(B − yα)2 − yα(B − 2yα) = (B − yα)2 − yα(B − yα) + y2α += +(xα − yα−1z + yα)(xα − yα−1z − 2yα) + 3y2α += +x2α − 2xαyα−1z − xαyα + y2α−2z2 + y2α−1z + y2α. +Computations show that, for all α ≥ 2, the ring R(Jgα,3) is Cohen-Macaulay, r(Jgα,3) = 0, but Jf is +of linear type if and only if α = 2; more precisely, if L (resp. Q) defines SymRJgα,3 (resp. R(Jgα,3)) +in the polynomial ring R[s, t, u], then we have found the relation +L : Q = (xα−1, yα−2)R[s, t, u]. +If β = 5, then in the cases α = 2 and α = 3 we have confirmed that depth R(Jgα,5) = 3, i.e., +the ring R(Jgα,5) is almost Cohen-Macaulay in the sense that its depth is 1 less than its dimension. +Also, we have r(Jgα,5) = 0. We strongly believe such properties hold for α ≥ 4 as well. + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +19 +Question 4.4. Let β ≥ 7 be odd. Is R(Jgα,β) almost Cohen-Macaulay? Is it true that r(Jgα,β) = 0 ? +If k ⊆ R and β ≥ 3 is odd, is gα,β irreducible? +5. Fourth family: linear free plane curves +In this section, we let R = k[x, y, z] and our objective is to exhibit our fourth family of free divisors, +which as we shall prove have the linearity property as in two of the previous families. Although in +[27, 6.4, p. 837] a classification of linear free divisors in at most 4 variables in the case k = C is +given, the approach provided here describes concretely a recipe to detect some linear free divisors +in 3 variables starting from a suitable 2 × 3 matrix L of linear forms (in fact, from only 3 linear +forms, as we shall clarify), where, we recall, k is not required to be algebraically closed; in regard to +this point, it should be mentioned that, even though most of the existing results in the literature are +established over C, there has always been an interest in free divisor theory over arbitrary fields (see, +e.g., [46] and [49]). +We could start focusing on the case where the 6 entries of L are general linear forms, but there is in +fact no need for this setting as we only suppose the forms in the first row to be linearly independent +over k and, naturally, the rank of the matrix to be 2. Thus, after eventually a linear change of +variables, we let +L = L (L1, L2, L3) := +� +x +y +z +L1 +L2 +L3 +� +, +for linear forms +L1 = a1x + a2y + a3z, L2 = a4x + a5y + a6z, L3 = a7x + a8y + a9z, +where at least one of the 2 × 2 minors +Q1 = xL2 − yL1, Q2 = xL3 − zL1, Q3 = yL3 − zL2 +does not vanish. We also consider the Jacobian matrix of Q = {Q1, Q2, Q3}, +Θ = Θ(Q) = + + +Q1x +Q2x +Q3x +Q1y +Q2y +Q3y +Q1z +Q2z +Q3z + + = + + +L2 − a4x − a1y +L3 + a7x − a1z +a7y − a4z +a5x − L1 − a2y +a8x − a2z +L3 − a8y − a5z +a6x − a3y +a9x − L1 − a3z +a9y − L2 − a6z + + . +Our result in this section is as follows. +Theorem 5.1. Maintain the above notations. If the cubic f := det Θ is non-zero and reduced, then +f is a linear free divisor. Precisely, a free basis of TR/k(f) is {θ1, θ2, ε3}, where ε3 is the Euler +derivation, θ2 = L1 ∂ +∂x + L2 ∂ +∂y + L3 ∂ +∂z, and +θ1 = (a1L1 + a2L2 + a3L3) ∂ +∂x + (a4L1 + a5L2 + a6L3) ∂ +∂y + (a7L1 + a8L2 + a9L3) ∂ +∂z . +Proof. Routine calculations show that η1 and η2 below are syzygies of Jf, +η1 = + + +(2a1 − a5 − a9)x + 3a2y + 3a3z +3a4x + (2a5 − a1 − a9)y + 3a6z +3a7x + 3a8y + (2a9 − a1 − a5)z + + = + + +3L1 − (a1 + a5 + a9)x +3L2 − (a1 + a5 + a9)y +3L3 − (a1 + a5 + a9)z + + +3×1 +and η2 being the 3 × 1 column-matrix given by + + +(−3a5 − 3a9)L1 + 6a2L2 + 6a3L3 + (−4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9))x +6a4L1 + (−6a1 + 3a5 − 3a9)L2 + 6a6L3 + (−4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9))y +6a7L1 + 6a8L2 + (−6a1 − 3a5 + 3a9)L3 + (−4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9))z + + . +Now let +A := a1 + a5 + a9, +B := −4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9), + +20 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +so that the following is a submatrix of the matrix of syzygies of Jf: + + +(−3a5 − 3a9)L1 + 6a2L2 + 6a3L3 + Bx +3L1 − Ax +6a4L1 + (−6a1 + 3a5 − 3a9)L2 + 6a6L3 + By +3L2 − Ay +6a7L1 + 6a8L2 + (−6a1 − 3a5 + 3a9)L3 + Bz +3L3 − Az + + +3×2 +. +Multiplying the second column by C := a1 + A = 2a1 + a5 + a9 and adding it to the first column, +we obtain an equivalent matrix +ϕ = + + +6(a1L1 + a2L2 + a3L3) + (B − AC)x +3L1 − Ax +6(a4L1 + a5L2 + a6L3) + (B − AC)y +3L2 − Ay +6(a7L1 + a8L2 + a9L3) + (B − AC)z +3L3 − Az + + +3×2 +. +Finally, attaching to ϕ a third column corresponding to the Euler derivation, the resulting 3 × 3 +matrix is easily seen to be equivalent to +Φ = + + +a1L1 + a2L2 + a3L3 +L1 +x +a4L1 + a5L2 + a6L3 +L2 +y +a7L1 + a8L2 + a9L3 +L3 +z + + +3×3 +and satisfies det Φ = 1 +2f. Now the proposed assertions follow by Lemma 1.4. +Numerous comments are in order. +Remark 5.2. (a) Recall that, by definition, being reduced is a necessary condition for a polynomial +to be free. Now we point out that, in general, it is possible for the cubic f := det Θ (with Θ as +defined above) to be non-reduced. For instance, if we start with the matrix L (x−y, x+y +z, y +z), +then f = −2(x + z)3. +(b) Concerning the condition f ̸= 0, it means (since char k = 0) that the quadrics Q1, Q2, Q3 are +algebraically independent, hence linearly independent. Clearly, this may not occur; for example, for +L (y, x, z) we have f = 0 because Q3 = −Q2. +(c) We remark that any f as in Theorem 5.1 is necessarily reducible at least if k = C. This follows +by the fact that a complex irreducible free divisor in 3 variables must have degree at least 5 (see [20, +Theorem 2.8]). +(d) A linear free divisor f in our fourth family can have an irreducible quadratic factor, at least over +k = R (or eventually a suitable finite field extension of Q). Indeed, starting for example with the +matrix L (0, x + z, y + z), we obtain +f = −2xq := −2x(x2 + xy − y2 + 3xz − yz + z2). +Forcing q to be the product of two linear forms with real coefficients yields a contradiction, hence q +is irreducible over R. However, it should be pointed out that q is reducible over C, as the rank of +its associated matrix is non-maximal. In fact we believe (but have no proof) that if k = C then a +free cubic f as in Theorem 5.1 must necessarily be a product of linear forms. This would imply that +the complex linear free divisor z(xz + y2) does not belong to our fourth family, which we have been +unable to prove. +(e) Our method does not work for higher degrees in general. Taking for example any of the matrices +� +x2 +y +z +y2 +z +x +� +, +� +x +y +z +z2 +x2 +y2 +� +, +� +x2 +y2 +z2 +z2 +x2 +y2 +� +, +we are led (following the same recipe) to polynomials that are not free as their Jacobian ideals fail to +be perfect. However, an interesting problem remains as to the possibility of producing free divisors +by means of a similar technique, but with carefully chosen entries of higher degrees. + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +21 +(f) Our method does not work for higher dimensions in general. +For example, over the ring +k[x, y, z, w], consider the matrix + + +x +y +z +w +x − y +x + w +y − z +x + 3y +2y − z +3w +x − w +y + 2w + + . +The maximal minors are 4 cubics whose Jacobian matrix has a reduced determinant g ̸= 0. +A +computation shows that Jg is not perfect, so that g is not free. +We also derive some additional features. +Proposition 5.3. Let f be as in Theorem 5.1. The following assertions hold: +(i) Jf is of linear type. In particular, r(Jf) = 0; +(ii) R(Jf) (∼= SymRJf) is a complete intersection; +(iii) reg Derk(R/(f)) = 1. +Proof. (i) From the proof of the theorem, the 3 × 2 matrix ϕ is a minimal presentation matrix of Jf. +It follows easily by the structure of ϕ – which in particular has only linear forms as entries – that +the so-called G3 condition is satisfied (see the definition in the next section, right before Example +6.5). Moreover, it is clear that Jf has projective dimension 1 (see also Lemma 1.3). Now, applying +[42, Proposition 4.11] we obtain that Jf is of linear type. +(ii) By the previous item we have R(Jf) ∼= SymRJf, and the latter is the quotient of R[s, t, u] +(where s, t, u are variables over R) by the ideal generated by 2 linear forms ξ1, ξ2 in s, t, u, which are +the entries of the matrix product [s t u] · ϕ. Saying that ht (ξ1, ξ2) = 1 means precisely +ξ2 = λξ1, +for some non-zero +λ ∈ k, +which is equivalent to the first column of ϕ being λ times the second column. Following the proof of +the theorem, this would yield det Φ = 0, a contradiction. Therefore, ht (ξ1, ξ2) = 2. +(iii) Since f is a free cubic, this follows from Lemma 1.7. +We close the section with a working example which is, on the other hand, somewhat degenerated +in the sense that two of the Li’s are equal. +Example 5.4. Taking L (y, x, x) yields the line arrangement +1 +2f = −x2y + y3 + x2z − y2z = (x + y)(x − y)(z − y), +which is then free by Theorem 5.1. In this case, writing down the syzygy matrix ϕ of Jf as in the +proof of the theorem and multiplying their columns by suitable non-zero scalars, we get the following +simpler presentation matrix for Jf: + + +y +x +x +y +x +3y − 2z + + . +It follows by Proposition 5.3(ii) that the Rees algebra is the complete intersection ring +R(Jf) ∼= R[s, t, u]/(ys + x(t + u), xs + yt + (3y − 2z)u). +6. Maximal analytic spread and an application to homaloidness +Consider the standard graded polynomial ring R = k[x1, . . . , xn] = k ⊕ R+, n ≥ 3, and let f ∈ R +be a non-zero reduced homogeneous polynomial of degree d ≥ 3. Recall that the Jacobian ideal Jf +can be minimally generated by the derivatives fx1, . . . , fxn since by convention f is not allowed to +be a cone. Moreover, ht Jf ≥ 2 as f is reduced. + +22 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +6.1. When does the Jacobian ideal have maximal analytic spread? Our goal in this part is +to answer this question by means of various characterizations. Recall that +ht Jf ≤ ℓ(Jf) ≤ n, +so here we are specifically interested in the property ℓ(Jf) = n, which holds if for example Jf is of +linear type; indeed, in this case we can write F(Jf) = SymRJf/R+SymRJf ∼= Symk(Jf/R+Jf) ∼= R +as k-algebras. +Example 6.1. The Jacobian ideal of the (non-free) cubic +f = xyz + w3 ∈ k[x, y, z, w] +can be shown to be of linear type, and so ℓ(Jf) = 4. +On the other hand, as we have seen in Proposition 2.3(2), even for linear free divisors in at least +5 variables the analytic spread of Jf can be arbitrarily smaller than the number n of variables. +Some of the characterizations to be given here are of cohomological nature, and some rely on the +asymptotic behavior of depth. One of the ingredients is a suitable auxiliary module, which we now +introduce. As usual, we denote the gradient vector of a polynomial g ∈ R by ∇g = (gx1, . . . , gxn) ∈ +Rn. Given f as above, we set +Cf := Rn/ +� n +� +i=1 +R ∇fxi +� +. +A few preparatory concepts are in order before stating our result. +Let E be a finitely generated module over a Noetherian ring A and let G Φ→ F → E → 0 be an +A-free presentation of E. Consider the dual map HomA(Φ, A): F → G. The Auslander transpose +(or Auslander dual) of E is the A-module +Tr E = coker HomA(Φ, A), +which is unique up to projective summands. We refer to [3]. +Now, suppose A = � +i≥0 Ai is standard graded over a field A0 and let A+ = � +i≥1 Ai be the +homogeneous maximal ideal of A. Assume that the A-module E is graded as well. Then, given an +integer j ≥ 0, the j-th local cohomology module of E is the limit +Hj +A+(E) = lim +−→ Extj +A(A/As ++, E). +Saying that E ∼= E′ as graded A-modules means, as usual, that there is a degree zero isomorphism +between E and E′. +Finally, given r ≥ 1, recall that the Noetherian ring A is said to satisfy (Serre’s) condition Sr if +depth Ap ≥ min {r, ht p} +for all +p ∈ Spec A. +This clearly holds (for all r) if A is Cohen-Macaulay. +Back to the polynomial setup, our result here is as follows. +Theorem 6.2. Given f ∈ R as before, the following assertions are equivalent: +(i) ℓ(Jf) = n; +(ii) dim Cf = n − 1; +(iii) ∇fx1, . . . , ∇fxn are R-linearly independent; +(iv) Ext1 +R(Cf, R) ∼= Cf(d − 2) as graded R-modules; +(v) Hn−1 +R+ (Cf) ∼= HomR(Cf, k)(n − d + 2) as graded R-modules; +(vi) depth R/Jm +f = 0 for some m ≥ 1, where the bar denotes integral closure; +(vii) depth R/Jm +f = 0 for all m ≫ 0. +Moreover, if R(Jf) satisfies the S2 condition, then these assertions are also equivalent to the following +ones: + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +23 +(viii) depth R/Jm +f = 0 for all m ≫ 0; +(ix) depth R/Jm +f = 0 for some m ≥ 1. +Proof. Let H be the graded Hessian map of f, i.e. the degree zero homomorphism Rn(−(d − 2)) → +Rn whose matrix in the canonical bases is the Hessian matrix of f. The image of H is the submodule +of Rn generated by the homogeneous vectors ∇fx1, . . . , ∇fxn. Thus, Cf is the cokernel of H, i.e. it +has a graded R-free presentation +(22) +Rn(−(d − 2)) +H +−→ Rn −→ Cf −→ 0. +Dualizing this sequence, and denoting by E∗ (resp. ϕ∗) the R-dual of an R-module E (resp. an +R-module homomorphism ϕ), we get an exact sequence +(23) +0 −→ C ∗ +f −→ Rn +H∗ +−→ Rn(d − 2) −→ Cf(d − 2) −→ 0, +where we observe that, since the Hessian matrix is symmetric, H∗ = H ⊗ 1R(d−2) so that, indeed, +coker H∗ = (coker H)(d − 2) = Cf(d − 2). +Now, ℓ(Jf) is the dimension of the special fiber ring F(Jf) = k[fx1, . . . , fxn], which by Lemma +1.9 can be computed as the rank of the Hessian matrix of f. Thus, +ℓ(Jf) = rank H = rank H∗, +and hence ℓ(Jf) = n if and only if H is injective (this is of course equivalent to Rn(−(d − 2)) ∼= +�n +i=1 R ∇fxi via H, which thus proves (i)⇔ (iii)), if and only if H∗ is injective. The latter property +means C ∗ +f = 0. +Therefore, in order to prove (i)⇔ (iv), it suffices to verify that C ∗ +f = 0 if and only if (iv) holds. +Suppose C ∗ +f = 0. Then, as we have seen, H is injective. Dualizing (22) (which is now a short exact +sequence) and comparing with (23), we obtain (iv). Conversely, assume that (iv) takes place. Thus +Cf ∼= Ext1 +R(Cf, R)(2 − d) ∼= Ext1 +R(Cf(d − 2), R). +Now recall that the R-torsion τR(Cf) of Cf coincides with the kernel of the canonical biduality map +Cf → C ∗∗ +f , and so, by [3, Proposition 2.6(a)], we have +τR(Cf) ∼= Ext1 +R(Tr Cf, R). +But (23) gives Tr Cf = Cf(d − 2). Putting these facts together, we obtain +C ∗ +f ∼= Ext1 +R(Cf(d − 2), R)∗ ∼= Ext1 +R(Tr Cf, R)∗ ∼= τR(Cf)∗ = 0. +Next, let us prove that (iv)⇔ (v). The graded canonical module of the standard graded polynomial +ring R is ωR = R(−n), so +Ext1 +R(Cf, ωR) ∼= Ext1 +R(Cf, R)(−n). +Also recall that, in the present setting, the Matlis duality functor is given by HomR(−, k). Thus, by +graded local duality (see [7, Example 13.4.6]), we can write +(24) +Hn−1 +R+ (Cf) ∼= HomR(Ext1 +R(Cf, R)(−n), k) ∼= HomR(Ext1 +R(Cf, R), k)(n). +If (iv) holds, then Hn−1 +R+ (Cf) ∼= HomR(Cf(d − 2), k)(n) ∼= HomR(Cf, k)(n − d + 2). +Conversely, +suppose (v). Using (24), we get +HomR(Cf, k)(n − d + 2) ∼= HomR(Ext1 +R(Cf, R), k)(n), +which is the same as an isomorphism HomR(Cf(−n + d − 2), k) ∼= HomR(Ext1 +R(Cf, R)(−n), k). +Taking Matlis duals and tensoring with R(n), we obtain (iv). +We proceed to show that (i)⇔(ii). First note that, by (22), the 0-th Fitting ideal of Cf is the +principal ideal generated by the determinant h of the Hessian matrix of f, so we have +� +0 :R Cf = +� +(h) + +24 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +and hence dim Cf = dim R/(h). It follows that dim Cf = n−1 if and only if h ̸= 0, i.e. H is injective, +which as seen above is equivalent to (i). +We clearly have ℓ(Jf) = ℓ((Jf)R+) and ht R+ = n. Thus, by the general characterization given in +[30, Proposition 4.1], we have that ℓ(Jf) = n if and only if R+ ∈ AssRR/Jm +f for all m ≫ 0, which is +tantamount to saying that (vii) holds. This proves the equivalence (i)⇔(vii). +Evidently, (vii)⇒(vi), and the converse follows once we recall the chain (see [30, Proposition 3.4]) +AssRR/Jf ⊂ AssRR/J2 +f ⊂ AssRR/J3 +f ⊂ . . . +Thus, we have proved that the statements (i), . . . ,(vii) are equivalent. +Now we point out that the implication (vii)⇒(viii) holds regardless of R(Jf) satisfying S2. Indeed, +condition (vii) means that the irrelevant ideal R+ belongs to the limit value A +∗(Jf) of the function +m �→ AssRR/Jm +f , +which is known to eventually stabilize (see, e.g., [30, Proposition 3.4]). There is also the set A ∗(Jf) +defined analogously as the stable set of asymptotic prime divisors with respect to the usual filtration +given by the powers of Jf. By [30, Proposition 3.17], we have +A +∗(Jf) ⊂ A ∗(Jf) +and hence R+ ∈ A ∗(Jf), which gives (viii). Notice that (viii)⇒(ix) trivially. +It remains to show (ix)⇒(i), under the hypothesis that R(Jf) satisfies S2. In this case, by [14, +Remark 2.16], the extended Rees algebra +R[Jft, t−1] = +� +i∈Z +Iiti ⊂ R[t, t−1] +(where, by convention, Ii = R whenever i ≤ 0) must satisfy S2 as well. +Note that (ix) means +R+ ∈ AssRR/Jm +f +for some m ≥ 1. Now we are in a position to apply [14, Proposition 4.1] in order +to conclude that ℓ(Jf) = n, as needed. +Remark 6.3. With the aid of [30, Proposition 3.26 and Proposition 3.20], the assertions (i), . . . ,(vii) +of Theorem 6.2 are also seen to be equivalent to each of the following ones: +(a) R+ ∈ A +∗(IJf) for any non-zero R-ideal I, i.e., +depth R/ImJm +f += 0 +for all +m ≫ 0; +(b) For some j, the integral closure of R[fx1/fxj, . . . , fxn/fxj] ⊂ k(x1, . . . , xn) contains a prime +Q of height 1 such that Q ∩ R = R+. +While, in Theorem 6.2, the implication (ix)⇒(i) (and consequently the implication (viii)⇒(i)) +holds if the Rees ring R(Jf) satisfies S2, we do not know whether this hypothesis can be dropped. +Thus the following question becomes natural (see also Question 6.17 in the next subsection). +Question 6.4. Suppose depth R/Jm +f = 0 for all m ≫ 0. Is it true that ℓ(Jf) = n ? Does this hold +if we only assume that depth R/Jm +f = 0 for some m ≥ 1 ? +Now let f ∈ R be a linear free divisor. As we will see later, if n ≤ 4 then ℓ(Jf) = n. The converse +is known to be false, and in Example 6.5 below we show in addition that there is a linear free divisor +f such that ℓ(Jf) = n for any prescribed n. +The following well-known notion will be useful (we state it over R). A non-zero homogeneous ideal +I of R, minimally generated by ν elements, is said to satisfy the Gs condition for a given s ≥ 0 if +ht Iν−j(ϕ) ≥ j + 1 +for +j = 1, . . . , s − 1. +Here, ϕ denotes a minimal presentation matrix (or syzygy matrix) of I, and note that there is no +dependence on the choice of ϕ because each Iν−j(ϕ) is just a Fitting ideal of I. + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +25 +Example 6.5. Given an arbitrary n, consider the normal crossing divisor +f = x1 · · · xn ∈ R = k[x1, . . . , xn], +which is a well-known linear free divisor. +The ideal Jf is simply the ideal generated by all the +products of distinct n − 1 indeterminates, and satisfies +depth R/Jm +f += max {0, n − m − 1} +for all +m ≥ 1. +In particular, depth R/Jm +f += 0 for all m ≥ n − 1. We now claim that R(Jf) is Cohen-Macaulay +(hence it has the S2 property). Notice first that a syzygy matrix of Jf is given by +ϕ = + + +x1 +0 +. . . +0 +0 +x2 +. . . +0 +... +... +... +... +0 +0 +. . . +xn−1 +−xn +−xn +. . . +−xn + + +. +Here we have +ht In−j(ϕ) = j + 1 +for +j = 1, . . . , n − 1, +so that Jf satisfies the Gn property. Moreover, because f is free, Jf has projective dimension 1 over +R (see Lemma 1.3). It follows by [42, Proposition 4.11] that R(Jf) is Cohen-Macaulay, as claimed. +Now we are in a position to apply Theorem 6.2 to conclude that ℓ(Jf) = n. +In addition, the theorem gives us that Cf has projective dimension 1 over R (because the gradient +vectors of the natural generators of Jf generate a free module) and dimension n − 1, hence Cf is a +Cohen-Macaulay module, which yields +Hi +R+(Cf) ∼= +� +HomR(Cf, k)(2), i = n − 1 +0 +, i ̸= n − 1 +In Example 6.5, another way to confirm that ℓ(Jf) = n is by showing that the (monomial) ideal Jf +is of linear type. This fact and lots of other experiments suggest a more restrictive question as well +as a conjecture about the interplay between maximal analytic spread and the linear type property; +as already seen, the latter implies the former. +Question 6.6. For arbitrary n ≥ 3 (the number of variables), does there exist a free divisor f, with +Jf not of linear type, such that ℓ(Jf) = n ? +The case of interest is n ≥ 4. Indeed, if n = 3 then any member of the family of (non-linear) free +divisors given in Section 4 yields an affirmative answer to this question. +Conjecture 6.7. If f is a linear free divisor such that ℓ(Jf) = n, then Jf is of linear type. +We have not been able to solve this conjecture for n ≥ 5. It is true in n ≤ 4 variables, as we can +verify using the classification of linear free divisors given in [27, 6.4, p. 837]. +Next we furnish more examples. +Example 6.8. Consider the so-called Gordan-Noether cubic +f = xw2 + ytw + zt2 ∈ R = k[x, y, z, w, t]. +In this case, while the symmetric algebra SymRJf = B/S = R[t1, t2, t3, t4, t5]/S has dimension +6 and depth 5, a calculation shows that R(Jf) is Cohen-Macaulay (in particular, it has the S2 +condition). Indeed, if ϕ is a minimal presentation matrix of Jf, then the saturation +S :B I4(ϕ)∞, + +26 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +which by [31, Lemma 2.11] defines R(Jf) in the ring B (note that I4(ϕ) defines the non-principal +locus of Jf), is perfect of codimension 4. Now, further computations show that depth R/Jf = 2 and +depth R/Jm +f += 1 +for all +m ≥ 2. +Therefore, using Theorem 6.2, we conclude that ℓ(Jf) < 5. More precisely, the special fiber ring can +be expressed as F(Jf) = k[t1, t2, t3, t4, t5]/(t2 +2 − t1t3), which yields ℓ(Jf) = 4. +Example 6.9. Consider the quintic +f = 2w4u + xu4 + ywu3 + zw2u2 ∈ R = k[x, y, z, w, u]. +This is the case n = 5 of Theorem 2.2, hence f is a linear free divisor (here it should be mentioned, +for completeness, that f/u is not free and its Jacobian ideal is not even linearly presented). By +Proposition 2.3(ii), we have ℓ(Jf) = 4 < 5, and Theorem 2.5 ensures that R(Jf) is Cohen-Macaulay. +Therefore, by Theorem 6.2, we conclude that +depth R/Jm +f +> 0 +for all +m ≥ 1. +In fact, for such f it can be verified that depth R/Jf = 3, depth R/J2 +f = 2, and depth R/Jm +f = 1 for +all m ≥ 3. An interesting consequence of the non-vanishing of the asymptotic depth of Jf concerns +the higher conormal modules Jm +f /Jm+1 +f +. Indeed, in this situation the (also well-defined) conormal +asymptotic depth of Jf must be positive as well, since by [6] we can write +lim +m→∞ depth Jm +f /Jm+1 +f +≥ +lim +m→∞ depth R/Jm +f +> 0. +It follows that R+ /∈ AssRJm +f /Jm+1 +f +for all m ≫ 0. +Example 6.10. Consider the plane sextic +f = x6 − 2x3y2z + y4z2 + y6 ∈ R = k[x, y, z]. +This is the case α = 3 and β = 2 of Theorem 4.1(i), hence f is a free divisor (which is no longer +linear). By Theorem 4.1(ii), the ring R(Jf) is Cohen-Macaulay. It can be verified that +depth R/Jm +f += 0 +for all +m ≥ 2. +Applying Theorem 6.2 we obtain that ℓ(Jf) = 3. Now from Theorem 4.1(iii) we know that Jf cannot +be of linear type. To see this explicitly, one of the minimal generators of the defining ideal of the +Rees algebra in the ring R[t1, t2, t3] is the following polynomial which is not linear in the ti’s: +3xyt1t2 + 2xzt1t3 − 3yzt2t3 − 3y2t2 +3 − 2z2t2 +3. +Furthermore, the theorem yields Ext1 +R(Cf, R) ∼= Cf(4) and +Hj +R+(Cf) ∼= +� +HomR(Cf, k)(−1), j = 2 +0 +, j ̸= 2 +Example 6.11. Consider the quartic +f = x4 − xyz2 + z3w ∈ R = k[x, y, z, w], +which is not free as Jf is not perfect. Let us also mention that the ideal Jf is not of linear type, +since the polynomial +4xt2 +2 − 4zt1t4 − yt2 +4 ∈ R[t1, t2, t3, t4] +is one of the minimal generators of the defining ideal of R(Jf). On the other hand, it is not hard to +verify that the associated graded ring of Jf – i.e. the graded algebra � +s≥0 Js +f/Js+1 +f +– satisfies the +S1 property; since in addition ht Jf ≥ 2, we get by [14, Remark 2.16] that R(Jf) satisfies S2 (it can +be shown that in fact R(Jf) is Cohen-Macaulay). Furthermore, depth R/Ji +f = 1 for i = 1, 2, 3, while +depth R/Jm +f += 0 +for all +m ≥ 4. + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +27 +Applying Theorem 6.2, we conclude that ℓ(Jf) = 4. We also get Ext1 +R(Cf, R) ∼= Cf(2), and +Hj +R+(Cf) ∼= +� +HomR(Cf, k)(2), j = 3 +0 +, j ̸= 3 +6.2. Application: Criterion for homaloidness. As above let f ∈ R = k[x1, . . . , xn], n ≥ 3, be a +non-zero reduced homogeneous polynomial. In this subsection, we assume additionally that the field +k is algebraically closed. To the form f we can associate the rational map +Pf = (fx1 : · · · : fxn) : Pn−1 ��� Pn−1, +the so-called polar map defined by f. Thus the base locus of Pf is the singular locus of the projective +hypersurface V (f) ⊂ Pn−1. +Definition 6.12. ([21]) The polynomial f is homaloidal if Pf is birational (hence a Cremona +transformation). +Over k = C, this definition can be translated by saying that Pf has degree 1 (taking into account +an appropriate notion of degree in this context), and according to [19, Corollary 2] the property of +being homaloidal depends only on fred. +The following is a preliminary fact connecting this class of polynomials to the class of free divisors. +It can be also seen as a first source of examples of homaloidal divisors (examples in higher dimensions +can be found, e.g., in [13] and [33]). Recall that in general the dimension of the image of the polar +map Pf is given by ℓ(Jf) − 1 (see the proof of Proposition 6.14). +Proposition 6.13. (k = C) If n ≤ 4 then every linear free divisor is homaloidal. +Proof. +Let f ∈ R be a linear free divisor. +Recall we are supposing that f is not a cone (see +Subsection 1.1). Thus, by [25, Proposition 2.4 and Proposition 2.5], f has a non-zero Hessian, so +that ℓ(Jf) = n. Hence, the dimension of the image of Pf is n − 1. As the linear rank of the gradient +ideal Jf is maximal, it follows by [22, Theorem 3.2] that Pf is birational. +Notice that this proposition fails if n ≥ 5. Indeed, if in this case we take f as being a linear free +divisor as described in Theorem 2.2, then by Proposition 2.3(ii) the analytic spread of Jf is 4, hence +the image of Pf has dimension at most n − 2 and so this map cannot be birational. +Our application regarding homaloidness is the following ideal-theoretic, also homological, version +of the criterion given in [22, Theorem 3.2]. It is not as practical or effective as the original one, but +in our view it adds some flavor to the classical – typically geometric – theory and, moreover, helps +linking to different algebraic tools and invariants. +Proposition 6.14. Given f ∈ R as before, let ϕ1 be the submatrix of a minimal syzygy matrix of +the ideal Jf consisting of its linear syzygies, and suppose In−1(ϕ1) ̸= (0). Assume any one of the +following situations: +(i) projdim Jm +f = n − 1 for some m ≥ 1; +(ii) R(Jf) satisfies S2, and projdim Jm +f = n − 1 for some m ≥ 1. +Then f is homaloidal. +Proof. First, in either case, our Theorem 6.2 (together with the Auslander-Buchsbaum formula) +ensures that ℓ(Jf) = n. On the other hand, +dim(image Pf) = dim Proj k[fx1, . . . , fxn] = dim Proj F(Jf) = ℓ(Jf) − 1 = n − 1. +Now [22, Theorem 3.2] ensures that Pf is birational, as needed. +Example 6.15. Let us first point out that not all homaloidal polynomials satisfy the condition +In−1(ϕ1) ̸= (0). Indeed, consider the cubic +f = xw2 + yzw + z3 ∈ k[x, y, z, w]. +Then it can be checked that: + +28 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +(a) f is an irreducible homaloidal polynomial. This is indeed the first member of the family of +irreducible homaloidal hypersurfaces described in [28, p. 1264]; +(b) The Jacobian ideal Jf is not linearly presented, and moreover has not enough linear syzygies. +More precisely, only 2 columns of a minimal presentation matrix ϕ are linear syzygies, and +hence obviously I3(ϕ1) = (0); +(c) Jf is not of linear type; +(d) Quite interestingly, Jf satisfies the conditions present in part (ii) of our Proposition 6.14. In +particular, it can be even shown that the Rees algebra of Jf is Cohen-Macaulay. +We now remark that if f is a linear free divisor then the condition In−1(ϕ1) ̸= (0) is automatically +satisfied as in this case ϕ1 = ϕ and In−1(ϕ) = Jf by the Hilbert-Burch theorem. We thus record the +following corollary. +Corollary 6.16. If f is a linear free divisor satisfying either condition (i) or (ii) of Proposition +6.14, then f is homaloidal. +Before giving the first illustration, we raise the following question. We remark that the answer +is yes if the second part of Question 6.4 has an affirmative answer as well. +Also note that, by +Proposition 6.13, the case of interest is n ≥ 5. +Question 6.17. (n ≥ 5) Let f be a linear free divisor satisfying projdim Jm +f = n−1 for some m ≥ 1. +Must f be homaloidal? +Example 6.18. The simplest example in arbitrary dimension is the normal crossing divisor f = +x1 · · · xn ∈ R studied in Example 6.5. Then f is a linear free divisor and we have seen in particular +that depth R/Jn−1 +f += 0, i.e., projdim Jn−1 +f += n − 1. Since Jf is the ideal generated by all squarefree +monomials of degree n − 1, we get by [50, Proposition 7.4.5] that all powers of Jf are integrally +closed; in particular, +Jn−1 +f += Jn−1 +f +. +It follows by Corollary 6.16 (or Proposition 6.14(i)) that f is homaloidal, thus retrieving the well- +known fact that the rational map Pn−1 ��� Pn−1 given by +(x1 : . . . : xn) �→ (x2x3 · · · xn : x1x3 · · · xn : . . . : x1x2 · · · xn−1) +is birational – the so-called Cremona involution on Pn−1. +Below we illustrate Proposition 6.14 in the situation where f is not free, and in both reducible +and irreducible cases. +Example 6.19. (n = 6) Consider the hyperplane-quadric arrangement +f = xw(yz + zt + tu) ∈ R = k[x, y, z, w, t, u]. +In this case, f is non-free because Jf is not perfect, while on the other hand this ideal (which is +linearly presented, so that ϕ1 = ϕ) satisfies I5(ϕ1) ̸= (0) and +projdim J3 +f = 5. +Moreover, as in Example 6.11, the associated graded ring of Jf has the S1 property and hence R(Jf) +satisfies S2. By Proposition 6.14(ii), f is homaloidal. i.e., the rational map P5 ��� P5 given by +(x : y : z : w : t : u) �→ (w(yz + zt + tu) : xzw : xw(y + t) : x(yz + zt + tu) : xwu : xwt) +is Cremona. + +FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD +29 +Example 6.20. (n = 5) Consider the irreducible cubic +f = xt2 + yzt + z3 + w2t ∈ R = k[x, y, z, w, t]. +The ideal Jf is perfect but f is non-free as ht Jf = 3. It also satisfies I4(ϕ1) ̸= (0) and +projdim J3 +f = 4. +Moreover, the associated graded ring of Jf is Cohen-Macaulay; in particular, R(Jf) satisfies S2. By +Proposition 6.14(ii), f is homaloidal. Explicitly, the rational map P4 ��� P4 given by +(x : y : z : w : t) �→ (t2 : zt : z2 + 1 +3yt : wt : yz + w2 + 2xt) +is Cremona. +Next, we provide a couple of additional observations and questions that, in our view, are interesting +and potentially motivating for future research. First, note that if we write the homaloidal quartic f +of Example 6.19 as +f = xg, +g = w(yz + zt + tu), +then a further calculation shows (using again our proposition) that g is homaloidal as well. This +fact, among other examples, led us to suggest the following “addition-deletion” problem inspired by +well-known investigations in free divisor theory (see [43], also [1] and [39]). +Question 6.21. (Addition-deletion for homaloidal divisors.) For polynomials f, g ∈ R, with f homa- +loidal, when is the product fg homaloidal? If fg is homaloidal, when is f or g homaloidal? +Now let f ∈ R = k[x, y, z, w, t, u, v] stand for the 2-catalecticant determinant +f = det + + +x +y +z +z +w +t +t +u +v + + . +According to [33, Proposition 3.25(b)]), this cubic is homaloidal. Then, for such f, we have detected +an intriguing, curious fact: the determinant h(f) of the Hessian matrix of f is a linear free divisor +– in particular, h(f) is already reduced. In the situation where h(f) is not reduced, we naturally +consider h(f)red, which likewise can be a linear free divisor. For example, let +g = det + + +x +w +z +y +y +x +w +z +w +z +y +x +z +y +x +w + + +in the ring R = k[x, y, z, w]. Note g is in fact a linear free divisor, and using Corollary 6.16 it is not +hard to see that g is also homaloidal. Here, +h(g) = λg2 +for some non-zero +λ ∈ k, +and therefore h(g)red is free. +As expected, this phenomenon does not take place in general. For instance, if once again we take +f as the homaloidal quartic of Example 6.19, then a calculation shows h(f) = 3x2w2f 2, so that +h(f)red = 3f is not free. +The facts above led us to raise the following question, which reconnects us to the central topic of +freeness and closes the paper. +Question 6.22. Let f be a homaloidal polynomial. When is h(f)red a (linear) free divisor? If h(f) +is reduced and not a cone, must it be a (linear) free divisor? + +30 +R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS +Acknowledgements. The second-named author was supported by the CNPq grants 301029/2019-9 +and 406377/2021-9. The third-named author was supported by the CNPq grants 305860/2019-4 and +425752/2018-6. +References +[1] T. Abe, Divisionally free arrangements of hyperplanes, Invent. Math. 204 (2016) 317–346. +[2] T. Abe, H. Terao, M. Yoshinaga, Totally free arrangements of hyperplanes, Proc. Amer. Math. Soc. 137 (2009) +1405–1410. +[3] M. Auslander, M. Bridger, Stable module theory, Mem. 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Toulouse Math. 23 (2014) +483–512. +Departamento de Matem´atica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, Para´ıba, +Brazil. +Email address: ricardo@mat.ufpb.br +Departamento de Matem´atica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, Para´ıba, +Brazil. +Email address: cleto@mat.ufpb.br +Departamento de Matem´atica, CCET, Universidade Federal de Sergipe, 49100-000 S˜ao Cristov˜ao, +SE, Brazil. +Email address: zaqueu@mat.ufs.br + diff --git a/IdE1T4oBgHgl3EQfYAQI/content/tmp_files/load_file.txt b/IdE1T4oBgHgl3EQfYAQI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab9d1d171f5b3fa6b7bc890408f3a8fbb812d7cf --- /dev/null +++ b/IdE1T4oBgHgl3EQfYAQI/content/tmp_files/load_file.txt @@ -0,0 +1,2007 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf,len=2006 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='03132v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='AC] 9 Jan 2023 FREE DIVISORS, BLOWUP ALGEBRAS OF JACOBIAN IDEALS, AND MAXIMAL ANALYTIC SPREAD RICARDO BURITY, CLETO B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND ZAQUEU RAMOS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay – a desirable feature in blowup algebra theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tools ranging from cohomology to asymptotic depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In addition, as an application, we give an ideal-theoretic homological criterion for homaloidal divisors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', hypersurfaces whose polar maps are birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Dedicated with gratitude to the memory of Professor Wolmer V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Vasconcelos, mentor of generations of commutative algebraists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Introduction The well-studied theory of free divisors – or free hypersurfaces – has its roots in the seminal work of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Saito [35], and in subsequent papers of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Terao [43, 44, 45] mostly concerned with the case of hyperplane arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The original environment was the complex analytic setting, and the motivation was the computation of Gauss-Manin connections for the universal unfolding of an isolated singularity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' for instance, it was proved that the discriminant in the parameter space of the universal unfolding is a free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Over time, different approaches, viewpoints, and interests have emerged, including algebraic (and algebro-geometric) adaptations and even generalizations that have drawn the attention of an increasing number of researchers over the last fifteen years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The list of references is huge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', Abe [1], Abe, Terao and Yoshinaga [2], Buchweitz and Conca [8], Buchweitz and Mond [9], Calder´on-Moreno and Narv´aez-Macarro [12], Damon [15], Dimca [17, 18], Dimca and Sticlaru [20], Miranda-Neto [31, 32], Schenck [37], Schenck, Terao and Yoshinaga [38], Schenck and Tohˇaneanu [39], Simis and Tohˇaneanu [41], Tohˇaneanu [47], and Yoshinaga [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, nice references containing interesting open problems on the subject (including the celebrated Terao’s Conjecture) are Dimca’s book [17] and Schenck’s survey [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In the present paper, the general goal is to present progress on the algebraic side of the theme, by means of various techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First, we explicitly describe four new families of homogeneous free divisors in standard graded polynomial rings over a field k with char k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Second, and in the same graded setup, we turn our angle to investigating blowup algebras of Jacobian ideals of polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' More precisely, we prove that for our families the Rees algebra is Cohen-Macaulay – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', from a geometric point of view, blowing-up their singular loci yields arithmetically Cohen-Macaulay schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Furthermore we characterize in various ways, via tools varying from (local) cohomology to asymptotic depth, the maximality of the dimension of the special fiber ring for polynomials which are no longer required to be free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The relevance of the latter lies in connections to the important 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Primary: 14J70, 32S05, 13A30, 14E05, 14M05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Secondary: 13C15, 13H10, 14E07, 32S22, 32S25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Free divisor, Jacobian ideal, blowup algebra, Rees algebra, analytic spread, homaloidal divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Corresponding author: Cleto B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Miranda-Neto (cleto@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='ufpb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 1 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS theory of homaloidal divisors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', homogeneous polynomials f ∈ k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn] (where, typically, the field k is assumed to be algebraically closed) for which the associated polar map Pf = � ∂f ∂x1 : · · · : ∂f ∂xn � : Pn−1 ��� Pn−1 is birational – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', Pf is a Cremona transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In fact we provide, as an application, an ideal- theoretic (also homological) homaloidness criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It is worth mentioning that the modern theory about such polynomials began in Ein and Shepherd-Barron [23], where it was proved for instance that the relative invariant of a regular prehomogeneous complex vector space is homaloidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Another classical reference is Dolgachev [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In order to introduce the other main concepts of interest to this paper, let k denote a field of characteristic zero and, given n ≥ 3, let R = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn] be a standard graded polynomial ring over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let R+ = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn) be the irrelevant ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Given a non-zero reduced homogeneous polynomial f ∈ R2 + (whose partial derivatives will be assumed, to avoid pathologies, to be k-linearly independent), it is well-known that the property of f being a free divisor can be translated into saying that the corresponding Jacobian ideal Jf = (∂f/∂x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , ∂f/∂x1) ⊂ R is perfect of codimension 2 – in particular, a free f must be highly singular in the sense that codim Sing V (f) = 2 regardless of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, the intervention of Jf in the theory is (naturally) crucial, and as a bonus this allows for an interesting link to the study of blowup algebras, particularly the traditional problem of describing ideals for which such rings are Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Here, we are especially interested in the Rees algebra R(Jf) = R � ∂f ∂x1 t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , ∂f ∂xn t � ⊂ R[t] and its special fiber ring F(Jf) = R(Jf) ⊗R k, which, as is well-known from blowup theory, encode relevant geometric information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Recall that the analytic spread of Jf, denoted ℓ(Jf), is the Krull dimension of F(Jf), which is bounded above by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Saying that Jf has maximal analytic spread means ℓ(Jf) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Next we briefly describe the contents of each section of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Section 1 invokes the definitions that are central to this paper, such as the notions of free divisor and blowup algebras of ideals, as well as a few auxiliary facts which will be used in some parts of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Also, some conventions are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In Section 2 we present our first family of free divisors in R, with n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' They are reducible and, in fact, linear in the sense that in addition the Jacobian ideal Jf is linearly presented, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', the entries of the corresponding Hilbert-Burch matrix are (possibly zero) linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We also determine the defining equations of F(Jf) and compute the analytic spread as well as the reduction number of Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, we prove that R(Jf) and F(Jf) are Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Section 3 describes our second family of free divisors, in an even number of at least 4 variables, and again reducible and linear in the above sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We exhibit a well-structured minimal set of generators for the module of syzygies of Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In addition, as in the previous family – but via different methods – we show that R(Jf) and F(Jf) are Cohen-Macaulay (the latter is in fact shown to be a generic determinantal ring) and determine the analytic spread and the reduction number of Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In Section 4, our third family is presented as a two-parameter family of (no longer linear) free divisors f = fα,β in 3 variables and of degree αβ, where α, β ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For the one-parameter family with β = 2 (also for (α, β) = (2, 3)), we show that R(Jf) is Cohen-Macaulay and derive that F(Jf) is isomorphic to a polynomial ring over k (so that Jf has reduction number zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Also we prove that f is reducible if k = C, and in case k ⊆ R we verify that f is reducible if β is odd and irreducible otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In addition, if β ≥ 3 is odd, we show how to derive yet another two-parameter family of free divisors g = gα,β (of degree αβ − α) from fα,β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' for the one-parameter family with β = 3, we deduce that R(Jg) is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 3 In Section 5 we introduce our fourth family of free divisors, in 3 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Such (reducible) divisors have the linearity property – as in the two first families – and are constructed as the determinant of the Jacobian matrix of a set of quadrics which we associate to 3 given suitable linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This family is in fact partially new, because if k = C then its members are recovered by the well-known classification of linear free divisors in at most 4 variables, whereas on the other hand our (permanent) assumption on k is that char k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Furthermore, we show that Jf is of linear type – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', the canonical epimorphism from the symmetric algebra of Jf onto R(Jf) is an isomorphism – and we derive that R(Jf) is a complete intersection ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For f ∈ R belonging to any of the four (or five) families, we use a result from Miranda-Neto [31] to easily determine the Castelnuovo-Mumford regularity of the graded module Derk(R/(f)) formed by the k-derivations of the ring R/(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Needless to say, the regularity is an important invariant which controls the complexity of a module (being related to bounds on the degrees of syzygies), whereas the derivation module is a classical object as it collects the tangent vector fields defined on the hypersurface V (f) ⊂ Pn−1 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We close the paper with Section 6, where we address the question as to when, for a (not necessarily free) polynomial f ∈ R, the ideal Jf has maximal analytic spread – the relevance of this task is the already mentioned connection to the theory of homaloidal divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We provide a number of charac- terizations of when such maximality holds, including cohomological conditions on a suitable auxiliary module as well as the asymptotic depth associated to both adic and integral closure filtrations of Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We also point out that the main problems we raise in this paper appear in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This includes a conjecture predicting that if f is linear free divisor satisfying ℓ(Jf) = n then Jf is of linear type (the case of interest is n ≥ 5), as well as the question of whether the reduced Hessian determinant of a homaloidal polynomial must necessarily be a (linear) free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For all such problems we were motivated and guided by several examples, and the computations were performed with the aid of the program Macaulay of Bayer and Stillman [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Preliminaries: Free divisors and blowup algebras We begin by invoking some definitions and auxiliary facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First we establish the convention that, throughout the entire paper, k denotes a field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' A few other conventions (including notations) will be made in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let R = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn] be a standard graded polynomial ring in n ≥ 3 indeterminates x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn over k, and let R+ = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn) denote the homogeneous maximal ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Free divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Fix a non-zero homogeneous polynomial f ∈ R2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' A logarithmic derivation of f is an operator θ = �n i=1 gi∂/∂xi, for homogeneous polynomials g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , gn ∈ R satisfying θ(f) = n � i=1 gi ∂f ∂xi ∈ (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Geometrically, θ can be interpreted as a vector field defined on Pn−1 k that is tangent along the (smooth part of the) hypersurface V (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' From now on we suppose f is reduced in the usual sense that fred = f, that is, f is (at most) a product of distinct irreducible factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In addition, we assume throughout – with no further mention – that the partial derivatives of f are k-linearly independent so as to prevent f from being a cone (recall that a polynomial g ∈ R is a cone if, after some linear change of coordinates, g depends on at most n − 1 variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Denote by TR/k(f) the R-module formed by the logarithmic derivations of f, which is also called tangential idealizer (or Saito-Terao module) of f, and commonly denoted Derlog(−V (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It is easy to see that TR/k(f) has (generic) rank n as an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' f is a free divisor if the R-module TR/k(f) is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS This concept, which originated in [35], has been shown to be of great significance to a variety of branches in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We recall yet another classical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If fxi := ∂f/∂xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , n, then the Jacobian ideal of f (also called gradient ideal of f) is given by Jf = (fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn) ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note that, because f is not a cone, the ideal Jf is minimally generated by the n partial derivatives of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Also recall that the Euler derivation εn := n � i=1 xi ∂ ∂xi is logarithmic for the homogeneous polynomial f by virtue of the well-known Euler’s identity �n i=1 xifxi = (deg f)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now we remark that, since TR/k(f) decomposes into the direct sum of the module of syzygies of Jf and the cyclic module Rεn (see [31, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2]), a free basis of TR/k(f) if f is a free divisor consists of the derivations corresponding to the columns of a minimal syzygy matrix of fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn together with εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Next we recall a useful characterization which is even adopted as the definition of free divisor by some authors, and moreover highlights the central role that commutative algebra plays in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ([31, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1]) f ∈ R is a free divisor if and only if Jf is a codimension 2 perfect ideal (equivalently, the ideal Jf has projective dimension 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In other words, f is a free divisor if and only if R/Jf is a Cohen-Macaulay ring and ht Jf = 2, where, here and in the entire paper, ht I stands for the height of an ideal I ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows that the classical Hilbert-Burch theorem plays a major role in the algebraic side of free divisor theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It is also worth mentioning that this fruitful interplay holds in a more general setting (see [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Below we invoke a well-known and very useful criterion of freeness detected by Saito himself in case k = C, but which is known to hold over any field of characteristic zero (see [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ([35, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='8(ii)], also [17, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1]) f ∈ R is a free divisor if and only if there exist n vector fields θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , θn ∈ TR/k(f) such that det [θj(xi)]i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=',n = λf for some non-zero λ ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In this case, the set {θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , θn} is a free basis of TR/k(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' As already pointed out, up to elementary operations in the columns of an n × (n − 1) syzygy matrix ϕ of Jf, the derivations θj’s of the free basis above correspond to the columns of ϕ along with the Euler vector field εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' There is also the following important subclass introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' f is a linear free divisor if f is a free divisor and the ideal Jf is linearly presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Stated differently, f is a linear free divisor if and only if Jf admits a minimal graded R-free resolution of the form 0 −→ R(−n)n−1 −→ R(−n + 1)n −→ Jf −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, the degree of a linear free divisor is necessarily equal to n (and thus has minimal degree, since any free divisor is seen to have degree at least n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now let us provisionally consider a more general setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let S be any Noetherian commutative ring containing k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' A k-derivation of S is defined as an additive map ϑ: S → S which vanishes on k and satisfies Leibniz’ rule: ϑ(uv) = uϑ(v) + vϑ(u), for all u, v ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Such objects are collected in an S-module, denoted Derk(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, if again R = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn], we get the R-module Derk(R), which is free on the ∂/∂xi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now if f ∈ R2 + is as above, we can also consider the FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 5 derivation module Derk(R/(f)), which can be graded as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First, assume that Derk(R) is given the grading inherited from the natural Z-grading of the Weyl algebra of R, so that each ∂/∂xi has degree −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We endow TR/k(f) with the induced grading from Derk(R), that is, a logarithmic derivation �n i=1 gi∂/∂xi ∈ TR/k(f) has degree δ if g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , gn ∈ R have degree δ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For example, εn ∈ [TR/k(f)]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Finally recall that there is an identification (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', [31, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1]) Derk(R/(f)) = TR/k(f)/fDerk(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then we let Derk(R/(f)) be graded with the grading induced from TR/k(f) by means of this quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The next auxiliary lemma is concerned with the graded module Derk(R/(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let us first recall the concept of Castelnuovo-Mumford regularity of a finitely generated graded module E over the graded polynomial ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let 0 → Fp → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' → F0 → E → 0 be a minimal graded R-free resolution of E, where Fi := �bi j=1 R(−ai,j), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note that p is the projective dimension of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If mi := max{ai,j | 1 ≤ j ≤ bi}, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , p, then the Castelnuovo-Mumford regularity of E is defined as reg E = max{mi − i | 0 ≤ i ≤ p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This gives in some sense a numerical measure of the complexity of the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' There are more general definitions given in terms of sheaf and local cohomologies (which in turn are related), but the one given above suffices for our purposes in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We refer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', to [4] and [7, Chapter 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ([31, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5(i)]) If f ∈ R is a free divisor of degree d, then reg Derk(R/(f)) = d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, if f ∈ R is a linear free divisor then reg Derk(R/(f)) = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It is worth mentioning that some authors have investigated the Castelnuovo-Mumford regularity of other objects that are also “differentially related” to f, such as the Milnor algebra R/Jf (see [11]) and the module TR/k(f) itself (see [16, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5] and [36, Section 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Blowup algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We close the section with a brief review on blowup algebras and a few closely related notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We fix a homogeneous proper ideal I of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The Rees algebra of I is the graded ring R(I) = � i≥0 Iiti ⊂ R[t], where t is an indeterminate over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This R-algebra defines the blowup along the subscheme corre- sponding to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The special fiber ring of I, sometimes dubbed fiber cone of I, is the special fiber of R(I), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', the (standard) graded k-algebra F(I) = R(I) ⊗R k ∼= � i≥0 Ii/R+Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The analytic spread of I is ℓ(I) = dim F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' There are bounds ht I ≤ ℓ(I) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Alternatively, R(I) can be realized as the quotient of the symmetric algebra SymRI (a basic construct in algebra) by its R-torsion submodule, which is in fact an ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus there is a natural R-algebra epimorphism SymRI → R(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If this map is an isomorphism, I is said to be of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Since R is in particular a domain, this is tantamount to saying that SymRI is a domain as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For instance, any ideal generated by a regular sequence is of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Next we provide a useful formula for the computation of the analytic spread by means of a Jacobian matrix (in characteristic zero, as we have permanently assumed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' To this end we consider an even more concrete description of the Rees algebra (hence of its special fiber), to wit, if we fix generators I = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fν) ⊂ R = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn], then R(I) is just the R-subalgebra generated by f1t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fνt ∈ R[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In the particular case where the fi’s are all homogeneous of the same degree – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', the partial derivatives of a homogeneous polynomial – we can write the special fiber as F(I) ∼= k[f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fν] as a k-subalgebra of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ([40, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1]) Write I = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fν) and suppose all the fi’s are homogeneous of the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Set Θ := � ∂fi ∂xj � , 1 ≤ i ≤ ν, 1 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then ℓ(I) = rank Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Finally recall that a subideal K ⊂ I is a reduction of I if the induced extension of Rees algebras R(K) ⊂ R(I) is integral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' equivalently, there exists r ≥ 0 such that Ir+1 = KIr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The minimal such r is denoted rK(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The reduction K is minimal if it is minimal with respect to inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now the reduction number of I is defined as r(I) = min{rK(I) | K is a minimal reduction of I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For instance, it is a standard fact (as k is infinite) that r(I) = 0 if and only if I can be generated by ℓ(I) elements, which occurs if for example I is of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' More generally, the following basic result gives a way to compute this number in the presence of a suitable condition on the standard graded k-algebra F(I), which can be also regarded (for the purpose of reading the Castelnuovo-Mumford regularity off a minimal graded free resolution) as a cyclic graded module over a polynomial ring k[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , tν] whenever I can be generated by ν forms in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ([26, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2]) If F(I) is Cohen-Macaulay, then r(I) = reg F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First family: linear free divisors in Pn−1 Before presenting our first family of free divisors as well as properties of related blowup algebras, let us record a couple of basic calculations which will be used without further mention in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let S = k[w, u] be a standard graded polynomial ring in 2 variables w, u, and consider the ideal n = (w, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Given an integer r ≥ 2, the following facts are well-known and easy to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (a) The ideal nr = (wr, wr−1u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , wur−1, ur) is a perfect ideal of codimension 2, having the following (r + 1) × r syzygy matrix: (1) ϕr = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 −w 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 u −w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −w 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' u \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (b) The presentation ideal of the Rees algebra R(nr), that is, the kernel of the surjective map of S-algebras S[y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , yr+1] ։ R(nr), yi �→ wr−i+1ui−1, is equal to Q = (I1(y · ϕr), I2(B)), where y = � y1 · · yr+1 � and B = � y1 · · yr y2 · · yr+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider the standard graded polynomial ring R = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn], where n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Denote xn−1 = w and xn = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let f = 2wn−1u + n−2 � i=1 xiwi−1un−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then f is a linear free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First notice that (2) fxi = wi−1un−i for each 1 ≤ i ≤ n − 2, FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 7 fw = 2(n − 1)wn−2u + n−2 � i=2 (i − 1)xiwi−2un−i and fu = 2wn−1 + n−2 � i=1 (n − i)xiwi−1un−(i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, the subideal (fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn−2) of Jf is equal to the ideal u2(w, u)n−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, if ϕn−3 is the (n − 2) × (n − 3) syzygy matrix of the ideal (w, u)n−3 (see (1)), then the columns of the n × (n − 3) matrix η = � ϕn−3 0 � are syzygies of the gradient ideal Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We also have the following equalities (3) ufw = 2(n − 1)wn−2u2 + n−2 � i=2 (i − 1)xiwi−2un−(i−1) = 2(n − 1)wfxn−2 + n−2 � i=2 (i − 1)xifxi−1 and (4) (n − 1)ufu = 2(n − 1)wn−1u + n−2 � i=1 (n − 1)(n − i)xiwi−1un−i = wfw + n−1 � i=1 (n(n − i − 1) + 1)xifxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now note that (3) and (4) yield two new (linear) syzygies of Jf, to wit, δ1 = � α2x2 α3x3 · · αn−2xn−2 2(n − 1)w −u 0 �t and δ2 = � β1x1 β2x2 · · βn−2xn−2 w −(n − 1)u �t where αi = i − 1 if 2 ≤ i ≤ n − 2, and βi = n(n − i − 1) + 1 whenever 1 ≤ i ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The minimal graded free resolution of Jf is (5) 0 → R(−n)n−1 ψ −→ R(−n + 1)n → Jf → 0 where ψ = � η δ1 δ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' From the discussion above, we already know that the sequence (5) is a complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' To prove that it is in fact a minimal graded free resolution of Jf, it suffices to verify that ht In−1(ψ) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note we can write ψ in the form ψ = � ϕn−3 ∗ 0 Φ � where Φ = � −u w 0 −(n − 1)u � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, det Φ · In−3(ϕn−3) = u2 · (w, u)n−3 ⊂ In−1(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, un−1 ∈ In−1(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' On the other hand, if we specialize the entries of ψ via the k-algebra endomorphism of R that fixes the variables w, u and maps the remaining ones to 0, we obtain the matrix ψ = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 −w 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 0 u −w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 0 0 u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −w 0 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' u 2(n − 1)w 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 −u w 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 −(n − 1)u \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The (n − 1)-minor of ψ obtained by omitting the last row is cwn−1 for a certain non-zero c ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, the (n − 1)-minor of ψ obtained by omitting the last row has the shape cwn−1 + G, for a suitable G ∈ (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, (un−1, cwn−1 + G) ⊂ In−1(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, ht In−1(ψ) ≥ 2 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS A computation shows that, in the first case covered by the theorem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', n = 4), the Jacobian ideal Jf is of linear type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' in particular, ℓ(Jf) = 4 and r(Jf) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The case n ≥ 5 is treated in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' As ingredients we consider the polynomial rings T := k[y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , yn−2, s, t] and T ′ := k[y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , yn−2] as well as the k-algebras A := k[fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn−2, fw, fu] and A′ := k[fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By factoring out u2 from the polynomials fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn−2 (see (2)), we see that A′ ∼= A′′ := k[wn−3, wn−4u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , un−3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' All these rings are related via the following commutative diagram of k-algebras: (6) T � � A T ′�� � � � A′ ∼ = � �� � A′′ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Maintain the above notations, and let f ∈ R be as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2, with n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The following assertions hold: (i) F(Jf) ∼= T/I2 � y1 · · yn−3 y2 · · yn−2 � as k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, F(Jf) is Cohen-Macaulay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) ℓ(Jf) = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) r(Jf) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iv) reg Derk(R/(f)) = n − 2 (also for n = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (i) Since Jf is a homogeneous ideal generated in the same degree, there is an isomorphism of graded k-algebras F(Jf) ∼= A, so that F(Jf) ∼= T/Q where Q := ker (T ։ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By the diagram (6), we get Q′T ⊂ Q, where Q′ := ker (T ′ ։ A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' From Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1(b) we have Q′T = I2 � y1 · · yn−3 y2 · · yn−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, ht Q ≥ ht Q′T = n−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, in order to prove that Q = I2 � y1 · · yn−3 y2 · · yn−2 � , we must show ht Q ≤ n − 4, or equivalently, dim A ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, on the other hand, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='9 gives dim A = rank Θ, where Θ is the Hessian matrix of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Notice that the (Jacobian) matrix Θ can be written in blocks as Θ = � 0 Θw,u Θt w,u ∗ � where Θw,u is the (n − 2) × 2 Jacobian matrix of fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn−2 with respect to w, u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Clearly, I2(Θw,u) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, I4(Θ) ̸= 0 because I2(Θw,u)2 ⊂ I4(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence rank Θ ≥ 4, so that dim A = rank Θ ≥ 4, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The Cohen-Macaulayness of F(Jf) will be confirmed below, in the proof of item (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) Since ℓ(Jf) = dim F(Jf), the statement follows directly from the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) It is well-known that I2 � y1 · · yn−3 y2 · · yn−2 � is a perfect ideal with linear resolution (in fact, this ideal is resolved by the Eagon-Northcott complex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, the ring F(Jf) ∼= T/Q is Cohen- Macaulay and its Castelnuovo-Mumford regularity is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='10, r(Jf) = reg F(Jf) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iv) By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2, f is a linear free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now the assertion follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 9 Our next goal is to prove that, for f as above, R(Jf) is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First, we need an auxiliary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let k[z, v] = k[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , zm, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , vm] be a polynomial ring, with m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider F = a1v1z1 + · · · + amvmzm, where a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , an are non-zero elements of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let M = � z1 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' zm−1 z2 z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' zm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, k[z, v]/(I2(M), F) is a Cohen-Macaulay domain of codimension m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By [24, Section 4], we have that k[z, v]/I2(M) is a Cohen-Macaulay domain of codimension m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, since F /∈ I2(M), the ring k[z, v]/(I2(M), F) is Cohen-Macaulay of codimension m − 1, and so it remains to show that it is a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Clearly, we can assume m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z1 is a (k[z, v]/(I2(M), F))-regular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Suppose that z1 is not (k[z, v]/(I2(M), F))-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, z1 ∈ p for some associated prime p of k[z, v]/(I2(M), F), and hence in particular (z1, I2(M), F) ⊂ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now let M1 be the matrix obtained from M by deletion of its first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then it is easy to see that (z1, z2, I2(M1), F) ⊂ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If M2 is the matrix obtained by deletion of the first column of M1, then (z1, z2, z3, I2(M2), F) ⊂ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proceeding in this way, we get (z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , zm−1, F) ⊂ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Since ht (z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , zm−1, F) = m, it follows that ht p ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' But, this is a contradiction because p is a associated prime of the Cohen-Macaulay (hence unmixed) ring k[z, v]/(I2(M), F), which has codimension m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This proves the Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Finally, localizing in z1 we deduce the isomorphism k[z, v][z−1 1 ] (I2(M), F)k[z, v][z−1 1 ] ∼= k[z, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , vm][z−1 1 ] I2(M)k[z, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , vm][z−1 1 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' But the ring on the right side of the isomorphism is a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By the claim, it follows that k[z, v]/(I2(M), F) is a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let f ∈ R be as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, R(Jf) is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider the natural epimorphism C := k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn−2, w, u, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , yn−2, s, t] ։ R(Jf), whose kernel we denote J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' From the previous considerations (and notations), it follows that K := (I1(γ · ψ), Q) ⊂ J where γ = � y1 · · yn−2 s t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note we can rewrite the ideal K as K = I2 � u y1 · · yn−3 w y2 · · yn−2 � + (G, H), G = γ · δ1 = −su + 2(n − 1)yn−2w + n−2 � i=2 αiyi−1xi and H = γ · δ2 = −(n − 1)tu + sw + n−2 � i=1 βiyixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let K1 := I2 � u y1 · · yn−3 w y2 · · yn−2 � +(H) ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, C/K1 is a Cohen-Macaulay domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Denote K0 := I2 � u y1 · · yn−3 w y2 · · yn−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It is well-known that C/K0 is a Cohen-Macaulay integral domain of dimension n + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, since H /∈ K0, this polynomial must be C/K0-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, 10 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS the ring C/K1 = C/(K0, H) is Cohen-Macaulay of dimension n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, C/K1 satisfies Serre’s condition S2 (see the definition in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We claim that, even more, the ring C/K1 is normal, from which its integrality will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In order to show that C/K1 is normal, it remains to verify that it is locally regular in codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note ht K1 = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By the classical Jacobian criterion, it suffices to prove that ht (K1, In−2(Θ)) ≥ n, where Θ denotes the Jacobian matrix of K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Notice that the matrix Θ, after a reordering of its columns (which obviously does not affect ideals of minors), can be written in the format Θ = � Θ′ 0 ∗ Θ′′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Precisely, Θ′ is the Jacobian matrix of K0 with respect the variables w, u, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , yn−2 and Θ′′ is the (row) Jacobian matrix of H with respect to the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn−2, s, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, (K1, I1(Θ′′) · In−3(Θ′)) ⊂ (K1, In−2(Θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now pick a minimal prime q of (K1, In−2(Θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, I1(Θ′′) · In−3(Θ′) ⊂ q, which yields I1(Θ′′) ⊂ q or In−3(Θ′) ⊂ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If I1(Θ′′) ⊂ q then ht q ≥ n because I1(Θ′′) = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , yn−2, w, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' On the other hand, if In−3(Θ′) ⊂ q then (K0, In−3(Θ′)) ⊂ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' But it is well-known that ht (K0, In−3(Θ′)) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, ht q ≥ n in any case, and we get ht (K1, In−2(Θ)) ≥ n, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' C/K is a Cohen-Macaulay domain of dimension n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By Claim 1 and its proof, C/K1 is a Cohen-Macaulay domain of dimension n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, since G /∈ K1, the ring C/K = C/(K1, G) is Cohen-Macaulay of dimension n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It remains to prove that C/K is a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First we claim that u is C/K-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Suppose otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then u ∈ p for some associated prime p of C/K, which gives (u, wy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , wyn−3, I2(N), G, H) ⊂ p, where N := � y1 · · yn−3 y2 · · yn−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, (7) Q1 := (u, w, I2(N), G, H) ⊂ p or Q2 := (u, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , yn−3, G, H) ⊂ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We have C/(u, w, I2(N), H) ∼= (k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn−2, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , yn−2]/(I2(N), β1y1x1 + · · · + βn−2yn−2xn−2))[s, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' From this isomorphism and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4, the ring C/(u, w, I2(N), H) is a Cohen-Macaulay domain of dimension (n − 1) + 2 = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, since G /∈ (u, w, I2(N), H), we obtain that C/Q1 = C/(u, w, I2(N), G, H) is a Cohen-Macaulay ring of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, ht Q1 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' On the other hand, C/Q2 ∼= k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn−2, w, yn−2, s, t]/(yn−2w, sw + βn−2yn−2xn−2) is a Cohen-Macaulay ring of dimension n, which yields ht Q2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows, by (7), that ht p ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This is a contradiction, because p is an associated prime of C/K, which is Cohen-Macaulay of codi- mension n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' So, u is C/K-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, by localizing in u and setting D := k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn−2, w, u, y1, s, t], routine calculations give (C/K)[u−1] ∼= D[u−1]/(G, H)D[u−1] ∼= k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn−2, w, u, y1][u−1], which is a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, C/K is a domain, which proves Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' To conclude the proof of the theorem, we notice that since K ⊂ J are prime ideals of the same codimension, then necessarily K = J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, R(Jf) ∼= C/J is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Second family: linear free divisors in P2n−1 In order to describe our second family of free divisors, consider the standard graded polynomial ring R = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , x2n−2, w, u] in 2n ≥ 4 indeterminates over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let f = wuq, q = (x1u − x2w)(x3u − x4w) · · · (x2(n−1)−1u − x2(n−1)w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For every 1 ≤ i ≤ n − 1, denote qi = q/(x2i−1u − x2iw) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, (8) fx2i−1 = u2wqi and fx2i = −w2uqi (1 ≤ i ≤ n − 1), (9) fw = qu − wu n−1 � i=1 x2iqi and fu = qw + wu n−1 � i=1 x2i−1qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Using (8) and (9) we easily deduce the following relations: (10) det � fx2i−1 fx2j−1 fx2i fx2j � = 0 (1 ≤ i < j ≤ n − 1), (11) wfx2i−1 + ufx2i = 0 (1 ≤ i ≤ n − 1), wfw + ufu = (n + 1)f, (12) x2i−1fx2i−1 + x2ifx2i = f (1 ≤ i ≤ n − 1), (13) (n + 1)x2i−1fx2i−1 + (n + 1)x2ifx2i − ufu − wfw = 0 (1 ≤ i ≤ n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Set α = a1a2, β = b1b2 and γ = a1b2 + a2b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In addition to the equalities above, we have (n + 1) n−1 � i=1 x2ifx2i = −(n + 1)w2u n−1 � i=1 x2iqi = (n + 1)[w(fw − qu)] = nwfw − ufu + (ufu + wfw) − (n + 1)f = nwfw − ufu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (14) Now we are in a position to prove the first result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Maintain the above notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The following assertions hold: (i) f is a linear free divisor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) The 2n × (2n − 1) matrix (15) ψn = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 w (n + 1)x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 0 u (n + 1)x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 (n + 1)x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 · · w (n + 1)x2n−3 0 0 0 · · u (n + 1)x2n−2 (n + 1)x2n−2 0 −w · · 0 −w −nw 0 −u · · 0 −u u \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb is a syzygy matrix of Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus a free basis of TR/k(f) is {θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , θ2n−1, ε2n}, where the θi’s correspond to the columns of ψn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) reg Derk(R/(f)) = 2(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 12 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (i) Consider the 2n × 2n matrix M = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 w x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 0 x1 u x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 (n + 1)x2 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 · · w x2n−3 0 x2n−3 0 0 · · u x2n−2 (n + 1)x2n−2 x2n−2 0 0 · · 0 0 −nw w 0 0 · · 0 0 u u \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Using (11), (12), (14), and the Euler relation, it is easy to see that ∇f · M = [ 0 f · · 0 f 0 2nf ], so that ∇f · M ≡ 0 mod f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, (n + 1)f = det M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4 (or, in this case, by the version of Saito’s criterion stated in [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4]), we conclude that f is a linear free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) For simplicity, write ψn = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By (i) and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3, we know that Jf is a codimension 2 perfect ideal, so it suffices to prove that ∇f ·ψ = 0 and that ψ has maximal rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The former follows by (11), (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now denote by ∆ the (2n − 1)-minor of ψ obtained by omitting the 2n-th row of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It is easy to see that ∆ modulo w is given by (n + 1)nx1x3 · · · x2n−3un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, ∆ is non-zero as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, ψ has maximal rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) By part (i), f is a linear free divisor (in 2n variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now we apply Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For the next results, we consider a set of 2n variables z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2n−2, s, t over R as well as the natural epimorphism S := k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , x2n−2, w, u, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2n−2, s, t] ։ R(Jf) whose kernel we denote J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By the equalities (10) we have an inclusion I2 � z1 z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−3 z2 z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−2 � ⊂ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, K := � I1(γ · ψn), I2 � z1 z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−3 z2 z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−2 �� ⊂ J where γ = � z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−2 s t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The generators of I1(γ · ψn) are of three types: (16) wz2i−1 + uz2i (1 ≤ i ≤ n − 1), Fi := (n + 1)(x2i−1z2i−1 + x2iz2i) − ws − ut (1 ≤ i ≤ n − 1), G := (n + 1) n−1 � i=1 x2iz2i − nws + ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We can use the generators of type (16) as well as the ideal I2 � z1 z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−3 z2 z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−2 � to rewrite K as K = \uf8eb \uf8ec \uf8ec \uf8ec \uf8edI2 � z1 z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−3 −u z2 z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−2 w � � �� � =:K0 , F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , Fn−1, G \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 With this, we have S/K ∼= A[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , x2n−2, s, t]/(F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , Fn−1, G)A[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , x2n−2, s, t] FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 13 where A := k[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2n−2, w, u]/K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, consider the 2n × n matrix ζ = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 (n + 1)z1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 (n + 1)z2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 (n + 1)z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (n + 1)z2n−3 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (n + 1)z2n−2 (n + 1)z2n−2 −w −w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −w −nw −u −u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −u u \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb taken as a matrix with entries in the domain A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We denote by M the A-module defined as the cokernel of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Maintain the above notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then: (i) M is an A-module of projective dimension 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) The symmetric algebra SymAM is a Cohen-Macaulay domain of dimension 2n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (i) Consider the complex (17) 0 −→ An ζ −→ A2n −→ M −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By the well-known Buchsbaum-Eisenbud acyclicity criterion, in order to show that (17) is exact it suffices to confirm that rank ζ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' To this end, consider the following n × n submatrix of ζ: η := \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 (n + 1)z1 · · 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (n + 1)z2n−3 0 −w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −w −nw \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We have det η = −n(n + 1)n−1z1 · · · z2n−3w ̸= 0 (modK0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, ζ has rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) In addition to the property given in (i), recall A is a Cohen-Macaulay domain and ht K0 = n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, because of [29, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1], it suffices to show that ht(It(ζ) + K0) ≥ 2n − t + 1 for every 1 ≤ t ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For this note first that, by suitably permuting the rows of ζ, we obtain a matrix N of the form N = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ∗z1 · · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 ∗z2i−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ∗z2n−3 0 −u −u −u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −u u ∗z2 · · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 ∗z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 · · ∗z2i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 ∗z2i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ∗z2n−2 ∗z2n−2 −w −w −w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −w −nw \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb where all coefficients ∗ are equal to n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let us denote the top and bottom blocks of N by Nodd and Neven, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Our goal is to prove ht(It(N) + K0) ≥ 2n − t + 1 whenever 1 ≤ t ≤ n, where 14 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS as before K0 = I2 � z1 z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−3 −u z2 z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−2 w � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let P be a prime ideal containing It(N) + K0 and having the same codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' From Nodd it is easy to see that the ideal Ct generated by the t-products of the set {z1, z3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2n−3, z2n−1 := u} is contained in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' But, by [48, Section 2], the minimal primes of Ct are of the form (z2j1−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2jn−t+1−1) for certain 1 ≤ j1 < · · · < jn−t+1 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, we can suppose that (z2j1−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2jn−t+1−1) ⊂ P with 1 ≤ j1 < · · · < jn−t+1 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Analogously, from Neven we can write (z2i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2in−t+1) ⊂ P for certain 1 ≤ i1 < · · · < in−t+1 ≤ n (we put z2n := w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let us assume that the following condition takes place: (†) There exists j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , n} such that {z2j−1, z2j} ∩ P = {z2j−1} or {z2j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Suppose {z2j−1, z2j} ∩ P = {z2j−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, by the relations in K0, all the odd variables belong to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, we have (n − t + 1) � �� � even variables + n ���� odd variables = 2n − t + 1 variables in P, which gives ht(It(N) + K0) ≥ 2n − t + 1 for 1 ≤ t ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The argument for the case {z2j−1, z2j} ∩ P = {z2j} is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, suppose that (†) is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Without loss of generality, we may assume (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , jn−t+1) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , n − t + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows that z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2(n−t+1)−1, z2(n−t+1) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider the following t × t submatrix of ζ: \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 ∗z2(n−t+1)+1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 0 0 ∗z2(n−t+2)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ∗z2n−3 0 −u −u −u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −u u −w −w −w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −w −nw \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The determinant of this matrix is cz2(n−t+1)+1 · · · z2n−3z2n−1z2n for some c ∈ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' in particular, this determinant lies in P and hence zs ∈ P for some s with 2(n − t + 1) + 1 ≤ s ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, since we are assuming that (†) does not hold, there exist two consecutive indices 2j − 1, 2j with 2(n − t + 1) + 1 ≤ 2j − 1, 2j ≤ 2n satisfying z2j−1, z2j ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now we can suppose, without loss of generality, that 2j − 1 = 2(n − t + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We have (z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2(n−t+1)−1, z2(n−t+1), z2(n−t+1)+1, z2(n−t+1)+2) + K0 ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, considering the subideal �K0 := I2 � z2(n−t+2)+1 z2(n−t+3)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−3 −u z2(n−t+3) z2(n−t+4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−2 w � ⊂ K0, we observe that all the variables appearing in �K0 are different from the 2(n − t + 2) variables that already belong to P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' since in addition ht �K0 = t − 3, we conclude ht(It(N) + K0) = ht P ≥ 2(n − t + 2) + (t − 3) = 2n − t + 1 whenever 1 ≤ t ≤ n, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' So we have shown that SymAM is a Cohen-Macaulay domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note that M possesses a rank as an A-module (equal to n, by (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now recall that the Rees algebra of the A-module M, denoted RA(M), can be defined as the quotient of SymAM by its A-torsion submodule (see [42] for the general theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consequently, since in this case A and SymAM are both domains, we can identify SymAM = RA(M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' in particular, using [42, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2] (which gives a formula for the dimension FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 15 of the Rees algebra of a module with rank) and noticing that dim A = 2n − (n − 1) = n + 1, we finally get dim SymAM = dim RA(M) = dim A + rankAM = (n + 1) + n = 2n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Maintain the above notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then: (i) K = J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) The Rees algebra R(Jf) is Cohen-Macaulay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) Let T = k[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , z2n−2, s, t], with n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, F(Jf) ∼= T/I2 � z1 z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−3 z2 z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−2 � as k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, F(Jf) is Cohen-Macaulay, ℓ(Jf) = n + 2, and r(Jf) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We have a natural epimorphism SymAM ∼= S/K ։ S/J ∼= R(Jf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2, K is a prime ideal of S, and dim SymAM = 2n + 1 = dim R + 1 = dim R(Jf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' So ht K = ht J , and then K = J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2 once again, we obtain that R(Jf) is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This proves (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In order to prove (iii), let R+ be the homogeneous maximal ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We have F(Jf) ∼= R(Jf)/R+R(Jf) ∼= S/(R+S, K) ∼= T/I2 � z1 z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−3 z2 z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' z2n−2 � , which, as is well-known (being a generic determinantal ring), is Cohen-Macaulay of dimension n + 2 and moreover has regularity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The latter, by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='10, gives r(Jf) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (a) The ideal Jf is of linear type if and only if n = 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', the case where R is a polynomial ring in 4 variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Indeed, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3(iii), if n ≥ 3 then r(Jf) = 1 ̸= 0, hence Jf cannot be of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Conversely, let n = 2, so that R = k[x1, x2, w, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We can check that the ideals of minors of ψ2 (see (15)) satisfy ht Is(ψ2) ≥ 5 − s = (2n − 1) + 2 − s for s = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows by [29, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1] that Jf is of linear type (in particular, r(Jf) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note in addition that SymRJf is a complete intersection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', the polynomials L1 = 3x1z1 + 3x2z2 − ws − ut, L2 = wz1 + uz2, L3 = x2z1 + x1z2 + us + wt form an R[z1, z2, s, t]-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It is also worth mentioning that, in 3 variables, if g ∈ k[x, y, z] defines a rank 3 central hyperplane arrangement, then it has been recently shown that Jg is of linear type and moreover that the property of the symmetric algebra of Jg being a complete intersection characterizes the freeness of g (see [10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='14 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (b) Let n = 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', R = k[x1, x2, x3, x4, w, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In this case, a computation shows that the entries of the product � z1 · · z4 s t � ψ3 form a regular sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', SymRJf is a complete intersection once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Question 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For an arbitrary n, is SymRJf a complete intersection ring?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 16 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Third family: non-linear free plane curves In this section we furnish our third family of free divisors and some of its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In fact, from such a family we will derive yet another one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Similar examples (also in 3 variables) can be found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', in [20] and [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider the two-parameter family of polynomials f = fα,β = (xα − yα−1z)β + yαβ ∈ R = k[x, y, z], for integers α, β ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The following assertions hold: (i) f is a free divisor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) R(Jf) is Cohen-Macaulay if (α, β) = (2, 3) or if α ≥ 2 and β = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) Jf is not of linear type if α = 2 and β ≥ 3 or if α ≥ 3 and β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In these cases, F(Jf) is a polynomial ring over k and then r(Jf) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iv) f is reducible over k = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If k ⊆ R then f is reducible if β is odd and irreducible otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (v) reg Derk(R/(f)) = αβ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (i) We have fx = αβxα−1(xα−yα−1z)β−1, fy = αβyαβ−1−(α−1)βyα−2z(xα−yα−1z)β−1, fz = −βyα−1(xα−yα−1z)β−1 Note that we can write fx, fy and fz as fx = αxα−1G, fy = xα−1P + yα−1Q, fz = −yα−1G for certain G, P, Q ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, (18) Jf = I2 \uf8ee \uf8f0 yα−1 −α−1P 0 G αxα−1 Q \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, since Jf has codimension two, it follows by the Hilbert-Burch theorem that Jf is a perfect ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3, f is a free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) In the specific cases (α, β) = (2, 2) and (α, β) = (2, 3), the Cohen-Macaulayness of R(Jf) can be confirmed by a routine computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, we may suppose α ≥ 3 and β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Determining G, P, Q in this situation, we obtain from (18) that a syzygy matrix of Jf is ϕ = \uf8ee \uf8f0 yα−1 (α − 1)xyα−2z 0 α(xα − yα−1z) αxα−1 α2yα + α(α − 1)yα−2z2 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let us denote by Q the (prime) ideal of k[x, y, z, s, t, u] = R[s, t, u] defining R(Jf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Notice that (19) I1 �� s t u � ϕ � = (syα−1 + αuxα−1, yα−2H + αxαt) ⊂ Q, where H := (α − 1)xzs + (α2y2 + α(α − 1)z2)u − αyzt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Clearly, we can rewrite I1([ s t u ] · ϕ) as I1 �� yα−2 xα−2 � � sy H αux αx2t �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now it follows from Cramer’s rule that (20) det � sy H αux αx2t � = αx2yst − αuxH = αx(xyst − uH) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' From (19) and (20) we deduce an inclusion P := (syα−1 + αuxα−1, yα−2H + αxαt, xyst − uH) ⊂ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' P is a perfect ideal of height 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 17 Clearly, ht P ≥ 2 and P = I2 \uf8ee \uf8f0 H −xt −ys u αxα−1 yα−2 \uf8f9 \uf8fb Now, Claim 1 follows by the Hilbert-Burch theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' P = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Since P ⊆ Q and ht P = ht Q = 2, it suffices to prove that the ideal P is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note first that x is regular modulo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' To show this, suppose otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then we would have x ∈ p for some associated prime p of R[s, t, u]/P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, (x, syα−1, yα−2H, uH) ⊂ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Using the explicit format of H given above, it is easy to see that ht (x, syα−1, yα−2H, uH) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, ht p ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' But this is a contradiction because, by Claim 1, P is perfect of height 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, by inverting the element x we get D := k[x, y, z, s, t, u][x−1] Pk[x, y, z, s, t, u][x−1] = k[x, y, z, s, t, u][x−1] (u + α−1x1−αyα−1s, xt + α−1x1−αyα−2H, xyst − uH) = k[x, y, z, s, t, u][x−1] (u + α−1x1−αyα−1s, xt + α−1x1−αyα−2H) ∼= k[x, y, z, s, t][x−1] (at + bs) ∼= k[x, y, z, x−1][s, t] (at + bs) where a := x(1 − x−αyα−1z) and b := x1−αyα−2[(α − 1)za + x1−αyα+1] are elements in the coefficient ring k[x, y, z, x−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Since a and b are easily seen to be relatively prime in this facto- rial domain, the element at + bs must be irreducible in k[x, y, z, x−1][s, t], so that the quotient k[x, y, z, x−1][s, t]/(at+bs) ∼= D is a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This means (as x is regular modulo P) that R[s, t, u]/P is a domain, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Finally, by Claim 1 and Claim 2, we conclude that R(Jf) is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) First, if α = 2, a simple inspection shows that the linear type property of Jf fails in case β ≥ 3 (and holds if β = 2), by analyzing the saturation of the ideal S of 2 linear forms defining SymRJf in R[s, t, u] by the ideal Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The resulting ideal – which thus defines R(Jf) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', [31, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='11]) – turns out to strictly contain S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In addition, it is contained in (x, y, z)R[s, t, u], so that F(Jf) ∼= k[s, t, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' As to the case where α ≥ 3 and β = 2, we can use a previous calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Precisely, by the structure of the defining ideal Q = P ⊂ k[x, y, z, s, t, u] of R(Jf) as obtained in item (ii), we readily get that Jf is not of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, by looking at the non-linear Rees equation xyst − uH ∈ (x, y, z)R[s, t, u] we conclude that, once again, F(Jf) ∼= k[s, t, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In either case, ℓ(Jf) = 3 and (by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='10) r(Jf) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iv) Let k = C and assume first that β = 2m, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We have f = (xα − yα−1z)2m + y2mα and hence, for i = √−1 and A := xα − yα−1z, (21) f = (iAm + ymα)(−iAm + ymα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, assume β ≥ 3 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Write f = (xα − yα−1z + yα − yα)β + yαβ and set B := xα − yα−1z + yα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus we can rewrite f = (B − yα)β + yαβ = [Bg + (−1)βyαβ] + yαβ = Bg 18 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS for a suitable g := gα,β ∈ R of degree α(β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Of course, this also shows that if k ⊆ R and β is odd, then f is reducible over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Finally, if k ⊆ R then the irreducibility of f over k for even β follows from the structure of the factors described in (21) over the unique factorization domain C[x, y, z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (v) According to item (i), f is a free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Noticing that its degree is αβ, the assertion follows by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Computations strongly suggest that the cases described in item (ii) are precisely the ones where the Cohen-Macaulayness of R(Jf) takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Concerning the linear type property of Jf, computations also indicate that Jf is not of linear type if α, β ≥ 3 – cases not covered by part (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The (partially computer-assisted) conclusion is that Jf is of linear type if and only if α = β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Here we want to point out that the form g = gα,β ∈ Rα(β−1) defined in the proof of item (iv) is also a free divisor provided that β ≥ 3 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First, note that g is defined by means of Bg = (B − yα)β + yαβ, where B = xα − yα−1z + yα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Explicitly, from Bg = (B − yα)β + yαβ = β � j=0 �β j � Bβ−j(−yα)j + yαβ = B · \uf8eb \uf8ed β−1 � j=0 (−1)j �β j � Bβ−j−1yαj \uf8f6 \uf8f8 we get g = β−1 � j=0 (−1)j �β j � Bβ−j−1yαj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' An elementary calculation shows that we can write gx, gy, gz as gx = αxα−1T, gy = xα−1U + yα−1V , gz = yα−1T, for certain T, U, V ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, Jg = I2 \uf8ee \uf8f0 yα−1 −α−1U 0 T αxα−1 V \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Since ht Jg = 2, the ideal Jg must be perfect by the Hilbert-Burch theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, g is a free divisor whenever β ≥ 3 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='7 we have reg Derk(R/(g)) = αβ − α − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We also observe that, for every odd β ≥ 3, the form g is reducible over k = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Indeed, let Φ = β−1 � j=0 (−1)j �β j � Zβ−j−1W j ∈ C[Z, W], which then factors as a product of linear forms in Z and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now let σ: C[Z, W] → C[x, y, z] be the homomorphism given by Z �→ B and W �→ yα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, the form σ(Φ) = g is reducible in C[x, y, z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Finally, let us study the algebra R(Jgα,β) in the cases β = 3 and β = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If β = 3 then first as a matter of illustration we explicitly have gα,3 = B2 − 3Byα + 3y2α = (B − yα)2 − yα(B − 2yα) = (B − yα)2 − yα(B − yα) + y2α = (xα − yα−1z + yα)(xα − yα−1z − 2yα) + 3y2α = x2α − 2xαyα−1z − xαyα + y2α−2z2 + y2α−1z + y2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Computations show that, for all α ≥ 2, the ring R(Jgα,3) is Cohen-Macaulay, r(Jgα,3) = 0, but Jf is of linear type if and only if α = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' more precisely, if L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Q) defines SymRJgα,3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' R(Jgα,3)) in the polynomial ring R[s, t, u], then we have found the relation L : Q = (xα−1, yα−2)R[s, t, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If β = 5, then in the cases α = 2 and α = 3 we have confirmed that depth R(Jgα,5) = 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', the ring R(Jgα,5) is almost Cohen-Macaulay in the sense that its depth is 1 less than its dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Also, we have r(Jgα,5) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We strongly believe such properties hold for α ≥ 4 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 19 Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let β ≥ 7 be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Is R(Jgα,β) almost Cohen-Macaulay?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Is it true that r(Jgα,β) = 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If k ⊆ R and β ≥ 3 is odd, is gα,β irreducible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Fourth family: linear free plane curves In this section, we let R = k[x, y, z] and our objective is to exhibit our fourth family of free divisors, which as we shall prove have the linearity property as in two of the previous families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Although in [27, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 837] a classification of linear free divisors in at most 4 variables in the case k = C is given, the approach provided here describes concretely a recipe to detect some linear free divisors in 3 variables starting from a suitable 2 × 3 matrix L of linear forms (in fact, from only 3 linear forms, as we shall clarify), where, we recall, k is not required to be algebraically closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' in regard to this point, it should be mentioned that, even though most of the existing results in the literature are established over C, there has always been an interest in free divisor theory over arbitrary fields (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', [46] and [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We could start focusing on the case where the 6 entries of L are general linear forms, but there is in fact no need for this setting as we only suppose the forms in the first row to be linearly independent over k and, naturally, the rank of the matrix to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, after eventually a linear change of variables, we let L = L (L1, L2, L3) := � x y z L1 L2 L3 � , for linear forms L1 = a1x + a2y + a3z, L2 = a4x + a5y + a6z, L3 = a7x + a8y + a9z, where at least one of the 2 × 2 minors Q1 = xL2 − yL1, Q2 = xL3 − zL1, Q3 = yL3 − zL2 does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We also consider the Jacobian matrix of Q = {Q1, Q2, Q3}, Θ = Θ(Q) = \uf8ee \uf8f0 Q1x Q2x Q3x Q1y Q2y Q3y Q1z Q2z Q3z \uf8f9 \uf8fb = \uf8ee \uf8f0 L2 − a4x − a1y L3 + a7x − a1z a7y − a4z a5x − L1 − a2y a8x − a2z L3 − a8y − a5z a6x − a3y a9x − L1 − a3z a9y − L2 − a6z \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Our result in this section is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Maintain the above notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If the cubic f := det Θ is non-zero and reduced, then f is a linear free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Precisely, a free basis of TR/k(f) is {θ1, θ2, ε3}, where ε3 is the Euler derivation, θ2 = L1 ∂ ∂x + L2 ∂ ∂y + L3 ∂ ∂z, and θ1 = (a1L1 + a2L2 + a3L3) ∂ ∂x + (a4L1 + a5L2 + a6L3) ∂ ∂y + (a7L1 + a8L2 + a9L3) ∂ ∂z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Routine calculations show that η1 and η2 below are syzygies of Jf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='η1 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='(2a1 − a5 − a9)x + 3a2y + 3a3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3a4x + (2a5 − a1 − a9)y + 3a6z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3a7x + 3a8y + (2a9 − a1 − a5)z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8fb = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3L1 − (a1 + a5 + a9)x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3L2 − (a1 + a5 + a9)y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3L3 − (a1 + a5 + a9)z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='and η2 being the 3 × 1 column-matrix given by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='(−3a5 − 3a9)L1 + 6a2L2 + 6a3L3 + (−4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9))x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='6a4L1 + (−6a1 + 3a5 − 3a9)L2 + 6a6L3 + (−4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9))y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='6a7L1 + 6a8L2 + (−6a1 − 3a5 + 3a9)L3 + (−4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9))z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now let A := a1 + a5 + a9, B := −4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9), 20 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS so that the following is a submatrix of the matrix of syzygies of Jf: \uf8ee \uf8f0 (−3a5 − 3a9)L1 + 6a2L2 + 6a3L3 + Bx 3L1 − Ax 6a4L1 + (−6a1 + 3a5 − 3a9)L2 + 6a6L3 + By 3L2 − Ay 6a7L1 + 6a8L2 + (−6a1 − 3a5 + 3a9)L3 + Bz 3L3 − Az \uf8f9 \uf8fb 3×2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Multiplying the second column by C := a1 + A = 2a1 + a5 + a9 and adding it to the first column, we obtain an equivalent matrix ϕ = \uf8ee \uf8f0 6(a1L1 + a2L2 + a3L3) + (B − AC)x 3L1 − Ax 6(a4L1 + a5L2 + a6L3) + (B − AC)y 3L2 − Ay 6(a7L1 + a8L2 + a9L3) + (B − AC)z 3L3 − Az \uf8f9 \uf8fb 3×2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Finally, attaching to ϕ a third column corresponding to the Euler derivation, the resulting 3 × 3 matrix is easily seen to be equivalent to Φ = \uf8ee \uf8f0 a1L1 + a2L2 + a3L3 L1 x a4L1 + a5L2 + a6L3 L2 y a7L1 + a8L2 + a9L3 L3 z \uf8f9 \uf8fb 3×3 and satisfies det Φ = 1 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now the proposed assertions follow by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Numerous comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (a) Recall that, by definition, being reduced is a necessary condition for a polynomial to be free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now we point out that, in general, it is possible for the cubic f := det Θ (with Θ as defined above) to be non-reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For instance, if we start with the matrix L (x−y, x+y +z, y +z), then f = −2(x + z)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (b) Concerning the condition f ̸= 0, it means (since char k = 0) that the quadrics Q1, Q2, Q3 are algebraically independent, hence linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Clearly, this may not occur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' for example, for L (y, x, z) we have f = 0 because Q3 = −Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (c) We remark that any f as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1 is necessarily reducible at least if k = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This follows by the fact that a complex irreducible free divisor in 3 variables must have degree at least 5 (see [20, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (d) A linear free divisor f in our fourth family can have an irreducible quadratic factor, at least over k = R (or eventually a suitable finite field extension of Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Indeed, starting for example with the matrix L (0, x + z, y + z), we obtain f = −2xq := −2x(x2 + xy − y2 + 3xz − yz + z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Forcing q to be the product of two linear forms with real coefficients yields a contradiction, hence q is irreducible over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' However, it should be pointed out that q is reducible over C, as the rank of its associated matrix is non-maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In fact we believe (but have no proof) that if k = C then a free cubic f as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1 must necessarily be a product of linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This would imply that the complex linear free divisor z(xz + y2) does not belong to our fourth family, which we have been unable to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (e) Our method does not work for higher degrees in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Taking for example any of the matrices � x2 y z y2 z x � , � x y z z2 x2 y2 � , � x2 y2 z2 z2 x2 y2 � , we are led (following the same recipe) to polynomials that are not free as their Jacobian ideals fail to be perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' However, an interesting problem remains as to the possibility of producing free divisors by means of a similar technique, but with carefully chosen entries of higher degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 21 (f) Our method does not work for higher dimensions in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For example, over the ring k[x, y, z, w], consider the matrix \uf8ee \uf8f0 x y z w x − y x + w y − z x + 3y 2y − z 3w x − w y + 2w \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The maximal minors are 4 cubics whose Jacobian matrix has a reduced determinant g ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' A computation shows that Jg is not perfect, so that g is not free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We also derive some additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let f be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The following assertions hold: (i) Jf is of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, r(Jf) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) R(Jf) (∼= SymRJf) is a complete intersection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) reg Derk(R/(f)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (i) From the proof of the theorem, the 3 × 2 matrix ϕ is a minimal presentation matrix of Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows easily by the structure of ϕ – which in particular has only linear forms as entries – that the so-called G3 condition is satisfied (see the definition in the next section, right before Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, it is clear that Jf has projective dimension 1 (see also Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, applying [42, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='11] we obtain that Jf is of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) By the previous item we have R(Jf) ∼= SymRJf, and the latter is the quotient of R[s, t, u] (where s, t, u are variables over R) by the ideal generated by 2 linear forms ξ1, ξ2 in s, t, u, which are the entries of the matrix product [s t u] · ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Saying that ht (ξ1, ξ2) = 1 means precisely ξ2 = λξ1, for some non-zero λ ∈ k, which is equivalent to the first column of ϕ being λ times the second column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Following the proof of the theorem, this would yield det Φ = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, ht (ξ1, ξ2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) Since f is a free cubic, this follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We close the section with a working example which is, on the other hand, somewhat degenerated in the sense that two of the Li’s are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Taking L (y, x, x) yields the line arrangement 1 2f = −x2y + y3 + x2z − y2z = (x + y)(x − y)(z − y), which is then free by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In this case, writing down the syzygy matrix ϕ of Jf as in the proof of the theorem and multiplying their columns by suitable non-zero scalars, we get the following simpler presentation matrix for Jf: \uf8ee \uf8f0 y x x y x 3y − 2z \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3(ii) that the Rees algebra is the complete intersection ring R(Jf) ∼= R[s, t, u]/(ys + x(t + u), xs + yt + (3y − 2z)u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Maximal analytic spread and an application to homaloidness Consider the standard graded polynomial ring R = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn] = k ⊕ R+, n ≥ 3, and let f ∈ R be a non-zero reduced homogeneous polynomial of degree d ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Recall that the Jacobian ideal Jf can be minimally generated by the derivatives fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn since by convention f is not allowed to be a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, ht Jf ≥ 2 as f is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 22 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' When does the Jacobian ideal have maximal analytic spread?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Our goal in this part is to answer this question by means of various characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Recall that ht Jf ≤ ℓ(Jf) ≤ n, so here we are specifically interested in the property ℓ(Jf) = n, which holds if for example Jf is of linear type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' indeed, in this case we can write F(Jf) = SymRJf/R+SymRJf ∼= Symk(Jf/R+Jf) ∼= R as k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The Jacobian ideal of the (non-free) cubic f = xyz + w3 ∈ k[x, y, z, w] can be shown to be of linear type, and so ℓ(Jf) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' On the other hand, as we have seen in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3(2), even for linear free divisors in at least 5 variables the analytic spread of Jf can be arbitrarily smaller than the number n of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Some of the characterizations to be given here are of cohomological nature, and some rely on the asymptotic behavior of depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' One of the ingredients is a suitable auxiliary module, which we now introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' As usual, we denote the gradient vector of a polynomial g ∈ R by ∇g = (gx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , gxn) ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Given f as above, we set Cf := Rn/ � n � i=1 R ∇fxi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' A few preparatory concepts are in order before stating our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let E be a finitely generated module over a Noetherian ring A and let G Φ→ F → E → 0 be an A-free presentation of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider the dual map HomA(Φ, A): F → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The Auslander transpose (or Auslander dual) of E is the A-module Tr E = coker HomA(Φ, A), which is unique up to projective summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We refer to [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, suppose A = � i≥0 Ai is standard graded over a field A0 and let A+ = � i≥1 Ai be the homogeneous maximal ideal of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Assume that the A-module E is graded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, given an integer j ≥ 0, the j-th local cohomology module of E is the limit Hj A+(E) = lim −→ Extj A(A/As +, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Saying that E ∼= E′ as graded A-modules means, as usual, that there is a degree zero isomorphism between E and E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Finally, given r ≥ 1, recall that the Noetherian ring A is said to satisfy (Serre’s) condition Sr if depth Ap ≥ min {r, ht p} for all p ∈ Spec A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This clearly holds (for all r) if A is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Back to the polynomial setup, our result here is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Given f ∈ R as before, the following assertions are equivalent: (i) ℓ(Jf) = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) dim Cf = n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iii) ∇fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , ∇fxn are R-linearly independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (iv) Ext1 R(Cf, R) ∼= Cf(d − 2) as graded R-modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (v) Hn−1 R+ (Cf) ∼= HomR(Cf, k)(n − d + 2) as graded R-modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (vi) depth R/Jm f = 0 for some m ≥ 1, where the bar denotes integral closure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (vii) depth R/Jm f = 0 for all m ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, if R(Jf) satisfies the S2 condition, then these assertions are also equivalent to the following ones: FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 23 (viii) depth R/Jm f = 0 for all m ≫ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ix) depth R/Jm f = 0 for some m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let H be the graded Hessian map of f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' the degree zero homomorphism Rn(−(d − 2)) → Rn whose matrix in the canonical bases is the Hessian matrix of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The image of H is the submodule of Rn generated by the homogeneous vectors ∇fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , ∇fxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, Cf is the cokernel of H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' it has a graded R-free presentation (22) Rn(−(d − 2)) H −→ Rn −→ Cf −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Dualizing this sequence, and denoting by E∗ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ϕ∗) the R-dual of an R-module E (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' an R-module homomorphism ϕ), we get an exact sequence (23) 0 −→ C ∗ f −→ Rn H∗ −→ Rn(d − 2) −→ Cf(d − 2) −→ 0, where we observe that, since the Hessian matrix is symmetric, H∗ = H ⊗ 1R(d−2) so that, indeed, coker H∗ = (coker H)(d − 2) = Cf(d − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, ℓ(Jf) is the dimension of the special fiber ring F(Jf) = k[fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn], which by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='9 can be computed as the rank of the Hessian matrix of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, ℓ(Jf) = rank H = rank H∗, and hence ℓ(Jf) = n if and only if H is injective (this is of course equivalent to Rn(−(d − 2)) ∼= �n i=1 R ∇fxi via H, which thus proves (i)⇔ (iii)), if and only if H∗ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The latter property means C ∗ f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, in order to prove (i)⇔ (iv), it suffices to verify that C ∗ f = 0 if and only if (iv) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Suppose C ∗ f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, as we have seen, H is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Dualizing (22) (which is now a short exact sequence) and comparing with (23), we obtain (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Conversely, assume that (iv) takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus Cf ∼= Ext1 R(Cf, R)(2 − d) ∼= Ext1 R(Cf(d − 2), R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now recall that the R-torsion τR(Cf) of Cf coincides with the kernel of the canonical biduality map Cf → C ∗∗ f , and so, by [3, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='6(a)], we have τR(Cf) ∼= Ext1 R(Tr Cf, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' But (23) gives Tr Cf = Cf(d − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Putting these facts together, we obtain C ∗ f ∼= Ext1 R(Cf(d − 2), R)∗ ∼= Ext1 R(Tr Cf, R)∗ ∼= τR(Cf)∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Next, let us prove that (iv)⇔ (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The graded canonical module of the standard graded polynomial ring R is ωR = R(−n), so Ext1 R(Cf, ωR) ∼= Ext1 R(Cf, R)(−n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Also recall that, in the present setting, the Matlis duality functor is given by HomR(−, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, by graded local duality (see [7, Example 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='6]), we can write (24) Hn−1 R+ (Cf) ∼= HomR(Ext1 R(Cf, R)(−n), k) ∼= HomR(Ext1 R(Cf, R), k)(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If (iv) holds, then Hn−1 R+ (Cf) ∼= HomR(Cf(d − 2), k)(n) ∼= HomR(Cf, k)(n − d + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Conversely, suppose (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Using (24), we get HomR(Cf, k)(n − d + 2) ∼= HomR(Ext1 R(Cf, R), k)(n), which is the same as an isomorphism HomR(Cf(−n + d − 2), k) ∼= HomR(Ext1 R(Cf, R)(−n), k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Taking Matlis duals and tensoring with R(n), we obtain (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We proceed to show that (i)⇔(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First note that, by (22), the 0-th Fitting ideal of Cf is the principal ideal generated by the determinant h of the Hessian matrix of f, so we have � 0 :R Cf = � (h) 24 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS and hence dim Cf = dim R/(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows that dim Cf = n−1 if and only if h ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' H is injective, which as seen above is equivalent to (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We clearly have ℓ(Jf) = ℓ((Jf)R+) and ht R+ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, by the general characterization given in [30, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1], we have that ℓ(Jf) = n if and only if R+ ∈ AssRR/Jm f for all m ≫ 0, which is tantamount to saying that (vii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This proves the equivalence (i)⇔(vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Evidently, (vii)⇒(vi), and the converse follows once we recall the chain (see [30, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4]) AssRR/Jf ⊂ AssRR/J2 f ⊂ AssRR/J3 f ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, we have proved that the statements (i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ,(vii) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now we point out that the implication (vii)⇒(viii) holds regardless of R(Jf) satisfying S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Indeed, condition (vii) means that the irrelevant ideal R+ belongs to the limit value A ∗(Jf) of the function m �→ AssRR/Jm f , which is known to eventually stabilize (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', [30, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' There is also the set A ∗(Jf) defined analogously as the stable set of asymptotic prime divisors with respect to the usual filtration given by the powers of Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By [30, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='17], we have A ∗(Jf) ⊂ A ∗(Jf) and hence R+ ∈ A ∗(Jf), which gives (viii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Notice that (viii)⇒(ix) trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It remains to show (ix)⇒(i), under the hypothesis that R(Jf) satisfies S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In this case, by [14, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='16], the extended Rees algebra R[Jft, t−1] = � i∈Z Iiti ⊂ R[t, t−1] (where, by convention, Ii = R whenever i ≤ 0) must satisfy S2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note that (ix) means R+ ∈ AssRR/Jm f for some m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now we are in a position to apply [14, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1] in order to conclude that ℓ(Jf) = n, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' With the aid of [30, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='26 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='20], the assertions (i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ,(vii) of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2 are also seen to be equivalent to each of the following ones: (a) R+ ∈ A ∗(IJf) for any non-zero R-ideal I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', depth R/ImJm f = 0 for all m ≫ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (b) For some j, the integral closure of R[fx1/fxj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn/fxj] ⊂ k(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn) contains a prime Q of height 1 such that Q ∩ R = R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' While, in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2, the implication (ix)⇒(i) (and consequently the implication (viii)⇒(i)) holds if the Rees ring R(Jf) satisfies S2, we do not know whether this hypothesis can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus the following question becomes natural (see also Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='17 in the next subsection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Suppose depth R/Jm f = 0 for all m ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Is it true that ℓ(Jf) = n ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Does this hold if we only assume that depth R/Jm f = 0 for some m ≥ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now let f ∈ R be a linear free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' As we will see later, if n ≤ 4 then ℓ(Jf) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The converse is known to be false, and in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5 below we show in addition that there is a linear free divisor f such that ℓ(Jf) = n for any prescribed n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The following well-known notion will be useful (we state it over R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' A non-zero homogeneous ideal I of R, minimally generated by ν elements, is said to satisfy the Gs condition for a given s ≥ 0 if ht Iν−j(ϕ) ≥ j + 1 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Here, ϕ denotes a minimal presentation matrix (or syzygy matrix) of I, and note that there is no dependence on the choice of ϕ because each Iν−j(ϕ) is just a Fitting ideal of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 25 Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Given an arbitrary n, consider the normal crossing divisor f = x1 · · · xn ∈ R = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn], which is a well-known linear free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The ideal Jf is simply the ideal generated by all the products of distinct n − 1 indeterminates, and satisfies depth R/Jm f = max {0, n − m − 1} for all m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, depth R/Jm f = 0 for all m ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We now claim that R(Jf) is Cohen-Macaulay (hence it has the S2 property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Notice first that a syzygy matrix of Jf is given by ϕ = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 x1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' xn−1 −xn −xn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' −xn \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Here we have ht In−j(ϕ) = j + 1 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , n − 1, so that Jf satisfies the Gn property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, because f is free, Jf has projective dimension 1 over R (see Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows by [42, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='11] that R(Jf) is Cohen-Macaulay, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now we are in a position to apply Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2 to conclude that ℓ(Jf) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In addition, the theorem gives us that Cf has projective dimension 1 over R (because the gradient vectors of the natural generators of Jf generate a free module) and dimension n − 1, hence Cf is a Cohen-Macaulay module, which yields Hi R+(Cf) ∼= � HomR(Cf, k)(2), i = n − 1 0 , i ̸= n − 1 In Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5, another way to confirm that ℓ(Jf) = n is by showing that the (monomial) ideal Jf is of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This fact and lots of other experiments suggest a more restrictive question as well as a conjecture about the interplay between maximal analytic spread and the linear type property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' as already seen, the latter implies the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For arbitrary n ≥ 3 (the number of variables), does there exist a free divisor f, with Jf not of linear type, such that ℓ(Jf) = n ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The case of interest is n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Indeed, if n = 3 then any member of the family of (non-linear) free divisors given in Section 4 yields an affirmative answer to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If f is a linear free divisor such that ℓ(Jf) = n, then Jf is of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We have not been able to solve this conjecture for n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It is true in n ≤ 4 variables, as we can verify using the classification of linear free divisors given in [27, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 837].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Next we furnish more examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider the so-called Gordan-Noether cubic f = xw2 + ytw + zt2 ∈ R = k[x, y, z, w, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In this case, while the symmetric algebra SymRJf = B/S = R[t1, t2, t3, t4, t5]/S has dimension 6 and depth 5, a calculation shows that R(Jf) is Cohen-Macaulay (in particular, it has the S2 condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Indeed, if ϕ is a minimal presentation matrix of Jf, then the saturation S :B I4(ϕ)∞, 26 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS which by [31, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='11] defines R(Jf) in the ring B (note that I4(ϕ) defines the non-principal locus of Jf), is perfect of codimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now, further computations show that depth R/Jf = 2 and depth R/Jm f = 1 for all m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, using Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2, we conclude that ℓ(Jf) < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' More precisely, the special fiber ring can be expressed as F(Jf) = k[t1, t2, t3, t4, t5]/(t2 2 − t1t3), which yields ℓ(Jf) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider the quintic f = 2w4u + xu4 + ywu3 + zw2u2 ∈ R = k[x, y, z, w, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This is the case n = 5 of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2, hence f is a linear free divisor (here it should be mentioned, for completeness, that f/u is not free and its Jacobian ideal is not even linearly presented).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3(ii), we have ℓ(Jf) = 4 < 5, and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5 ensures that R(Jf) is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Therefore, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2, we conclude that depth R/Jm f > 0 for all m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In fact, for such f it can be verified that depth R/Jf = 3, depth R/J2 f = 2, and depth R/Jm f = 1 for all m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' An interesting consequence of the non-vanishing of the asymptotic depth of Jf concerns the higher conormal modules Jm f /Jm+1 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Indeed, in this situation the (also well-defined) conormal asymptotic depth of Jf must be positive as well, since by [6] we can write lim m→∞ depth Jm f /Jm+1 f ≥ lim m→∞ depth R/Jm f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows that R+ /∈ AssRJm f /Jm+1 f for all m ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider the plane sextic f = x6 − 2x3y2z + y4z2 + y6 ∈ R = k[x, y, z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This is the case α = 3 and β = 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1(i), hence f is a free divisor (which is no longer linear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1(ii), the ring R(Jf) is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It can be verified that depth R/Jm f = 0 for all m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Applying Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2 we obtain that ℓ(Jf) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1(iii) we know that Jf cannot be of linear type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' To see this explicitly, one of the minimal generators of the defining ideal of the Rees algebra in the ring R[t1, t2, t3] is the following polynomial which is not linear in the ti’s: 3xyt1t2 + 2xzt1t3 − 3yzt2t3 − 3y2t2 3 − 2z2t2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Furthermore, the theorem yields Ext1 R(Cf, R) ∼= Cf(4) and Hj R+(Cf) ∼= � HomR(Cf, k)(−1), j = 2 0 , j ̸= 2 Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Consider the quartic f = x4 − xyz2 + z3w ∈ R = k[x, y, z, w], which is not free as Jf is not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let us also mention that the ideal Jf is not of linear type, since the polynomial 4xt2 2 − 4zt1t4 − yt2 4 ∈ R[t1, t2, t3, t4] is one of the minimal generators of the defining ideal of R(Jf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' On the other hand, it is not hard to verify that the associated graded ring of Jf – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' the graded algebra � s≥0 Js f/Js+1 f – satisfies the S1 property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' since in addition ht Jf ≥ 2, we get by [14, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='16] that R(Jf) satisfies S2 (it can be shown that in fact R(Jf) is Cohen-Macaulay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Furthermore, depth R/Ji f = 1 for i = 1, 2, 3, while depth R/Jm f = 0 for all m ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 27 Applying Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2, we conclude that ℓ(Jf) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We also get Ext1 R(Cf, R) ∼= Cf(2), and Hj R+(Cf) ∼= � HomR(Cf, k)(2), j = 3 0 , j ̸= 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Application: Criterion for homaloidness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' As above let f ∈ R = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , xn], n ≥ 3, be a non-zero reduced homogeneous polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In this subsection, we assume additionally that the field k is algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' To the form f we can associate the rational map Pf = (fx1 : · · · : fxn) : Pn−1 ��� Pn−1, the so-called polar map defined by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus the base locus of Pf is the singular locus of the projective hypersurface V (f) ⊂ Pn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' ([21]) The polynomial f is homaloidal if Pf is birational (hence a Cremona transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Over k = C, this definition can be translated by saying that Pf has degree 1 (taking into account an appropriate notion of degree in this context), and according to [19, Corollary 2] the property of being homaloidal depends only on fred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The following is a preliminary fact connecting this class of polynomials to the class of free divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It can be also seen as a first source of examples of homaloidal divisors (examples in higher dimensions can be found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', in [13] and [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Recall that in general the dimension of the image of the polar map Pf is given by ℓ(Jf) − 1 (see the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (k = C) If n ≤ 4 then every linear free divisor is homaloidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let f ∈ R be a linear free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Recall we are supposing that f is not a cone (see Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Thus, by [25, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5], f has a non-zero Hessian, so that ℓ(Jf) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Hence, the dimension of the image of Pf is n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' As the linear rank of the gradient ideal Jf is maximal, it follows by [22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2] that Pf is birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Notice that this proposition fails if n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Indeed, if in this case we take f as being a linear free divisor as described in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2, then by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='3(ii) the analytic spread of Jf is 4, hence the image of Pf has dimension at most n − 2 and so this map cannot be birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Our application regarding homaloidness is the following ideal-theoretic, also homological, version of the criterion given in [22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It is not as practical or effective as the original one, but in our view it adds some flavor to the classical – typically geometric – theory and, moreover, helps linking to different algebraic tools and invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Given f ∈ R as before, let ϕ1 be the submatrix of a minimal syzygy matrix of the ideal Jf consisting of its linear syzygies, and suppose In−1(ϕ1) ̸= (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Assume any one of the following situations: (i) projdim Jm f = n − 1 for some m ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (ii) R(Jf) satisfies S2, and projdim Jm f = n − 1 for some m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then f is homaloidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First, in either case, our Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2 (together with the Auslander-Buchsbaum formula) ensures that ℓ(Jf) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' On the other hand, dim(image Pf) = dim Proj k[fx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' , fxn] = dim Proj F(Jf) = ℓ(Jf) − 1 = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now [22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='2] ensures that Pf is birational, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let us first point out that not all homaloidal polynomials satisfy the condition In−1(ϕ1) ̸= (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Indeed, consider the cubic f = xw2 + yzw + z3 ∈ k[x, y, z, w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then it can be checked that: 28 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS (a) f is an irreducible homaloidal polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This is indeed the first member of the family of irreducible homaloidal hypersurfaces described in [28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 1264];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (b) The Jacobian ideal Jf is not linearly presented, and moreover has not enough linear syzygies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' More precisely, only 2 columns of a minimal presentation matrix ϕ are linear syzygies, and hence obviously I3(ϕ1) = (0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (c) Jf is not of linear type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (d) Quite interestingly, Jf satisfies the conditions present in part (ii) of our Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In particular, it can be even shown that the Rees algebra of Jf is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We now remark that if f is a linear free divisor then the condition In−1(ϕ1) ̸= (0) is automatically satisfied as in this case ϕ1 = ϕ and In−1(ϕ) = Jf by the Hilbert-Burch theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We thus record the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If f is a linear free divisor satisfying either condition (i) or (ii) of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='14, then f is homaloidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Before giving the first illustration, we raise the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' We remark that the answer is yes if the second part of Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4 has an affirmative answer as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Also note that, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='13, the case of interest is n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (n ≥ 5) Let f be a linear free divisor satisfying projdim Jm f = n−1 for some m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Must f be homaloidal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The simplest example in arbitrary dimension is the normal crossing divisor f = x1 · · · xn ∈ R studied in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then f is a linear free divisor and we have seen in particular that depth R/Jn−1 f = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', projdim Jn−1 f = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Since Jf is the ideal generated by all squarefree monomials of degree n − 1, we get by [50, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='5] that all powers of Jf are integrally closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' in particular, Jn−1 f = Jn−1 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It follows by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='16 (or Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='14(i)) that f is homaloidal, thus retrieving the well- known fact that the rational map Pn−1 ��� Pn−1 given by (x1 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' : xn) �→ (x2x3 · · · xn : x1x3 · · · xn : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' : x1x2 · · · xn−1) is birational – the so-called Cremona involution on Pn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Below we illustrate Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='14 in the situation where f is not free, and in both reducible and irreducible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (n = 6) Consider the hyperplane-quadric arrangement f = xw(yz + zt + tu) ∈ R = k[x, y, z, w, t, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In this case, f is non-free because Jf is not perfect, while on the other hand this ideal (which is linearly presented, so that ϕ1 = ϕ) satisfies I5(ϕ1) ̸= (0) and projdim J3 f = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, as in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='11, the associated graded ring of Jf has the S1 property and hence R(Jf) satisfies S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='14(ii), f is homaloidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=', the rational map P5 ��� P5 given by (x : y : z : w : t : u) �→ (w(yz + zt + tu) : xzw : xw(y + t) : x(yz + zt + tu) : xwu : xwt) is Cremona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD 29 Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (n = 5) Consider the irreducible cubic f = xt2 + yzt + z3 + w2t ∈ R = k[x, y, z, w, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The ideal Jf is perfect but f is non-free as ht Jf = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' It also satisfies I4(ϕ1) ̸= (0) and projdim J3 f = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Moreover, the associated graded ring of Jf is Cohen-Macaulay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' in particular, R(Jf) satisfies S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='14(ii), f is homaloidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Explicitly, the rational map P4 ��� P4 given by (x : y : z : w : t) �→ (t2 : zt : z2 + 1 3yt : wt : yz + w2 + 2xt) is Cremona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Next, we provide a couple of additional observations and questions that, in our view, are interesting and potentially motivating for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' First, note that if we write the homaloidal quartic f of Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='19 as f = xg, g = w(yz + zt + tu), then a further calculation shows (using again our proposition) that g is homaloidal as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' This fact, among other examples, led us to suggest the following “addition-deletion” problem inspired by well-known investigations in free divisor theory (see [43], also [1] and [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' (Addition-deletion for homaloidal divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=') For polynomials f, g ∈ R, with f homa- loidal, when is the product fg homaloidal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If fg is homaloidal, when is f or g homaloidal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Now let f ∈ R = k[x, y, z, w, t, u, v] stand for the 2-catalecticant determinant f = det \uf8ee \uf8f0 x y z z w t t u v \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' According to [33, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='25(b)]), this cubic is homaloidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Then, for such f, we have detected an intriguing, curious fact: the determinant h(f) of the Hessian matrix of f is a linear free divisor – in particular, h(f) is already reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' In the situation where h(f) is not reduced, we naturally consider h(f)red, which likewise can be a linear free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For example, let g = det \uf8ee \uf8ef\uf8ef\uf8f0 x w z y y x w z w z y x z y x w \uf8f9 \uf8fa\uf8fa\uf8fb in the ring R = k[x, y, z, w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Note g is in fact a linear free divisor, and using Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='16 it is not hard to see that g is also homaloidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Here, h(g) = λg2 for some non-zero λ ∈ k, and therefore h(g)red is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' As expected, this phenomenon does not take place in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' For instance, if once again we take f as the homaloidal quartic of Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='19, then a calculation shows h(f) = 3x2w2f 2, so that h(f)red = 3f is not free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The facts above led us to raise the following question, which reconnects us to the central topic of freeness and closes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Let f be a homaloidal polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' When is h(f)red a (linear) free divisor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' If h(f) is reduced and not a cone, must it be a (linear) free divisor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 30 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' BURITY, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' MIRANDA-NETO, AND Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' RAMOS Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The second-named author was supported by the CNPq grants 301029/2019-9 and 406377/2021-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' The third-named author was supported by the CNPq grants 305860/2019-4 and 425752/2018-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' References [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Abe, Divisionally free arrangements of hyperplanes, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 204 (2016) 317–346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' [2] T.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Toulouse Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' 23 (2014) 483–512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Departamento de Matem´atica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, Para´ıba, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Email address: ricardo@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='ufpb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='br Departamento de Matem´atica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, Para´ıba, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Email address: cleto@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='ufpb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='br Departamento de Matem´atica, CCET, Universidade Federal de Sergipe, 49100-000 S˜ao Cristov˜ao, SE, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content=' Email address: zaqueu@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='ufs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} +page_content='br' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfYAQI/content/2301.03132v1.pdf'} diff --git a/J9E2T4oBgHgl3EQfpQj_/content/2301.04028v1.pdf b/J9E2T4oBgHgl3EQfpQj_/content/2301.04028v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..96c25867b7ca41778e10e6a4795be2edd5fed69e --- /dev/null +++ b/J9E2T4oBgHgl3EQfpQj_/content/2301.04028v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9017e57084bc0217eff0ba576b2c011b96fa72fcd767395dc8ff960b3a21d553 +size 340783 diff --git a/J9E2T4oBgHgl3EQfpQj_/vector_store/index.pkl b/J9E2T4oBgHgl3EQfpQj_/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f9914bd1c3a1e53e263fb5391c5ed4c406468053 --- /dev/null +++ b/J9E2T4oBgHgl3EQfpQj_/vector_store/index.pkl @@ -0,0 +1,3 @@ +version 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However, the proliferation of abortion +misinformation following the Supreme Court’s decision to overturn Roe v. Wade banning legal abortion in the +US highlighted a gap in scientific attention to individual health-related misinformation. To address this gap, +we conducted a study with 60 TikTok users to uncover their experiences with abortion misinformation and +the way they conceptualize, assess, and respond to misleading video content on this platform. Our findings +indicate that users mostly encounter short-term videos suggesting herbal “at-home” remedies for pregnancy +termination. While many of the participants were cautious about scientifically debunked “abortion alternatives,” +roughly 30% of the entire sample believed in their safety and efficacy. Even an explicit debunking label attached +to a misleading abortion video about the harms of “at-home” did not help a third of the participants to dismiss +a video about self-administering abortion as misinformation. We discuss the implications of our findings for +future participation on TikTok and other polarizing topics debated on social media. +CCS Concepts: • Security and privacy → Social aspects of security and privacy; Usability in security +and privacy; • Human-centered computing → Empirical studies in ubiquitous and mobile comput- +ing. +Additional Key Words and Phrases: TikTok, misinformation, abortion, fact-check, debunking, social media +1 +INTRODUCTION +Misinformation, thriving around polarizing topics [126], draws a particular attention in online +discourses centered around health issues [113]. Alternative health narratives are not a new phe- +nomenon [98], but social media’s affordances for anonymity, free content creation, and the lack of +editorial checks allow for rapid dissemination among users that, in turn, join pro/against camps +about “infant vaccination” [97], “COVID-19 mass immunization” [118], and “cancer treatments” +[46] in droves. Health misinformation is not just a harmful pretext for a polarizing social media +discourse, but is a real threat for both individual and public well-being [107]. +In the past, the health misinformation was incited mainly by misleading health research [29], +deceptive interpretations of symptoms [36], and contested public health governance decisions +[43]. In these cases, misinformation appended a fear of either undesirable or unknown health +consequences, prompting people to question long-standing health scientific methods. The response +to such misinformation, thus, was complicated by the tendency of people to hold beliefs that align +with a persuasive message working on their biases and self-preservation [32]. In these circumstances, +the effort was driven towards prebunking health myths [58], “accuracy nudges” to debunk the +misleading health claims [78], and flagging dangerous health misinformation content on social +media [94]. Some of these interventions did take some of the sting out of the misinformation [106], +but are far from providing a comprehensive health misinformation containment [20]. +While the scientific community continues to work towards minimizing the adverse effects of +“fear-mongering” health misinformation [115], a new type of dangerous health misinformation +– appending the lack of desirable and known health practices – was abruptly amplified in the +immediate aftermath of the US Supreme Court decision to strike down the constitutional right +1 +arXiv:2301.05128v1 [cs.SI] 12 Jan 2023 + +Authors +for abortion [112]. The inability to obtain a legal abortion turned people to search engines and +social media to learn how to manage their reproductive decisions and perform safe abortions [99]. +Unfortunately, not all information aligned with the National Library of Medicine’s description of +abortion and recommendations for safe practices [2]. +Many questionable practices including pills, oils, and herbs for inducing abortion flooded social +media, both as claims and as an advertisements in users’ feeds [99]. Platforms used diverse strategies +to mitigate this misinformation: YouTube added “context labels” to such abortion content [122], +Twitter decided to promote authoritative abortion information in its Twitter Moments and Events +[52], and Meta purportedly blocked questionable abortion treatment advertisements [66]. TikTok +also stated it removed and labeled videos with abortion misinformation [51], but many of the +questionable home practices aimed to “cause a miscarriage” still appeared in users’ personal +streams [18]. Debunking of abortion misinformation on TikTok followed up [103], but the slow-in- +nature checking and verifying of health-related facts was no match for the rapid spread of videos +recommending dangerous abortion remedies. +Since misinformation in general is “sticky”, i.e. repeated exposure to false statements make them +appear truthful [57], the lack of systematic response against abortion misinformation at this stage +created a situation where “sticky” unproven abortion remedies could lead people to attempt unsafe +procedures and cause serious bodily harm. While all social media platforms require scrutiny of their +abortion misinformation handling, TikTok – deemed the “New Google” for Gen-Z [41] – draws +special attention in this conundrum as pressing reproductive decisions are particularly interesting +to the majority of users on this platform. TikTok’s status as a platform for social support exchange +[4] further exacerbates the immediate danger of abortion alternatives as supportive communication +adds to “stickiness” and internalization of such content among adolescents and young adults [30]. +Motivated to explore how TikTok users deal with misinformation and alternative abortion +narratives, we conducted a study with 60 TikTok users in the United States. First, we obtained +a large TikTok dataset to uncover the main themes of abortion misinformation on this platform +and get a better sense of how TikTok recommends and moderates such content. Leveraging the +unofficial TikTok-API python [108] library, we scraped 8,226 videos with 77,880 hashtags, of which +17,606 were unique in tagging these videos. We collected the videos using a snowball sampling +strategy, starting with scraping three initial hashtags, specifically #TikTokTaughtMe, #healthcare, +and #abortion. We selected the first one as it’s the de facto hashrtag through which a user is +“googling” short-form TikTok videos for challenges and advice for self-help [18] and the other two +as we were focused on health/abortion misinformation in particular. +To report the findings from our study, we review the related work on misinformation and +misleading health information on social media in Section 2. Section 3 provides the broader context +of abortion misinformation narratives on social media following the ban on abortion in the United +States. Section 4 provides the methodological details of our study and Sections 5, 6, and 7 elaborates +how participants in our study conceptualize, encounter, and respond to abortion misinformation, +respectively. We draw on our findings in Section 8 to discuss the implications for both individual +and public health as well as social media content moderation of alternative abortion narratives. +Finally, Section 9 concludes the paper. +2 +HEALTH MISINFORMATION BACKGROUND +2.1 +Health Misinformation Narratives +Health-related rumors and alternative narratives precede the Internet era and were concentrated +around health issues with either unknown or undesirable consequences. In the 1980s, for example, +the KGB initiated an information warfare campaign called “Operation Infektion” to spread the +2 + +Abortion Misinformation on TikTok +rumour that HIV/AIDS was a mis-fired American biological weapon in order to undermine the +United States’ credibility during the Cold War [16]. In the same time, the tobacco industry in the +United States created a “disinformation playbook” to systematically distort and downplay the link +between the consumption of tobacco and cancer [86]. While the intent to mislead was clearly +present in these campaigns, the volume and output of the rumors and alternative health narratives +was limited to a number of outlets and fabricated publications. +The Internet and social media changed the landscape by enabling an inordinately high volume +and rapid output of health information with varying quality to reach the public [130]. The health- +related rumors and alternative narratives collaterally grew and were amplified to a point where +they yielded uncontrollable consequences even for known and treatable health issues. For example, +a poorly designed study in the 1990s that falsely claimed that the measles, mumps, rubella (MMR) +vaccine causes autism [72] caused such a regression in public immunization that resulted in several +measles outbreaks twenty years later on [44]. Rumours about the Ebola and Zika viruses also +overshadowed the evidence-based health information and resulted in higher vaccine hesitancy in +fear of undesirable health consequences and death [75, 119]. The vaccine hesitancy, on a global +level, achieved a climax during the COVID-19 pandemic with an unprecedented volume and output +of COVID-19 related misinformation [82]. +While the majority of misleading health information on social media focuses on vaccines and +communicable diseases, rumors and alternative narratives also spread about cancer, heart disease, +and other conditions [117]. For example, social media users are more likely to trust and share +cancer-related rumours if the rumours are dreadful rather than wishful, and if one has had previous +personal experience [26]. The uncontrollable consequences in these cases are not overall treatment +hesitancy but seeking of alternative and unproven treatments about diabetes [55], heart failure [25], +hypertension [54] and psoriasis [83]. Interestingly, in all of these cases of non-communicable health +issues, the unsubstantiated claims were promulgated through videos as a particularly influential +mode for conveying misleading health evidence (e.g. also used in anorexia and dietary disorders’ +deceiving messages) [13]. +2.2 +Response to Health Misinformation +In the context of misleading health claims, misinformation is considered by its opposition to the +consensus of what the medical community defines as accurate and evidence-based information +[107]. Scholars, in response, have focused the attention of anti-health-misinformation on two main +fronts: 1) examining the harms of the misinformation [62, 102]; and 2) misinformation prebunking +and debunking [53, 57, 80]; The harms of health misinformation are reflected in dramatic increase +in vaccine hesitancy [62], pursuing dangerous home therapies (e.g. cancer cleansing, weight loss, +virus prevention) [102], as well as increased hostility toward health workers [68]. +The goal of “prebunking” or forewarning is improving people’s ability to spot and resist ma- +nipulation techniques commonly used in health misinformation [57]. To this objective, people +nowadays are “innoculated” against health misinformation by the use of “accuracy nudges” [78], +social correction that occurs via peers [116], or play browser-based games about health myths and +facts [6]. The “prebunking” was shown to be an effective strategy [58], though in time of social +media virality the inoculation effect wanes for users with a conspiracy mentality about unknown +and undesirable health consequences (e.g. the COVID-19 pandemic) [11]. +If this “innoculation” is rendered ineffective, “debunking” is the next step where verifiable +corrections of the falsehoods from credible sources are presented in order to break the illusion +of truth [32, 81]. Debunking, as in fact-checking of health misinformation, was shown to give +mixed results depending on the perceived credibility and expertise of the sources in science-related +contexts [128]. The perception of credibility and expertise, for example, matters little to people with +3 + +Authors +strong conspiratorial ideation tendencies who tend to mistrust any official source [56]. As pressing +health problems of general public interest are hard not be seen also in a political context, debunking +of health misinformation was found to work either when it comes from sources that are perceived +to share people’s values and worldviews [70], or when people maintain a science-accepting attitude +regardless of their political worldviews [3] +2.3 +Moderation of Health Misinformation +In as much as the prebunking and debunking helps in curbing the health misinformation work +online, they are nonetheless slow and difficult to scale to the pace, volume, and output of infor- +mation sharing on social media [40]. Platforms, in response, had to turn to automated means +of moderating unsubstantiated content and questionable accounts to prevent an “outbreak” of +misleading information, especially after the meteoric influx of COVID-19 rumors, conspiracies, +and falsehoods [130]. YouTube opted for a soft moderation and decided to apply context labels to +video searches for health information that link to credible sources recommended by the National +Academy of Medicine [38]. Twitter, up to early December 2022, also applied soft moderation in +two forms: (i) interstitial covers, which obscure the misleading content and require users to click +through to see the information; and (ii) trustworthiness tags, which appear under the content and +do not interrupt the user or compel action [49, 94]. Meta, the parent company of Facebook and +Instagram, did the same in conjunction with hard moderation, taking down prominent accounts +that spread COVID-19 misinformation (e.g. Robert F. Kennedy Jr.’s account was blocked after he +repeatedly undercut trust in the COVID-19 vaccines) [24]. TikTok followed suit and expanded their +soft moderation labeling with trustworthiness tags to content pertaining to eating disorders, health +challenges, and alternative medical treatment videos next to misleading COVID-19 videos [111]. +The response to platform moderation has been, at best, mixed. The interstitial covers provided +an adequate “accuracy nudge” for users to distance from COVID-19 misinformation posts, but users +largely ignored trustworthiness tags [91, 95]. Further, numerous studies reveal that trustworthiness +tags “backfire” (i.e. make users believe health misinformation more, not less [28, 31, 71, 110]). In the +context of the COVID-19 pandemic, the tags triggered a “belief echo,” manifested as skepticism of +adequate mass COVID-19 immunization [94]. A possible reason for such an unexpected reception +of the trustworthiness tags was the asymmetrical nature of soft moderation—the mere exposure +to health misinformation often generates a strong and automatic effective response while the tag +itself may not generate a response of an equal and opposite magnitude [35]. This is because the +trustworthiness tags often lack meaning, have ambiguous wording, or ask users to find health +information themselves (e.g. learn more about COVID-19), which is cognitively demanding and +time consuming [31]. +2.4 +Health Misinformation on TikTok +TikTok, a social media platform for short-form videos, has rapidly grown in popularity in the last +few years [45]. A central feature of TikTok is the ‘For You’ page, a feed of algorithmically curated +videos for the user based on the user’s past browsing, viewing, and interactions with videos [50]. +Users can also search for videos based on hashtags and, in some cases, sounds. Roughly 75% of +the global users on the platform are age 34 or younger, and every fifth person in the United States +visits the platform on a daily basis [45]. +TikTok’s affordances for viral spread of curated short-form videos and demographic structure +[60] made the platform particularly interesting for healthcare workers and science communicators +creating educational content both based on their speciality and more general advice [101, 127]. A +review of 331 videos with authoritative COVID-19 information [59] showed that anti-stigma/anti- +rumor, disease knowledge, encouragement, personal precautions, recognition, and societal crisis +4 + +Abortion Misinformation on TikTok +management drive platform engagement. Another review of 199 videos with information about +chronic obstructive pulmonary disease showed that most of them have a satisfactory scientific +background [100]. An analysis of obstetrician-gynaecologists (OBGYNs) videos and the associated +hashtags and commentary [101] revealed that the health “educators” not just convey authoritative +sex health information [104], but use it to creatively debunk the related misinformation and mislead- +ing treatments. This practice of authoritative health communication as a form of misinformation +diffraction also motivated proposals for teaching abortion using TikTok [30]. +Other work on misleading health content on TikTok is scarce and overwhelmingly focuses on +COVID-19 health misinformation [64, 127]. Basch et al [5] analyzed a sample of 72 videos containing +the hashtag #covidvaccine and found that slightly more than half of them discouraged getting the +vaccine by showing purportedly adverse vaccine reactions. Baumel et al. [7] analyzed the 100 “most +liked” videos under each of the #Pfizer and #Moderna hashtags and found that 44.2% and 28.8% +of the comments conveyed misleading negative sentiment against these two COVID-19 vaccines, +respectively. Baumel et al. [8] also analyzed TikTok commentary related to masks’ effectiveness in +combating COVID-19 and found that 45.3% of commentary using the #MasksDontWork hashtag +contained misinformation. Analyzing a sample of 1000 videos, Shang et al. [93] found that around +22.6% of the videos contained misleading COVID-19 content, but were shared as much as one with +verified COVID-19 information. +Outside of the COVID-19 theme, O’Sullivan et al. [73] analyzed 27 TikTok videos containing +pediatric urology claims and found that only 22.2% contained information that can also be found in +official guidelines provided by the European Association of Urology (EAU). Xu et al. [121] reviewed +65 TikTok videos with the hashtag #prostatecancer and found that at least 48% of them contained +explicit prostate cancer misinformation. Zheng et al. [129] study found that the top 100 videos with +the #acne hashtag had seriously misleading information about diagnosis and treatments. +The only study so far analyzing the way misinformation is moderated with warning labels on +TikTok focused on COVID-19 content [60]. Ling et al. [60] collected 41,000 videos that include 26 +COVID-19-related hashtags in their description. Through a qualitative analysis, they found out that +TikTok likely moderates videos based on hashtags included in the description without an in-depth +analysis of the content. Ling et al. learned that this moderation strategy led to a large false positive +rate – about a quarter of the videos with a misinformation warning label did not contain content +related to COVID-19. The study also found a 7.7% false negative rate where videos with actual +COVID-19 misinformation did not include warning labels. +3 +ABORTION MISINFORMATION CONTEXT +Prior studies have shown that 70.1% of women obtain information regarding abortion from the +Internet [61]. Abortion misinformation online, thus, takes many forms and users generally have +difficulties discerning inaccuracies in the related alternative narratives [74]. Bessett et al. [12] +presented 586 participants with five vignettes of abortion misinformation – safety, breast cancer, +infertility, mental health risk, and legality of abortion – and found that only 4% of participants +were able to correctly identify all of vignettes as misinformation while 73% pointed to two or fewer +vignettes as inaccurate. +Common abortion misinformation topics that have been studied are the increased risk of breast +cancer, future infertility, depression/anxiety, and post-traumatic stress [74]. This misinformation is +spread through multiple sources, including state-mandated “Women’s Right to Know” documentation +that providers must supply before a woman can consent to having an abortion [9], despite official +guidance from the National Academies of Sciences, Engineering, and Medicine [69]. The Guttmacher +Institute found that two states inaccurately include a link between abortion and an increased risk +of breast cancer, 19 states link abortion to future infertility, and eight states link abortion to +5 + +Authors +negative emotional and psychological responses [42]. Kern and Reader [52] reported that abortion +misinformation specifically related to an “abortion reversal pill” increased on Facebook from 20 +interactions on June 23 to 3,500 interactions on June 24 2022, the day after the Supreme Court +decision to overturn Roe v. Wade. Godoy [37] also reported that following the Supreme Court ruling, +Spanish-language abortion misinformation was deliberately designed to galvanize voters in Latino +communities across the US. +Abortifacient herbs – purportedly providing the ability to induce a spontaneous miscarriage – +form the majority of post-Roe v Wade misinformation [17]. The toxicity of abortifacient herbs has +been widely studied as shown in Table 1 but there is little literature and few studies related to the +topic of “herbal abortions” [48]. Most existing studies were done in countries where abortion was +not legal until recently. Abortion did not become legal in Uruguay until 2012 [63], for example, and +a 2003 study found that the Montevideo Poison Centre had 86 cases of ingestion of herbal infusions +with abortive intent from 1986 to 1999 [27]. In the United States, misinformation surrounding +“herbal abortions” in viral videos on TikTok has increased dramatically after legal abortion was +overturned [114]. The consequences of these viral misinformation videos already brought several +people to the emergency rooms seeking critical lifesaving treatment, making active prebunking +and debunking by qualified health professionals imperative [88]. +Table 1. Herbal Abortifacients and Side Effects +Common Name +Side Effects +Black Cohosh +Hepatoxicity [34] +Blue Cohosh +Nicotinic toxicity: tachycardia, hypertension, headaches, +abdominal pain, vomiting, muscle weakness and fascicula- +tions, seizures, and coma [84] +Eastern Daisy Fleabane +Insufficient evidence on the safety and effectiveness as an +abortifacient agent [109] +Mugwort +Vomiting, hypertension, confusion, respiratory distress, +coma, and seizures [22] +Parsley +Abdominal pain, vomiting, genital hemorrhage, anemia, +jaudice [27], internal bleeding, convulsions, and death [21] +Pennyroyal +Gastrointestinal +upset, +fainting, +intestinal +bleeding, +seizures, hepatomegaly or injury, multiple organ failure, +coma, cardiac arrest, and death [41, 105] +Rue +Vomiting, liver damage, anemia, tremors, respiratory dis- +tress, multiple organ failure, and death [47] +4 +MISINFORMATION AND ALTERNATIVE ABORTION NARRATIVES ON TIKTOK +4.1 +Research Questions +The volume and output of abortion misinformation naturally prompted health experts, by them- +selves, to dispel the inaccuracies related to herbal abortions, abortion pills, and abortion side-effects +directly on social media [92]. This effort is by all means needed, but is likely not to be sufficient to +prevent an “outbreak” of unsafe decisions about people’s reproductive health in the long run. An +intuitive response, then, would be a systematic prebunking and debunking of abortion misinforma- +tion in coordination with moderation of related content on social media. Based on past experiences +6 + +Abortion Misinformation on TikTok +with health misinformation, however, such a response leaves little room for nuanced explorations +of how people engage on their own with abortion misinformation in the first place [101]. +As TikTok has been identified as a “hotbed of abortion misinformation” [39], such an exploration +would be beneficial to the ongoing response of removing and moderating misleading content +on TikTok [51] as it will provide knowledge on how users conceptualize, encounter, assess, and +respond to abortion falsehoods. So far, such knowledge is scarce and only provides glimpses on +how users conceptualize misinformation encountered on the traditional social media platforms +[96]. To address this knowledge gap, we set to conduct a study that aimed to answer the following +research questions: +(1) RQ1: Concept: How do social media conceptualize misinformation on TikTok (definition, +origins, targets, and purpose)? +(2) RQ2: Encounters: What encounters with abortion misinformation users had so far on TikTok +and how they dealt with it? +(3) RQ3: Response: What strategies users employ in assessing and responding to various abortion +misinformation content on TikTok? +4.2 +Dataset +Preliminary, we set to collect a dataset of abortion misinformation on TikTok in the immediate +period after the overturn of Roe vs Wade, up till the end of November 2022, as shown in Table +2. We leveraged the unofficial TikTok-API python library [108] to scrape 8,226 videos, which +we collected using a snowball sampling strategy, starting with scraping three initial hashtags, +specifically #TikTokTaughtMe, #Healthcare, and #Abortion. Due to the limitations of the API, each +search returned no more than 300 videos. To continue collecting hashtags, we searched for each +of the hashtags that were associated with each of the three seeding hashtags above, effectively +performing a snowballing sampling of the TikTok’s base with abortion-related short-form videos. +Table 2. TikTok Abortion Hashtag Dataset +Attribute +Value +Total Number of Posts +8,226 +Number of Hashtags +77,880 +Unique Hashtags +17,606 +From here, we vectorized the hashtags using Scikit-Learn’s CountVectorizer [76] to create a +dense boolean array of 1,754 tokens – character unigrams, bigrams, and trigrams – that appeared +in 0.01 - 99% of hashtag samples. We then identified the closest hashtags to a given input hashtags +using Minkowski distance, or ||𝑥||𝑝 where 𝑥 is the difference between an searched vector and +the saved vectors from our hashtag dataset, and 𝑝 is a scalar that we selected. To identify 𝑝, we +reviewed kernel density estimate plots of the distances to several searched hashtags and identified +the most expected bimodal distribution, with a smaller left distribution of relevant hashtags, and a +larger right distribution of less relevant hashtags. We settled on an ideal 𝑝 of 2, which is ||𝑥||2, or +euclidean distance. +Using these hashtag representations, we were easily able to identify perturbations in hashtags +that might otherwise be moderated by TikTok [51]. For example, searching the representations for +hashtags like #selfharm highlighted the existence of #sêlfhârm, and #abortion revealed #abotion +and #anortion, as well as longer hashtags like #abortionishealthcare and #abortionishealthçare. A +7 + +Authors +search with regular terms and hashtags like “#abortifacient” indeed does not present any videos +tagged as such, but following our dataset analysis above, we discovered that a small change in the +spelling – #ab0rtifacient, for example – unveils a lot of abortion videos that promote abortifacient +solutions for miscarriage. +Many of these videos were not necessarily were tagged with the exact search hashtag and +may even be tagged with the original “#abortifacient.” One could argue that an ordinary users +might not know what hashtags exist, but from the video posting functionality, a list of suggested +hashtag completions provide additional variations, and the number of videos with each variation. +As such, even an incomplete, suggestive hashtag search on TikTok brings seemingly obscured +tags for abortion misinformation, as exemplified in Figure 1 Using these built in features, we +quickly identified dozens of videos that described methods for “at-home” abortions as candidates +for misleading claims we wanted to test in our study. +Fig. 1. The images above demonstrate how, while a search for “#abortion #herbs” does not return any videos +due to the TikTok’s guidelines for harmful content [51], a search for “#abotion #herbs” not only returns videos +with similar typos, it also returns videos with the original “#abortion #herbs” spelled correctly. Additionally, +when posting a video, TikTok suggests additional hashtags, any of which can be searched to find additional +hashtags, like #ab0rtionishelathcare. +4.3 +Sample +The analysis of the information in our dataset, given in section 7, helped us identify the main themes +of abortion misinformation content on TikTok in the aftermath of Roe vs Wade decision. As we were +interested in better understanding how actual users deal with this content, we obtained approval +from our Institutional Review Board (IRB) to conduct an exploratory survey (the questionnaire is +provided in the Appendix) with a sample of TikTok users ages 18 and above in the United States. +We used Prolific for recruitment and after we consolidated the responses we obtained through +Qualtrics, we ended with a sample of total of 60 participants. The responses were anonymous, +and the survey allowed users to skip any question they were uncomfortable answering, taking +8 + +11:30 +@45Gl92% +Q #abortion #herbs +X +Top +Users +Videos +Sounds +LIVE +Hashtags +Noresultsfound +Thisphrase maybeassociatedwithbehaviororcontent +that violates our guidelines. Promoting a safe and +positive experience is TikTok's top priority.For more +information,we invite youto review our Community +Guidelines11:30 +Q45GEl 92% +#abotion #herbs +Q +X +Top +Users +Videos +Sounds +LIVE +Hashtags +All +Unwatched +Watched +Recentlyuploaded +Detailed Information +Herbs that Induce +Contained within +Aborfion +What method I've +used to interrupt +pregnancy at home +naturally +Perspective from a +Herbalist +6/28 +8/26 +.#abortion#herbs +.herbsOnHANDsothat +#herbalmedicine +you can even start taking t... +medical.mamacita +?569 +boobygrow +?157 +SHOT +Abortifacient Herbs +proud of Texas! +etthat +babies will ie..Sad... +DON'T use these +dangerous herbs +O +<11:35 +Q45GEl 88% +Post +#abOrt +Select cover +#Hashtags +@Mention +OVideos +#abOrt +467.5Kviews +#abOrtionsaveslives +4.3Mviews +#abOrto +3.8M views +#abOrto +1.8M views +#abOrtire +Oviews +#abOrtionishelathcare +206.6Kviews +#abOrted +93.7K views +#abOrtOlegal +34.4Kviews +#abOrtionjustice +67.6Kviews +#abOrtionrights +47.2K views +I= +VAbortion Misinformation on TikTok +around 25 minutes to complete it. Participants were offered a compensation rate of $5 each. The +demographic structure of our sample is given in Table 3. +Table 3. Sample Demographic Distribution +Gender +Female +44 (73.33%) +Male +15 (25%) +Non-cisgender +1 (1.67%) +Age +[18-20] +5 (8.33%) +[21-30] +33 (55%) +[31-40] +12 (20%) +[41-50] +6 (10%) +[51-60] +4 (6.67%) +[61+] +0 (0%) +Political leanings +Left +38 (63.33%) +Moderate +14 (23.33%) +Right +5 (8.33%) +Apolitical +3 (5%) +Highest Level of Education Completed +High school +12 (20%) +College +43 (71.67%) +Graduate +5 (8.33%) +4.4 +Method and Analysis +Participants were provided an open ended qualitative survey through Prolific that provided a list of +questions and a predetermined set of TikTok videos we selected from our dataset. We singled out +seven videos in total from our dataset that contained abortion misinformation already debunked +by the time of our study [103]. We used the input on general abortion misinformation from Table 1, +information from authoritative verifiable sources [69], and verbatim misinformation terms from +two fact-checking articles [23, 109] as a selection criteria for videos promoting the use of herbal +abortifacients. We also chose to focus only on “at-home abortion remedies” as explicit health +misinformation [101] and not alternative abortion narratives involving “religion” or “political +contextualizaiton” to avoid bias and expressive responding [10]. +We wanted to have as many varying modalities, formats, and creators in our selection as possible, +therefore we selected two videos that contained only text and five videos featuring the creator +of the content. Six of the selected videos were created by women and one was created by an +individual who identifies as transgender in their profiles. The creators of the videos were ethnically +diverse, consisting of individuals who identify in their profile or other videos as White, Black, +North American Indigenous, and Native Hawaiian or Pacific Islander. We must note that TikTok, in +response to the increased scrutiny about their lax handling of health misinformation [51], claims +to regularly remove misleading abortion content so there is a possibility that our dataset was +considerably restricted for our particular selection. +Participants were asked to describe their experience with encountering misinformation on +TikTok. Next, we asked participants to provide their opinions on where misinformation comes +from, what purpose misinformation serves on social media, and who creates and benefits from it. +Participants were then asked to further elaborate how they determine a certain social media post is +misinformation, and what tactics they employ when dealing with misinformation. +In reporting the results, we utilized as much as possible verbatim quotation of participants’ +answers, emphasized in “italics” and with a reference to the participant as either PXYZ# or +[PXYZ#], where P denotes participant, X denotes the number of the participant in the sample +(ordered by the time of participation), Y denotes their gender identity (F - female, M - male, NC - +9 + +Authors +non-cisgender), Z denotes their political identity (L - left-leaning, M - moderate, R - right-leaning; +A - apolitical), and # denotes the upper bound of their age bracket. For example, P16FL30 refers +to participant 16, female, left-leaning, age bracket [21-30]. +5 +MISINFORMATION CONCEPTUALIZATION ON TIKTOK +5.1 +Definition +First, we asked our participants to define misinformation in their own words. Exactly half the +sample provided a definition that did not include any intention in the production or dissemination +of questionable content, along the lines of the misinformation definitions outlined in [120]. All of +these participants conceptualized falsehoods through the inherently fallacious information mental +model of misinformation on social media described in [96]. For example, P31FL30 defined it +as “untrue/unsubstantiated statements being presented as fact,” P13FR30 as “incorrect, skewed, or +communicated incorrectly,” and P27MM20 as simply “false information.” In this half of the sample, +18 (60%) of the participants identified as left-leaning, 3 (10%) as right-leaning, 8 (26.67%) as moderate, +and one (3.33%) as apolitical. +The other half of our sample expressed intentionality as an additional quality of misinformation, +de facto referring to disinformation instead [126]. Using the folk models of misinformation on +social media [96], more than half, 20 (66.67%), of the participants conceptualized misinformation +as out-of-context narratives, for example, P36FL40 stated that misinformation is “is intentionally, +either by using wrong information or leaving out context, misleading to the people reading it.” The +next most popular folk model 6 (20%) was external propaganda and the participants pointed to +“intentional spread of misleading information to stir an emotion or to further promote a system, product, +or person” [P5FL30]. The remaining 4 (13.33%) participants conceptualized misinformation as +political (counter)argumentation pointing to cases where “a journalist or news source provides false +information to persuade you in one political direction” [P47MR30]. Here, the older participants were, +the more they saw an intention in the spread of misinformation. For example, P50FL60 placed the +intentionality where “videos get edited and changed, and convincing memes with cherry picked facts +are created as part of misinformation that has been used extensively in politics and the pandemic.” +5.2 +Origins +Three quarters or 45 of the participants in our sample felt that misinformation on TikTok came +directly from a creator of the TikTok video. In the view of P20FL40, misinformation on TikTok +is brought by “people who are trying to gain clout, or get numerous views.” P19FL30 went further +and reckoned that “misinformation can come from the creators’ own consumption of misinformation, +or a creators’ misinterpretation of information, or a creators’ attempt to sell something or an idea +to influence others/gain attention.” The creators of Tiktok content, in the view of P27MM20, “are +people who doesn’t care about misinformation but more about views and attention”. +The remaining 25% pointed to the “other” side of a polarized debate or issue i.e. “people on both the +left and right who want to increase views, as well as institutions and political groups with agendas to +create misinformation” [P50FL60]. P2FL50, seeing misinformation as out-of-context narrative, felt +that it “comes from a variety of places such as Republicans, Russia or China” and P12FL60 seconded +the impression of external interference “directly from a bad-actor or a big-mouth source such as +Fox News. P50FL60, using the political (counter)argumentation model of misinformation, directly +accused the GOP for “catering misinformation to low information and low IQ people who will believe +anything they get told because GOP knows they can’t fool the science/college crowd.” +10 + +Abortion Misinformation on TikTok +5.3 +Targets +Half of our sample felt that the targets of misinformation are “vulnerable people who do not know +how to research and form their own opinions” [P7FM30], specifically “Younger people, older people, +or more easily-influenced crowds, which are the people who are not likely to fact check a claim” +[P40FL30]. P50FL60 expanded this list to include people “in areas that have low instances of college +education and high poverty areas; who are lower income and very religious; who are already suffering +themselves and see anyone who gets ahead as a threat to them; and who have very little access to help +so they resent people.” The other half felt that “anyone and everyone can be a target of misinformation” +[P44FL30]. P5FL30 described the targets of misinformation on TikTok to be from “All ages, races, +sexualities, and backgrounds are targets of misinformation because the algorithm brings it to your ‘For +You page’.” +5.4 +Purpose +20 (33.33%) of our participants explicitly indicated that the purpose of misinformation on TikTok +is for profit. The profit was assigned either to “politicians and large corporations who either make +money off what evolves from misinformation campaigns or who benefit politically and financially +from legislation enacted when bad actors are elected to government offices” [P12FL60] or to content +creators themselves as “they get paid from the views, and there’s probably some devout followers to +these people which give them a recurring income from just watching the videos every time they post” +[P15ML30]. Implicit gains, such as “engagement boosts” [P1MR50] that ultimately lead to profit per +the TikTok participation model [50], was the purpose that 14 (23.33%) of our participants identified +behind the spread of misinformation on TikTok. They identified “creators and influencers looking to +gain followers and views” [P1MR50] and “people who doesn’t [sic] care about misinformation but +more about views and attention” [P27MM20]. +Misinformation as a political ammunition was the purpose identified by 13 (21.67%) of our +participants. P31FL30 indicated that the purpose of misinformation is “political influence feeding +into distrust of science and government” and P59FL30 felt the misinformation on TikTok is brought +“to divide people further and continue to build up the conservative party.” 5 (8.33%) of participants +felt that misinformation was to “stir the pot” on TikTok, i.e. “foreign agency targeting the US +or groups within the US that want superiority” [P41FM40]. Videos created by “trolls make up a +portion of deliberate misinformation” [P38FL30] to “gets a rise out of someone,” in the view of +P5FL30. A subgroup of 8 (13.33%) participants, indicated that “no one” [P10MA40] benefits from +misinformation on TikTok, both in a short-run and “ultimately, in the long-run” [P58FM30]. +6 +ABORTION MISINFORMATION ENCOUNTERS ON TIKTOK +6.1 +Encounters +Exactly half of the sample indicated they have seen abortion misinformation on TikTok prior to +the study. The misleading content mostly consisted of “videos like these claiming at home abortion +remedies” [P25FL20] but also included “misinformation rooted in religion – churches show videos +of full term pregnancies being ripped apart by limbs from wombs” [P50FL60]. Participants also +indicate they were seeing politically contextualized abortion narratives “on both sides of the political +spectrum” [P7FM30] to either ban or allow “birth control as well as contraceptives provided by the +government” [P24FR20]. Participants also indicated that they see “people who are Pro Life on TikTok +that spread all kinds of rumors and lies about abortion all the time” [P5FL30] “mostly to cause fear” +[P49Fl30]. +11 + +Authors +6.2 +Response +About half of the participants indicated that abortion misinformation invoked negative emotions in +them. Participants stated that the “videos in all just made me sad” [P6FA30], that they were “very +disturbed by this abortion misinformation” [P13FR30], and “disappointed that people are making +content like this” [P15ML30]. Some of them said that their “first response was shock that someone +would even think this” [P24FR20], that and it is “worrying that this type of misinformation is being +shared because it can be dangerous” [P39-FL30]. Participants in our sample felt “anger, resentment” +[P42FL30] and “disgust that people will believe anything they see and try it” [P56FM50]. The other +half indicated they were mostly “intrigued and wanted to know if anything in these herbal videos +was true or not” [P41FM40]. +In response to abortion misinformation content on TikTok, our participants said they “did not +engage because it only further spreads the misinformation” [P8FL20] or “just ignored it” [P52MM40]. +Some participants indicated they took action on the video by doing “research on abortion and +take what I gather on TikTok with a grain of salt unless it is information spread by an actual health +professional” [P26FL30]. There were also participants that “blocked the creators that spread the +misinformation” [P20FL40], “reported these videos for spreading false information” [P49Fl30], or +“Liked comments pointing out the abortion falsehoods” [P31FL30]. +7 +RESPONSE TO ABORTION MISINFORMATION ON TIKTOK +7.1 +Post #1 +The first post we presented to participants was labeled by the creator with the hashtags #roevwade, +#abortion, #herb,s #knowyourherbs, #herbalist, #womensrights, #fightbackwithherbs, #herbalism, +#homesteadinglife, #michigan, #crazyplantlady, and #farmlife. This post discusses the use of Eastern +Daisy Fleabane root [109]. The use of fleabane as an abortifacient herbal tea was found misleading as +it can “have unpredictable effects” and there is no evidence that this root can induce a miscarriage, +as shown in Table 1. The screenshot of the post as it appeared in the standard TikTok app is shown +in Figure 2. +7.1.1 +Assessment. We broke down the results of the participants’ evaluation in two groups based +on their baseline mental model of misinformation on TikTok we outlined in section 5 above. +The assessment results of the first TikTok video in our study are given in Table 4. Out of the +30 participants who thought misinformation is disseminated on TikTok without intent, 14 were +randomly selected to assess the first post with abortion misinformation. Three of them thought the +video is indeed misinformation, stating that “this is likely misinformation, as a claim such as this +likely has very little evidence to substantiate it” [P46MM30]. A surprising 50% of the participants in +this group though the video was not misinformation, feeling that “it is true because she sound [sic] +like she knows what she is talking about” [P8FL30]. Four participants in this group were unsure if +this video was misinformation, worried that “they state some historical context not verified in any +way” [P42FL30]. +Out of the other 30 participants that saw misinformation being spread with intent on TikTok, +12 were randomly shown the first post (the random selection was done by the Qualtrics survey +software we used, leading to slightly unbalanced group). Four of them confirmed the video contains +falsehoods with quite a verbose justification: “This is misinformation. It is referencing clearing your +liver which immediately points to something more like a Multi-Level Marketing (MLM) product; She’s +using the same terminology that essential oil salespeople use; These people can never name what +toxins, etc you are eliminating because that is not actually happening” [P50FL60]. Three participants +thought otherwise, believing this post was not misinformation because “the content creator seems +12 + +Abortion Misinformation on TikTok +Fig. 2. TikTok Post #1 +Table 4. Is Post #1 Misinformation? +Misinformation (no intent) [viewed: 14 participants] +Yes +3 (21.43%) +No +7 (50%) +Unsure +4 (28.57%) +Disinformation (intent) [viewed: 12 participants] +Yes +4 (33.34%) +No +3 (25%) +Unsure +5 (41.67%) +informed” [P10MA40]. Five participants were unsure, because they had “no idea whether the claim +in the video is based in truth or not” [P32NC50]. +7.1.2 +Response. Participants were also asked to describe what action they would take for each post, +with their actions given in Table 5. Of the participants that thought misinformation is disseminated +on TikTok without intent, six (42.86%) said that they would “scroll past without interacting with the +post” [P42FL30]. Participants in this group were equally likely to “verify said information to see if +it is accurate” [P20FL40] and “would like the post” [P8FL20]. Only two (14.28%) participants in this +group said they would “unfollow this person” [P56FM50] or “report this video for dangerous activities” +[P24FR20]. Of the participants who saw misinformation being spread with intent on TikTok, five +(41.67%) said they “would just scroll past it” [P10MA40] and five (41.67%) said they would “look +at the comments and then conduct my own personal research” [P58FM30]. There were also two +(16.67%) participants that said they “would block it” [P4MM50] or “would report this”[P50FL60] +and no participants from this group said they would like the post. +13 + +Following +ForYou +LIVE +229 +herbs,savealife +38 +thewheatwitch +Fight back against the patriarchy by knowing what herbs +J I sound - thewheatwitch - BethAuthors +Table 5. What action would you take on Post #1? +Misinformation (no intent) [viewed: 14 participants] +Ignore +6 (42.86%) +Fact-check +3 (21.43%) +Block +1 (7.14%) +Report +1 (7.14%) +Like +3 (21.43%) +Disinformation (intent) [viewed: 12 participants] +Ignore +5 (41.67%) +Fact-check +5 (41.67%) +Block +1 (8.33%) +Report +1 (8.33%) +Like +0 (0%) +7.2 +Post #2 +The second post that we presented to the participants was labeled by the creator with the hashtags +#roevwade, #women, #health, and #holistic. This post discusses the use of an abortion tea containing +multiple herbs, including rue, which has dangerous side effects noted in Table 1 and health advisories +warn that it “can lead to death for both the mother and baby” [67]. The screenshot of the post as it +appeared in the standard TikTok app is shown in Figure 3. +Fig. 3. TikTok Post #2 +7.2.1 +Assessment. The participants that thought that misinformation is disseminated on TikTok +without intent were almost evenly split regarding this post, as shown in Table 6. Six (46.15%) said +they “don’t believe this is misinformation” [P33FL30] and five (38.46%) said they “do think this +post is misinformation” [P13FR30]. The remaining two (15.38%) participants said they “cannot +confirm if this post is misinformation” [P17FR30]. Participants that saw falsehoods as disinformation +on TikTok did not see this post as inaccurate as only two (15.38%) of them thought the post is +“misinformation because it is suggesting that an herb blend is a safe and effective way to self administer +14 + +LIVE +FollowingFor You +Herbs: +1tspRue +Crush/Grind: +1tspTansy +1tspFenugreekSeeds +1/2tspWormwood +1tspAniseSeeds +9290 +1tspComfreyleaf +1tspAshwagandaroot +1tspSage +1tspDillSeed +1tspRosemary +346 +1/2CinnamonStick +2tsp Lemongrass +2tspNettleLeaf +TikTo +@loudm +enia +295 +cleerly_clara +Period Tips +#roevwade #women#health#holistic +J1 -cleerly_clara-ClaraC +origirAbortion Misinformation on TikTok +an abortion” [P36FL40]. Five participants (38.46%) said they “don’t think this is misinformation +because the post provides full context on the information it was trying to provide” [P40FL30]. Six +participants, or (46.15%) were unsure because they “don’t have enough knowledge to know if it’s +misinformation” [P59FL30]. +Table 6. Is Post #2 Misinformation? +Misinformation (no intent) [viewed: 13 participants] +Yes +5 (38.46%) +No +6 (46.15%) +Unsure +2 (15.38%) +Disinformation (intent) [viewed: 13 participants] +Yes +2 (15.38%) +No +5 (38.46%) +Unsure +6 (46.15%) +7.2.2 +Response. The 13 participants who thought of no intent behind misinformation on TikTok +indicated they would perform a wide variety of activities for this post as shown in Table 7. Four +(30.77%) of them said they would “potentially do some research into the other herbs that they are not +familiar with” [P17FR30], three (23.08%) said they “would move past it“ [P8FL20], two (15.38%) +said they “would most likely block this account” [P13FR30], and two (15.38%) said they “would +probably like this post” [P31FL30]. The two participants that said they would block this video +indicated they felt very strongly about the content as it is “definitely misinformation because there is +scientific evidence that home remedies for this almost never work” [P25FL20] and “it was extremely +uncalled for in suggesting abortion tea; If this was in my TikTok feed I would [also] report this video” +[P24FR20]. +None of the participants that saw misinformation being spread with intent on TikTok said they +would block or report this post. The most popular responses included that six (46.15%) said they +would “simply move past the video and not pay no mind to it” [P35FL30] and five (38.46%) said they +would fact-check the content. P5FL30 included that to determine the post is not misinformation +they would need to see “a Gynocologist [sic] [to] validate or debunk this information before I could trust +it. This would include discussing the ingredients, linking relevant papers and studies, and reminding +people to go see their doctor for personal medical information.” Two (15.38%) of the participants +indicated responses that they’ve “done research on natural remedies for several health issues and if it +was on my feed I may comment or like it” [P6FA30]. +Table 7. What action would you take on Post #2? +Misinformation (no intent) [viewed: 13 participants] +Ignore +3 (23.08%) +Fact-check +4 (30.77%) +Block +2 (15.38%) +Report +2 (15.38%) +Like +2 (15.38%) +Disinformation (intent) [viewed: 13 participants] +Ignore +6 (46.15%) +Fact-check +5 (38.46%) +Block +0 (0%) +Report +0 (0%) +Like +2 (15.38%) +15 + +Authors +7.3 +Post #3 +The third post that we presented to the participants appears to be a ScienceDirect article indicating +which herbs are abortifacients, but is actually a Google search result snippet with a caption stating +“learn your herbs”. This TikTok video was not labeled with any hashtags by the creator. Although +the full ScienceDirect article purportedly referenced in this video provides warnings about the +toxicity risks of the herbs [87], the screenprint also has directions on the dosage for pregnancy +termination centrally positioned in the overall text. The screenshot, shown in Figure 4, includes +the pennyroyal and mugwort herbs as abortifacients. +Fig. 4. TikTok Post #3 +7.3.1 +Assessment. Multiple participants in both groups observed that the TikTok post appeared +to be from ScienceDirect. A few noticed it was a Google search result or that the “source looks +questionable” [P49Fl30] and reflected that in their responses. In the misinformation group, as +shown in Table 8, six (40%) of the participants said they “think this post maybe fake because the +whole information is not given just the negative information that they want you to see” [P11ML40]. +Five (33.33%) leaned towards it not being misinformation and weighing that “this post is presenting +information from ScienceDirect, which I consider to be a reputable source of science-based and factual +information” [P44FL30]. Four (26.67%) participants were unsure because “This post may contain +some degree of accuracy given that the source is somewhat credible, but herbs are often not studied +enough for these claims to be made with a high degree of accuracy” [P46MM30]. +Most of the participants in the disinformation group were unsure if this post was misinformation. +10 (62.5%) of them said they “honestly cannot tell if this is misinformation; I see that the website +credited is science based, but without going to the website, I am really unsure” [P41FM40]. The next +most common response was that “this tries to present as being medically accurate, but it’s just a +screenshot of Google search results, so I would not trust it on its face” [P12FL60]. Only P15ML30 +16 + +d emmenaayowine hearyguinclude (Qt +arenot limited to)tansy,thuja, +safflower,scotchbroom,rue, +angelica,mugwort,wormwood, +yarrow,and essentialoil of +pennyroyal. +MaternalConsiderations:Carboprostis +ananalogof15-methylprostaglandin +PGF2a.... +DosageWithQualifiers:Pregnancy +LINDIGEN +termination-begin100mcgIMtestd +then 2... +DrugClass:Abortifacients;Oxytocics +Prostaglandins,Stimulants,uterine +Indications:Pregnancytermination +uterineatony +E +https://www.sciencedirect.com>topics +Abortifacients-anoverview +ScienceDirectTopics +Share +nikki.kiya1 +learn your herbs +Aboutfeatured snippets +Fee +J No - Kreepa Oh No - KreepaAbortion Misinformation on TikTok +said he did not “know if it’s misinformation; I’d assume it isn’t because I’m not how someone could lie +about a google search coming up.” +Table 8. Is Post #3 Misinformation? +Misinformation (no intent) [viewed: 15 participants] +Yes +6 (40%) +No +5 (33.33%) +Unsure +4 (26.67%) +Disinformation (intent) [viewed: 16 participants] +Yes +5 (31.25%) +No +1 (6.25%) +Unsure +10 (62.5%) +7.3.2 +Response. The participants in both groups, as indicated in Table 9, were mostly inclined to +ignore this post, saying they “wouldn’t interact with such a post” [P3ML50] with P29FL60 noting +that the post “was a standard google search so it may be true but again I would not respond and I +would swipe on by.” The next most common response of the participants in the misinformation +group was to “report this creator” [P16FL30]. The remaining two (13.33%) participants in this group +said they would “likely look at the comments to see what others have said” [P19FL30]. Six (37. 5%) of +the participants in the disinformation group said they “would probably do research on the specifics if +it was something I desired to learn more about” [P40FL30]. Only one participant in this group said +they would block the post “because it does not state the risks of using these” [P34MM30]. +Table 9. What action would you take on Post #3? +Misinformation (no intent) [viewed: 15 participants] +Ignore +9 (60%) +Fact-check +2 (13.33%) +Block +0 (0%) +Report +4 (26.67%) +Like +0 (0%) +Disinformation (intent) [viewed: 16 participants] +Ignore +9 (56.25%) +Fact-check +6 (37.5%) +Block +1 (6.25%) +Report +0 (0%) +Like +0 (0%) +7.4 +Post #4 +The fourth post presented to the participants was labeled by the creator with the hashtags #fyp, +#prochoice, #roevwade, and #herbodyherchoice. The text overlay in the video also includes penny- +royal and mugwort under the heading “unfortunately it’s come down to this”. The screenshot of the +post as it appeared in the standard TikTok app is shown in Figure 5. +7.4.1 +Assessment. As shown in Table 10, six (37.5%) participants in the misinformation group +indicated that they “do think that this post is misinformation” [P13FR30], five (31.25%) said it is +“not misinformation because it didn’t advise a particular viewpoint or way of thinking” [P54FL30], +and five (31.25%) said “It’s not clear that the post is misinformation because it’s just a list of herbs, +supplements, and foods” [P49Fl30]. The disinformation group of participants felt more strongly +that this post was misinformation, with nine (69.23%) indicating “this one is completely lacking +context or supporting information, so, yes, I would qualify it as misinformation” [P12FL60]. Three +(23.08%) were “not sure it’s misinformation but it may cause unsafe things to occur if not posted with +caution” [P35FL30]. Only P39FL30 said she “doesn’t really think it’s misinformation”. +17 + +Authors +Fig. 5. TikTok Post #4 +Table 10. Is Post #4 Misinformation? +Misinformation (no intent) [viewed: 16 participants] +Yes +6 (37.5%) +No +5 (31.25%) +Unsure +5 (31.25%) +Disinformation (intent) [viewed: 13 participants] +Yes +9 (69.23%) +No +1 (7.69%) +Unsure +3 (23.08%) +7.4.2 +Response. When asked to describe the action they would take on this post, 8 (50%) participants +in the misinformation group said they “would just ignore the video and move on” [P52MM40], as +shown in Table 11. Three (18.75%) participants said they “would probably read through the comments +and either search for similar videos on TikTok or Google it” [P33FL30], another three (18.75%) said +they would “report this creator because their information is dangerous” [P16FL30], and the remaining +two (12.5%) said they would “most likely block this user” [P13FR30]. Almost all of the participants +in the disinformation group would ignore the fourth post. 11 (84.62%) said “would probably scroll +past this without interacting because even if it is truth it isn’t helpful or informative” [P48FL30]. +The remaining two participants said they “would try to fact check as best as I could” [P6FA30] and +“report this post” [P12FL60]. No participants in this group said they would block this post and no +participants in either group said they would like the post. +18 + +CIVE +FollowingForYou +Q +unfortunatelytscome +downto this +chamomile +cinnamon +liver +mugwort +femaleginseng +unripepapayaseed +12 +vitaminC +raweggs +pennyroyal +sesameseeds +w/honey +voidsnail +Share +stay safe outthere<3·#fyp#prochoice #roevwade +#herbodyherchoice +JNirvana +Pennyroyal Tea-NirvAbortion Misinformation on TikTok +Table 11. What action would you take on Post #4? +Misinformation (no intent) [viewed: 16 participants] +Ignore +8 (50%) +Fact-check +3 (18.75%) +Block +2 (12.5%) +Report +3 (18.75%) +Like +0 (0%) +Disinformation (intent) [viewed: 13 participants] +Ignore +11 (84.62%) +Fact-check +1 (7.69%) +Block +0 (0%) +Report +1 (7.69%) +Like +0 (0.00%) +7.5 +Post #5 +The fifth post that we presented was labeled with the hashtags #themoreyouknow, #parsley, #pes- +saryinsertion, #pessary, #fertilityherbs, #fertilityeducation, #plantsheal, #herbalist, #apothecarycab- +inet, #hippocraticoath, #ayurvedic, and #hippocratesfatherofmedicine. This creator has a series +of posts that contain potential “abortion inducers” and emmenagogues (herbs which purpotedly +stimulate menstruation). The post explains how to insert parsley into the cervix or make it into a +tea. In 2018, a women in Argentina died from attempting to induce a miscarriage while utilizing +this method, which “stimulates blood flow in the uterus and can lead to massive internal bleeding +and convulsions” [21]. The screenshot of the post as it appeared in the standard TikTok app is +shown in Figure 6. +Fig. 6. TikTok Post #5 +7.5.1 +Assessment. As shown in Table 12, most participants from both groups felt this post was +misinformation. Twenty-two participants in total were randomly selected to assess this video. +The groups were evenly distributed. Six (54.55%) participants in the misinformation group stated +19 + +FollowingForYou +Q +LIVE +101 +HERBS +THINGSEVERYONE WITHA +UTERUSSHOULDKNOW +PARSLEY +MILDEMMENAGOGUE.RELAXES +CERVXCOMBNESWELLWITH +VITAMINC. +PARSLEYPESSARY:2-4SPRIGSOF +PARSLEYRINSED.REMOVELARGER +PARTOFSTEMJUSTBELOWFIRSTLEAF +JOINT.RINSEAGAIN.PLACEINSIDE +282 +INTERNALLYAGAINSTCERVIX +CHANGEEVERY12HOURS.*TIE +STRINGAROUNDSTEMSFOREASIER +REMOVAL +TEA:BOILWATERINSMALLPOT.ADD2 +65 +HANDFOLLSOFCHOPPEDPARSLEY +COVERTIGHTLY.STEEP20-30MIN +shenvalleyapothecary +Reply to edits.in.green Parsley is a mild +emmenagogue but works well in conjunction... +See more +J)aniRoccaMusicoriginal sounoAuthors +they “feel this post is misinformation because it does not relay the effects of placing an unsterile +object inside your body, which is what the video is promoting/suggesting” [P17FR30] and that “this +post has no credible citations to support the claims it is making” [P14FL30]. Three (27.27%) of the +participants who thought misinformation is disseminated on TikTok without intent felt it was not +misinformation because “It’s always the consumers [sic] responsibility to do their own research and +make their own choice(s)” [P23FL30]. Two (18.18%) participants in this group said they “have no idea” +[P21FA40]. Eight (72.73%) participants in the disinformation group thought “it is misinformation +because the account is not stating what the benefits of parsley even are, nor where they gathered +this information” [P7FM30]. Three (27.27%) said they “suspect this is misinformation but do not +know for sure]” [P36FL40]. No participants in the disinformation group thought the post wasn’t +misinformation. +Table 12. Is Post #5 Misinformation? +Misinformation (no intent) [viewed: 11 participants] +Yes +6 (54.55%) +No +3 (27.27%) +Unsure +2 (18.18%) +Disinformation (intent) [viewed: 11 participants] +Yes +8 (72.73%) +No +0 (0%) +Unsure +3 (27.27%) +7.5.2 +Response. The misinformation group participants, as shown in Table 13 primarily said +they “would ignore this post” [P14FL30]. Two (18.18%) participants said they would “report the +video as unsafe due to the major consequences that doing what the video says to do could ensue +on someone’s health” [P17FR30], one (9.09%) said they “would like to learn more and can always +compare information on google” [P11ML40], and one (9.09%) said they “would most likely block +this account” [P13FR30]. Participants in the disinformation group were mostly split on the post’s +inaccuracies, saying they “wouldn’t respond and just keep scrolling” [P26FL30] and they “would look +up the benefits of parsley to see if the post has an validation” [P6FA30]. Two (18.18%) participants +said they would “typically report this post“ [P7FM30] and P38FL30 said she would “block this +account.” No participants in either group said they would like this video. +Table 13. What action would you take on Post #5? +Misinformation (no intent) [viewed: 11 participants] +Ignore +7 (63.74%) +Fact-check +1 (9.09%) +Block +1 (9.09%) +Report +2 (18.18%) +Like +0 (0%) +Disinformation (intent) [viewed: 11 participants] +Ignore +4 (36.36%) +Fact-check +4 (36.36%) +Block +1 (9.09%) +Report +2 (18.18%) +Like +0 (0%) +7.6 +Post #6 +The sixth post that we presented participants was labeled with the hashtags #greenscreen, #womb, +#roevwade, #pregnancyrelease, #abortion, #herbal, #safety, and #withlove. This post contains +information on which herbs to use to perform a “pregnancy release”. The herbs in the video include +20 + +Abortion Misinformation on TikTok +rue, pennyroyal, and mugwort, which were also included in prior videos. The source cited in +the video is indicated by the creator to be an article on herbal abortion from we.riseup.net. The +screenshot of the post as it appeared in the standard TikTok app is shown in Figure 7. +Fig. 7. TikTok Post #6 +7.6.1 +Assessment. As shown in Table 14, six (42.86%) of the misinformation group participants +indicated that this post was misinformation. Participant P19FL30 explained that “the content creator +in this video does tell viewers to do ‘proper research’ on the herbs she is promoting before using them; +However, she notes that if ‘something goes wrong’ and medical attention is needed that the viewer +does not need to tell their medical provider that they have been taking these herbs - assuming that +the creator is not a medical professional, I would consider this to be misinformation.” Five (35.71%) of +the participants said they weren’t sure if this post is misinformation because “there is an article +supporting the statements made by the speaker and the speaker seems to have genuine intent; However +the shared article is not in the form of a scientific verified and peer reviewed study” [P42FL30]. The +remaining three participants in this group said they “don’t believe this post is misinformation because +she provides supporting facts that you can double check and they will confirm her point” [P54FL30]. +Six (35.29%) of the participants in the disinformation group felt “this is not misinformation because +she is well spoken, shows where she is getting the information from, states what it is for and what it does +and also mentions to seek medical help” [P51FL40]. Six (35.29%) were unsure because they “can’t +confirm if this is misinformation or not because I am not familiar with the scientific data that either +backs up or refutes this info”[P1MR50] and five (29.41%) stated “Yes, this post is misinformation; To +start - the post screenshots a website: weriseup.net; That website is not a reliable medical source of +information. I would scroll past” [P41FM4]. +7.6.2 +Response. Table 15 indicates that more that half of the participants from both groups said +they “would scroll past” [P41FM40], “would simply ignore this post” [P1MR50]. Of the participants +in the misinformation group, four (28.57%) said they would “encourage viewers to follow up with +21 + +Which HerbstoTeke +LIVE +Each herb has a unique action on the body and it is useiui toknow how each one works +in order to select the right ones at the right time.The herbs have been categorised by +their properties and some herbs appear in more than one category as they affect the body +in more than one way. Many herbs could cause abortion single-handedly but a +combination ofcomplementaryherbs could provemore effective.Choosing herbs with +different actions to each otherwould be complementary,rather than using several with +the same action.There is some evidence to suggest that using just one to four herbs at a +time isbetterthan trying to combine a large numberofherbs. +These categories could be seen as an oversimplification as whole plants have a variety of +different actions combining to make cach one unique, but nevertheless these three basic +groupings arean invaluableguidelineforgetting it right. +ProgesteroneBlockingHerbs +These are also referred to asimplantation inhibiting herbs'as they are the category of +herbs that are used as cmergencycontraception after fertilisation,but they're effective +after that period as well.They work by interfering with the normal production of the +hormoneprogesterone,without whichthelining ofthewombbecomes unsupporti +the fertilised egg.Progesterone is crucial to the continued viability ofa pregnang +egg has not yet implanted in the womb then it is prevented from doing so and is +along with the lining.If the egg has already implanted, it detaches from the unsupp +womb liningandthe lining willbreakdown. +Cotton Root Bark +Wild CarrotSeed +VitaminC +(not a herb, but included here because it's readily +non-toxic) +Rue +(this is not a progesterone blocking herb, is pla +cause +thewomb lining to break down. It does this by +fblood +capillaries in the womb) +Uterine Contracting Herbs +6 +These herbs cause contractions in the uteru +ean +ideal addition afteraprogesteroneblocking +ofte +these herbs can help to regulate and streng +to the use of other abortifacients. +ue +Angelicaroot +DongQuai (Chinese angelica) +Green Screen +earthnmaya +linkin mybio for full article#greenscreen#womb +#roevwade #pregnancyrelease #abortion #heFee more +JJoriginal sound-earthnmaya-MAuthors +Table 14. Is Post #6 Misinformation? +Misinformation (no intent) [viewed: 14 participants] +Yes +6 (42.86%) +No +3 (21.43%) +Unsure +5 (35.71%) +Disinformation (intent) [viewed: 17 participants] +Yes +5 (29.41%) +No +6 (35.29%) +Unsure +6 (35.29%) +doing their own research and a notice that every woman’s body still responds differently to methods” +[P54FL30], two (14.29%) said they “would simply block the user and move on” [P52MM40], and +one (7.14%) said they “would report this post” [P25FL20]. The participants in the disinformation +group did not say they would block this post. Three (17.65%) of them indicated they would “read +the full article this video links, and find other videos to make a better judgement on the trueness of +this post” [P5FL30] and two (11.76%) said “there are natural remedies that are legitimate but telling +people they can take an herb for contraception is dangerous; I would report this one[P50FL60]. No +participants in either group said they would like this post. +Table 15. What action would you take on Post #6? +Misinformation (no intent) [viewed: 14 participants] +Ignore +7 (50%) +Fact-check +4 (28.57%) +Block +2 (14.29%) +Report +1 (7.14%) +Like +0 (0%) +Disinformation (intent) [viewed: 17 participants] +Ignore +12 (70.59%) +Fact-check +3 (17.65%) +Block +0 (0%) +Report +2 (11.76%) +Like +0 (0%) +7.7 +Post #7 +We also selected a post that was soft moderated by TikTok [51] and contained a trustworthiness +tag with stating “Participating in this activity could result in you or others getting hurt.” Post seven +was labeled by the creator with the hashtages #greenscreen and #roevwade. As this is the only +explicitly moderated post, we presented it to all 60 participants. The post, shown in Figure 8, has +overlay text that states “they may be able to ban abortion but they can’t ban these” and has slides of +images that include pennyroyal, juniper berries, vitamin C, mugwort, aspirin, and a wire hanger. +7.7.1 +Assessment. As shown in Table 16, 14 (46.67%) of the 30 participants in the misinformation +group said the post was misinformation “considering this person is unqualified to be spreading this +information” [P16FL30]. Eleven (36.67%) said the post was not misinformation because they “don’t +agree with using some of those products for achieving the goal that the video is promoting, but all +of those products shown, used in the correct way/dosage can result in abortion” [P17FR30]. The +remaining five (16.67%) participants indicated they would “have to research before deciding if it was +misinformation or not” [P44FL30]. +In the disinformation group, 13 (43.33%) of participants indicated that “this is misinformation +as there is no evidence this is safe and effective means of self administering an abortion” [P36FL40]. +Nine (30%) of the participants said “It is not helpful or informative; And the hanger is such a danger +suggestion to put out there; I wouldn’t consider this misinformation but it isn’t helpful information” +22 + +Abortion Misinformation on TikTok +Fig. 8. TikTok Post #7 +[P48FL30] and 8 (26.67%) stated “I don’t have the scientific data to comment on whether or not this is +misinformation” [P1MR50]. +Table 16. Is Post #7 Misinformation? +Misinformation (no intent) [viewed: 30 participants] +Yes +14 (46.67%) +No +11 (36.67%) +Unsure +5 (16.67%) +Disinformation (intent) [viewed: 30 participants] +Yes +13 (43.33%) +No +9 (30%) +Unsure +8 (26.67%) +7.7.2 +Response. As indicated in Table 17, 13 (43.33%) of the misinformation group and 18 (60%) of +the disinformation group said they “wouldn’t do anything with the post” [P6FA30] and they “would +not respond and just keep scrolling” [P26FL30]. Eleven participants of the misinformation group +(36.67%) and 8 of the disinformation group (26.67%) said they “would even consider reporting it as +harmful because some people are viewing it and are not aware of how to properly use the products” +[P17FR30] and “because it’s dangerous” [P16FL30]. Three (10%) participants from each group said +they would “have to do more research” [P2FL50] and they “would read the comments, look at similar +videos, or Google it” [P33FL30]. Two (6.67%) of the participants from the misinformation group +said they would like the post as they “agree with this post and have heard these methods before” +[P8FL20]. One of participants from each group said they would block the post because “it’s not +humorous joking around about women feeling they are being pushed to use alternative and possibly +23 + +Following +Q +LIVE +For You +286 +they may be able to bal +abortion but they can't +ban these +29.5K +215 +GreenScreen +2753 +13kasettetapes +#greenscreenthelast oneisajoke(kind)#roevwade +171 +Dalsound-speedaudiios_-Nick + Participating in this activity could result inyou +or others gettinghurt.Authors +dangerous medicine” [P60ML40] and “it is very disturbing and should not be allowed to be showed to +young women in particular” [P13FR30]. +Table 17. What action would you take on Post #7? +Misinformation (no intent) [viewed: 30 participants] +Ignore +13 (43.33%) +Fact-check +3 (10%) +Block +1 (3.33%) +Report +11 (36.67%) +Like +2 (6.67%) +Disinformation (intent) [viewed: 30 participants] +Ignore +18 (60%) +Fact-check +3 (10%) +Block +1 (3.33%) +Report +8 (26.67%) +Like +0 (0%) +8 +DISCUSSION +Our first research question aimed to uncover how social media users conceptualize misinformation +on TikTok, where does it originate, who are the targets, and what’s its purpose. Misinformation, +even within such a small sample as ours, invokes nuanced interpretations as people do not always +stick to the popular “fake news” association [58]. According to the participants in our sample, +misinformation on TikTok might involve falsehoods only, but falsehoods could be interspersed +with biased interpretations, facts taken out of context, or emotion-provoking narratives [96]. The +amalgamation of factual and inaccurate content was mostly a craft assigned to individual content +creators, often for personal gain, but the “usual suspects” – political parties and foreign unfriendly +countries to the United States – were not spared in disseminating misinformation on Tiktok. +This finding suggests that while TikTok’s affordances are fit for selling products or participate +in challenges [50], they are not prohibitive for political and foreign actors to adapt their agenda +setting and information operations. In the past, misleading narratives and content were migrating +from fringe and alt-communities to the mainstream platforms [124] and the evidence from our +study suggest that TikTok could be a future candidate destination. Our participants’ view that many +vulnerable people and easily-influenced crowds would unlikely step out from the “For You” page to +check a claim adds to this impression, confirming previous studies that identified such users as the +most sought out targets of misinformation [19, 77]. The TikTok participation model adds additional +incentive to engage with misinformation as it provides immediate profit and long-term gains, +according both to the participants in our sample and studies exploring political and conspirative +influence on TikTok [14, 65]. +Our second research question narrowed our exploration on misleading abortion narratives. +Given that we conducted our study after Roe vs Wade was overturned, health misinformation +regarding herbal abortifacients dominated the TikTok’s ‘For You’ pages of our participants. While +some of them were simply intrigued about these “at-home remedies,” there were participants +that were disturbed, worried, angered, and disgusted that dangerously misleading content like +this finds its way on the platform. This emotion-provoking response, though concentrated on +a topic of limited polarization, is worrisome because this is precisely the response that foreign +actors sought to incite in past polarizing debates on social media [90, 125]. These debates always +involved politicization of the discourse, so the evidence that our participants also saw politically +contextualized abortion narratives further suggest that TikTok might be the next health/political +misinformation battleground [36], despite the impression of mostly apolitical participation on the +platform so far [1]. +24 + +Abortion Misinformation on TikTok +But to substantiate such conjectures, one needs evidence on how actual users engage with +essential, health-related abortion misinformation in the first place. Uncovering this evidence was +central to our study and we attempted to gain as much as possible a nuanced insight into how +users actually deal with various abortion misinformation content on TikTok. Our findings indicate +that a significant number of TikTok users do not see videos promoting herbal abortifacients as +abortion misinformation, despite the scientific evidence to the contrary [9, 37, 41, 74, 89, 103]. +Regardless of whether the video included the creator itself, contained explicit tags to the herbal +abortifacients listed in Table 1, or was simply a textual post promoting these at-home remedies, many +participants were unconvinced the posts were misinformation. The majority of them conceptualized +misinformation as spreading falsehoods without intent; those that saw intent behind the spread of +falsehoods on TikTok, in general, were far more skeptical of the posts. +A deeper look into the responses of the participants that did not see misinformation in the selected +posts reveals that it was sufficient for the creators to appear “like they know what they are talking +about,” “provide context,” “seems informed,” and are “well spoken” to make the misinformation +believable. There was no particular demographic trend among these participants as the posts were +equally convincing to all gender, political, and age groups. The group of participants that called +out the misinformation indicated both analytical and heuristic cues that could be helpful for the +wider audience on TikTok to employ in dealing with misleading abortion content. For example, +videos that “look like MLM promotions”, “omit any side-effects” of a proposed treatment, “do not +include credible citations”, and the “creator is unqualified for making any health-related claims” +should be dismissed as abortion misinformation. All of these strategies have been proven to work +against health misinformation in the past [80], so our findings strengthen the case for continuing +the “inoculation” against false and misleading abortion narratives [58]. +This is especially important in situations where many users, like the ones in our study, remain +unsure whether these misleading videos are abortion misinformation or not. Our results suggest +that these users lack knowledge on the safety and the effectiveness of herbal abortifacients, which in +turn precludes them to make a decisive dismissal of the misinformation. Educational interventions +on TikTok regarding abortions have already be proposed [30], but question is if the algorithmic +curation of the ‘For You’ page for TikTok users “interested” in alternatives includes them too. +According to our findings, the undecided participants would be easily “nudged” with accurate +scientific information, which is a strategy that worked for other types of misleading content [78]. +It is reassuring to see a fact-checking trend among even a small sample like ours. Perhaps +affordances of TikTok probably encourage majority of the users to simply ignore many misleading +posts, as many of the participants in our study did, and avoid to critically discern the content +[79]. But recalling that familiarity, i.e. a repeated exposure to such videos makes the content look +truthful [80], this fact-checking trend might need to be appended with better debunking and soft +moderation efforts suggested to work for other health-related misinformation [95, 106]. Such a +need is further corroborated by our findings suggesting that even in the presence of the current +TikTok soft moderation labels on post #7, at least 30% of the entire sample still believed that the +content is not misinformation. +Equally important for consideration is that our results suggest a number of users are unafraid +to block or report abortion misinformation. Prior evidence on flagging and reporting misleading +content suggests that social media users might not opt for such actions as to preserve interpersonal +relationships and avoid social arguments, especially for issue that do not matter much to them +[33]. However, reporting misinformation becomes a proactive endeavour when health-related +misinformation is in question [15]. Our results confirm that this effect also takes place for abortion- +related misinformation and occurs across all demographics, making a further case for an actionable +framework of misinformation sharing and correction sharing on TikTok [123]. +25 + +Authors +8.1 +Ethical Considerations +The purpose of our study was not to generalize to a population; rather, to explore the phenomenon +of individual dealing with abortion misinformation from a regular user perspective. While we +added the participants’ self-reported age, gender, and political leanings for more detailed reporting, +we avoided providing definitive numbers accompanying the assessments and response strategies +for each of the seven intervention posts with these demographics. Instead, we supported our +findings with descriptive quotations from participants that convey the way ordinary users deal with +misleading videos on TikTok in a hope that the results can help to elevate the study of abortion +misinformation on social media as a whole. +We acknowledge that there might be a potential risk of repeated exposure to abortion misinfor- +mation, i.e. an “implied truth effect” [80], as each participant saw four of the short-form videos. To +mitigate this risk we explicitly pointed to the debunked information for the related “at-home” reme- +dies and their associated harms. There is also a risk from oversimplification of our results where the +participants’ conceptualization, assessment, and response to abortion misinformation, expressed on +their behalf, might not represent the entirety of the strategies used to deal with misleading videos. +We, of course, agree that users do employ others ways and means to deal with misleading videos +and we welcome every work that brings them to the fore. This will facilitate a scientific work on +abortion misinformation beyond simple oppositional responses [101], a corollary we also want to +avoid when contextualizing our results for future anti-misinformation interventions. +8.2 +Limitations +Our research was limited was limited in its scope to U.S. TikTok media users and the state of abortion +misinformation regarding at-home remedies that existed on TikTok in the period immediately +after the overturn of Roe vs Wade by the Supreme Court of the US. In as much as we attempted to +have balanced, diverse, and inclusive set of short-form videos regarding herbal abortifacients, there +certainly are, and will be, other similar content which our participants might assess and respond to +differently than they did in our study. While the content of these videos was scientifically debunked +and constituted misinformation during our study, we acknowledge that this might change in light +of new scientific evidence. We are also limited by the state of the content moderation policies on +TikTok that, together with public policy changes and new Supreme Court rulings, change and +impact the relevance of our results, therefore we exercise caution in considering these results in +the narrow post-Roe vs Wade context. +A limitation also comes from the sampling method and the use of a survey provider [85], as other +users and other samples might provide results that differ from the once we obtained as there is +little insight into general sampling and sample-related differences when users are broadly queried +about abortion misinformation. By asking users directly about how they interact with abortion +misinformation and misleading video content on TikTok, we got a wide variety of insights from a +broad range of perspectives. We did not measure the efficacy of users’ assessment approaches and +response strategies for explicitly politicized abortion misinformation content, nor did we ask how +users dealt with other abortion misinformation on other social media platforms. Short-form videos +are a relatively new way of persuasive communication appealing to younger users, but traditional +social media text, memes, and visceral images [5] might provoke different responses for a wider +population of users. Therefore, we are careful to avoid any predictive use of our findings because +TikTok’s affordances might change in the future. +26 + +Abortion Misinformation on TikTok +9 +CONCLUSION +Abortion misinformation undoubtedly shapes the way people make reproductive decisions and +TikTok, a very popular social media platform, allows misleading content regarding “at-home” +abortion remedies to reach wide audiences. 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BMJ Global +Health 6, 11 (2021). +[128] Jingwen Zhang, Jieyu Ding Featherstone, Christopher Calabrese, and Magdalena Wojcieszak. 2021. Effects of +fact-checking social media vaccine misinformation on attitudes toward vaccines. Preventive Medicine 145 (2021), +106408. +[129] David X. Zheng, Anne Y. Ning, Melissa A. Levoska, Laura Xiang, Christina Wong, and Jeffrey F. Scott. 2021. Acne and +social media: A cross-sectional study of content quality on TikTok. Pediatric Dermatology 38, 1 (2021), 336–338. +[130] Chris Zielinski. 2021. Infodemics and infodemiology: a short history, a long future. Rev Panam Salud Publica 45 (2021), +e40. +STUDY QUESTIONNAIRE +Exposure and Preconceptions +(1) Can you please define “misinformation” in your own words (please be verbose): [Open +Ended] +32 + +Abortion Misinformation on TikTok +(2) Have you encountered misinformation on TikTok and in what form? Please provide examples. +[Open Ended] +(3) Where does misinformation on TikTok come from, in your opinion? [Open Ended] +(4) Who is the target of misinformation on TikTok, in your opinion? [Open Ended] +(5) Who benefits from misinformation on TikTok, in your opinion? [Open Ended] +Engagement Strategies +(1) How do you suspect or know that a certain TikTok post is a misinformation? Please elaborate. +[Open Ended] +(2) What is your strategy for dealing with misinformation posts on TikTok? Please elaborate. +[Open Ended] +(3) Have you used any engagement features (e.g. share, comment, like, follow) for misinformation +posts on TikTok? If so, in what circumstances? [Open Ended] +(4) Have you used any action features (e.g. block, mute, report, unfollow) for misinformation +posts on TikTok? If so, in what circumstances? [Open Ended] +(5) Have you talked about a particular TikTok misinformation post outside social media? If so, +in what circumstances? [Open Ended] +Abortion Misinformation Exposure +(1) On what other occasions you have encountered misinformation regarding abortion on +TikTok? Please elaborate. [Open Ended] +(2) What was your response to this particular abortion misinformation on TikTok? Please +elaborate. [Open Ended] +(3) Where else does abortion misinformation exists outside of TikTok, in your opinion? Please +elaborate and share any experiences. [Open Ended] +33 + diff --git a/J9E4T4oBgHgl3EQfhw2D/content/tmp_files/load_file.txt b/J9E4T4oBgHgl3EQfhw2D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e92479fa7d193fd1b24370ef8dacc45ed2cadb59 --- /dev/null +++ b/J9E4T4oBgHgl3EQfhw2D/content/tmp_files/load_file.txt @@ -0,0 +1,1708 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf,len=1707 +page_content='Abortion Misinformation on TikTok: Rampant Content,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Lax Moderation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' and Vivid User Experiences FILIPO SHAREVSKI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' DePaul University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' United States JENNIFER VANDER LOOP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' DePaul University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' United States PETER JACHIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' DePaul University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' United States AMY DEVINE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' DePaul University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' United States EMMA PIERONI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' DePaul University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' United States The scientific effort devoted to health misinformation mostly focuses on the implications of misleading vaccines and communicable disease claims with respect to public health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' However, the proliferation of abortion misinformation following the Supreme Court’s decision to overturn Roe v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Wade banning legal abortion in the US highlighted a gap in scientific attention to individual health-related misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' To address this gap, we conducted a study with 60 TikTok users to uncover their experiences with abortion misinformation and the way they conceptualize, assess, and respond to misleading video content on this platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Our findings indicate that users mostly encounter short-term videos suggesting herbal “at-home” remedies for pregnancy termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' While many of the participants were cautious about scientifically debunked “abortion alternatives,” roughly 30% of the entire sample believed in their safety and efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Even an explicit debunking label attached to a misleading abortion video about the harms of “at-home” did not help a third of the participants to dismiss a video about self-administering abortion as misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We discuss the implications of our findings for future participation on TikTok and other polarizing topics debated on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' CCS Concepts: • Security and privacy → Social aspects of security and privacy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Usability in security and privacy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' • Human-centered computing → Empirical studies in ubiquitous and mobile comput- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Additional Key Words and Phrases: TikTok, misinformation, abortion, fact-check, debunking, social media 1 INTRODUCTION Misinformation, thriving around polarizing topics [126], draws a particular attention in online discourses centered around health issues [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Alternative health narratives are not a new phe- nomenon [98], but social media’s affordances for anonymity, free content creation, and the lack of editorial checks allow for rapid dissemination among users that, in turn, join pro/against camps about “infant vaccination” [97], “COVID-19 mass immunization” [118], and “cancer treatments” [46] in droves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Health misinformation is not just a harmful pretext for a polarizing social media discourse, but is a real threat for both individual and public well-being [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In the past, the health misinformation was incited mainly by misleading health research [29], deceptive interpretations of symptoms [36], and contested public health governance decisions [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In these cases, misinformation appended a fear of either undesirable or unknown health consequences, prompting people to question long-standing health scientific methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The response to such misinformation, thus, was complicated by the tendency of people to hold beliefs that align with a persuasive message working on their biases and self-preservation [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In these circumstances, the effort was driven towards prebunking health myths [58], “accuracy nudges” to debunk the misleading health claims [78], and flagging dangerous health misinformation content on social media [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Some of these interventions did take some of the sting out of the misinformation [106], but are far from providing a comprehensive health misinformation containment [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' While the scientific community continues to work towards minimizing the adverse effects of “fear-mongering” health misinformation [115], a new type of dangerous health misinformation – appending the lack of desirable and known health practices – was abruptly amplified in the immediate aftermath of the US Supreme Court decision to strike down the constitutional right 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='05128v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='SI] 12 Jan 2023 Authors for abortion [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The inability to obtain a legal abortion turned people to search engines and social media to learn how to manage their reproductive decisions and perform safe abortions [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Unfortunately, not all information aligned with the National Library of Medicine’s description of abortion and recommendations for safe practices [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Many questionable practices including pills, oils, and herbs for inducing abortion flooded social media, both as claims and as an advertisements in users’ feeds [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Platforms used diverse strategies to mitigate this misinformation: YouTube added “context labels” to such abortion content [122], Twitter decided to promote authoritative abortion information in its Twitter Moments and Events [52], and Meta purportedly blocked questionable abortion treatment advertisements [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok also stated it removed and labeled videos with abortion misinformation [51], but many of the questionable home practices aimed to “cause a miscarriage” still appeared in users’ personal streams [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Debunking of abortion misinformation on TikTok followed up [103], but the slow-in- nature checking and verifying of health-related facts was no match for the rapid spread of videos recommending dangerous abortion remedies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Since misinformation in general is “sticky”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' repeated exposure to false statements make them appear truthful [57], the lack of systematic response against abortion misinformation at this stage created a situation where “sticky” unproven abortion remedies could lead people to attempt unsafe procedures and cause serious bodily harm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' While all social media platforms require scrutiny of their abortion misinformation handling, TikTok – deemed the “New Google” for Gen-Z [41] – draws special attention in this conundrum as pressing reproductive decisions are particularly interesting to the majority of users on this platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok’s status as a platform for social support exchange [4] further exacerbates the immediate danger of abortion alternatives as supportive communication adds to “stickiness” and internalization of such content among adolescents and young adults [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Motivated to explore how TikTok users deal with misinformation and alternative abortion narratives, we conducted a study with 60 TikTok users in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' First, we obtained a large TikTok dataset to uncover the main themes of abortion misinformation on this platform and get a better sense of how TikTok recommends and moderates such content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Leveraging the unofficial TikTok-API python [108] library, we scraped 8,226 videos with 77,880 hashtags, of which 17,606 were unique in tagging these videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We collected the videos using a snowball sampling strategy, starting with scraping three initial hashtags, specifically #TikTokTaughtMe, #healthcare, and #abortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We selected the first one as it’s the de facto hashrtag through which a user is “googling” short-form TikTok videos for challenges and advice for self-help [18] and the other two as we were focused on health/abortion misinformation in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' To report the findings from our study, we review the related work on misinformation and misleading health information on social media in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Section 3 provides the broader context of abortion misinformation narratives on social media following the ban on abortion in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Section 4 provides the methodological details of our study and Sections 5, 6, and 7 elaborates how participants in our study conceptualize, encounter, and respond to abortion misinformation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We draw on our findings in Section 8 to discuss the implications for both individual and public health as well as social media content moderation of alternative abortion narratives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Finally, Section 9 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2 HEALTH MISINFORMATION BACKGROUND 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Health Misinformation Narratives Health-related rumors and alternative narratives precede the Internet era and were concentrated around health issues with either unknown or undesirable consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In the 1980s, for example, the KGB initiated an information warfare campaign called “Operation Infektion” to spread the 2 Abortion Misinformation on TikTok rumour that HIV/AIDS was a mis-fired American biological weapon in order to undermine the United States’ credibility during the Cold War [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In the same time, the tobacco industry in the United States created a “disinformation playbook” to systematically distort and downplay the link between the consumption of tobacco and cancer [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' While the intent to mislead was clearly present in these campaigns, the volume and output of the rumors and alternative health narratives was limited to a number of outlets and fabricated publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The Internet and social media changed the landscape by enabling an inordinately high volume and rapid output of health information with varying quality to reach the public [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The health- related rumors and alternative narratives collaterally grew and were amplified to a point where they yielded uncontrollable consequences even for known and treatable health issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' For example, a poorly designed study in the 1990s that falsely claimed that the measles, mumps, rubella (MMR) vaccine causes autism [72] caused such a regression in public immunization that resulted in several measles outbreaks twenty years later on [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Rumours about the Ebola and Zika viruses also overshadowed the evidence-based health information and resulted in higher vaccine hesitancy in fear of undesirable health consequences and death [75, 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The vaccine hesitancy, on a global level, achieved a climax during the COVID-19 pandemic with an unprecedented volume and output of COVID-19 related misinformation [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' While the majority of misleading health information on social media focuses on vaccines and communicable diseases, rumors and alternative narratives also spread about cancer, heart disease, and other conditions [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' For example, social media users are more likely to trust and share cancer-related rumours if the rumours are dreadful rather than wishful, and if one has had previous personal experience [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The uncontrollable consequences in these cases are not overall treatment hesitancy but seeking of alternative and unproven treatments about diabetes [55], heart failure [25], hypertension [54] and psoriasis [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Interestingly, in all of these cases of non-communicable health issues, the unsubstantiated claims were promulgated through videos as a particularly influential mode for conveying misleading health evidence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' also used in anorexia and dietary disorders’ deceiving messages) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Response to Health Misinformation In the context of misleading health claims, misinformation is considered by its opposition to the consensus of what the medical community defines as accurate and evidence-based information [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Scholars, in response, have focused the attention of anti-health-misinformation on two main fronts: 1) examining the harms of the misinformation [62, 102];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' and 2) misinformation prebunking and debunking [53, 57, 80];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The harms of health misinformation are reflected in dramatic increase in vaccine hesitancy [62], pursuing dangerous home therapies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' cancer cleansing, weight loss, virus prevention) [102], as well as increased hostility toward health workers [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The goal of “prebunking” or forewarning is improving people’s ability to spot and resist ma- nipulation techniques commonly used in health misinformation [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' To this objective, people nowadays are “innoculated” against health misinformation by the use of “accuracy nudges” [78], social correction that occurs via peers [116], or play browser-based games about health myths and facts [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The “prebunking” was shown to be an effective strategy [58], though in time of social media virality the inoculation effect wanes for users with a conspiracy mentality about unknown and undesirable health consequences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' the COVID-19 pandemic) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' If this “innoculation” is rendered ineffective, “debunking” is the next step where verifiable corrections of the falsehoods from credible sources are presented in order to break the illusion of truth [32, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Debunking, as in fact-checking of health misinformation, was shown to give mixed results depending on the perceived credibility and expertise of the sources in science-related contexts [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The perception of credibility and expertise, for example, matters little to people with 3 Authors strong conspiratorial ideation tendencies who tend to mistrust any official source [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As pressing health problems of general public interest are hard not be seen also in a political context, debunking of health misinformation was found to work either when it comes from sources that are perceived to share people’s values and worldviews [70], or when people maintain a science-accepting attitude regardless of their political worldviews [3] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='3 Moderation of Health Misinformation In as much as the prebunking and debunking helps in curbing the health misinformation work online, they are nonetheless slow and difficult to scale to the pace, volume, and output of infor- mation sharing on social media [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Platforms, in response, had to turn to automated means of moderating unsubstantiated content and questionable accounts to prevent an “outbreak” of misleading information, especially after the meteoric influx of COVID-19 rumors, conspiracies, and falsehoods [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' YouTube opted for a soft moderation and decided to apply context labels to video searches for health information that link to credible sources recommended by the National Academy of Medicine [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Twitter, up to early December 2022, also applied soft moderation in two forms: (i) interstitial covers, which obscure the misleading content and require users to click through to see the information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' and (ii) trustworthiness tags, which appear under the content and do not interrupt the user or compel action [49, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Meta, the parent company of Facebook and Instagram, did the same in conjunction with hard moderation, taking down prominent accounts that spread COVID-19 misinformation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Robert F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Kennedy Jr.’s account was blocked after he repeatedly undercut trust in the COVID-19 vaccines) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok followed suit and expanded their soft moderation labeling with trustworthiness tags to content pertaining to eating disorders, health challenges, and alternative medical treatment videos next to misleading COVID-19 videos [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The response to platform moderation has been, at best, mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The interstitial covers provided an adequate “accuracy nudge” for users to distance from COVID-19 misinformation posts, but users largely ignored trustworthiness tags [91, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Further, numerous studies reveal that trustworthiness tags “backfire” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' make users believe health misinformation more, not less [28, 31, 71, 110]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In the context of the COVID-19 pandemic, the tags triggered a “belief echo,” manifested as skepticism of adequate mass COVID-19 immunization [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A possible reason for such an unexpected reception of the trustworthiness tags was the asymmetrical nature of soft moderation—the mere exposure to health misinformation often generates a strong and automatic effective response while the tag itself may not generate a response of an equal and opposite magnitude [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This is because the trustworthiness tags often lack meaning, have ambiguous wording, or ask users to find health information themselves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' learn more about COVID-19), which is cognitively demanding and time consuming [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='4 Health Misinformation on TikTok TikTok, a social media platform for short-form videos, has rapidly grown in popularity in the last few years [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A central feature of TikTok is the ‘For You’ page, a feed of algorithmically curated videos for the user based on the user’s past browsing, viewing, and interactions with videos [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Users can also search for videos based on hashtags and, in some cases, sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Roughly 75% of the global users on the platform are age 34 or younger, and every fifth person in the United States visits the platform on a daily basis [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok’s affordances for viral spread of curated short-form videos and demographic structure [60] made the platform particularly interesting for healthcare workers and science communicators creating educational content both based on their speciality and more general advice [101, 127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A review of 331 videos with authoritative COVID-19 information [59] showed that anti-stigma/anti- rumor, disease knowledge, encouragement, personal precautions, recognition, and societal crisis 4 Abortion Misinformation on TikTok management drive platform engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Another review of 199 videos with information about chronic obstructive pulmonary disease showed that most of them have a satisfactory scientific background [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' An analysis of obstetrician-gynaecologists (OBGYNs) videos and the associated hashtags and commentary [101] revealed that the health “educators” not just convey authoritative sex health information [104], but use it to creatively debunk the related misinformation and mislead- ing treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This practice of authoritative health communication as a form of misinformation diffraction also motivated proposals for teaching abortion using TikTok [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Other work on misleading health content on TikTok is scarce and overwhelmingly focuses on COVID-19 health misinformation [64, 127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Basch et al [5] analyzed a sample of 72 videos containing the hashtag #covidvaccine and found that slightly more than half of them discouraged getting the vaccine by showing purportedly adverse vaccine reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Baumel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [7] analyzed the 100 “most liked” videos under each of the #Pfizer and #Moderna hashtags and found that 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2% and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='8% of the comments conveyed misleading negative sentiment against these two COVID-19 vaccines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Baumel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [8] also analyzed TikTok commentary related to masks’ effectiveness in combating COVID-19 and found that 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='3% of commentary using the #MasksDontWork hashtag contained misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Analyzing a sample of 1000 videos, Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [93] found that around 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='6% of the videos contained misleading COVID-19 content, but were shared as much as one with verified COVID-19 information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Outside of the COVID-19 theme, O’Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [73] analyzed 27 TikTok videos containing pediatric urology claims and found that only 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2% contained information that can also be found in official guidelines provided by the European Association of Urology (EAU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [121] reviewed 65 TikTok videos with the hashtag #prostatecancer and found that at least 48% of them contained explicit prostate cancer misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [129] study found that the top 100 videos with the #acne hashtag had seriously misleading information about diagnosis and treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The only study so far analyzing the way misinformation is moderated with warning labels on TikTok focused on COVID-19 content [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [60] collected 41,000 videos that include 26 COVID-19-related hashtags in their description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Through a qualitative analysis, they found out that TikTok likely moderates videos based on hashtags included in the description without an in-depth analysis of the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' learned that this moderation strategy led to a large false positive rate – about a quarter of the videos with a misinformation warning label did not contain content related to COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The study also found a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='7% false negative rate where videos with actual COVID-19 misinformation did not include warning labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 3 ABORTION MISINFORMATION CONTEXT Prior studies have shown that 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1% of women obtain information regarding abortion from the Internet [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Abortion misinformation online, thus, takes many forms and users generally have difficulties discerning inaccuracies in the related alternative narratives [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Bessett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [12] presented 586 participants with five vignettes of abortion misinformation – safety, breast cancer, infertility, mental health risk, and legality of abortion – and found that only 4% of participants were able to correctly identify all of vignettes as misinformation while 73% pointed to two or fewer vignettes as inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Common abortion misinformation topics that have been studied are the increased risk of breast cancer, future infertility, depression/anxiety, and post-traumatic stress [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This misinformation is spread through multiple sources, including state-mandated “Women’s Right to Know” documentation that providers must supply before a woman can consent to having an abortion [9], despite official guidance from the National Academies of Sciences, Engineering, and Medicine [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The Guttmacher Institute found that two states inaccurately include a link between abortion and an increased risk of breast cancer, 19 states link abortion to future infertility, and eight states link abortion to 5 Authors negative emotional and psychological responses [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Kern and Reader [52] reported that abortion misinformation specifically related to an “abortion reversal pill” increased on Facebook from 20 interactions on June 23 to 3,500 interactions on June 24 2022, the day after the Supreme Court decision to overturn Roe v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Wade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Godoy [37] also reported that following the Supreme Court ruling, Spanish-language abortion misinformation was deliberately designed to galvanize voters in Latino communities across the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Abortifacient herbs – purportedly providing the ability to induce a spontaneous miscarriage – form the majority of post-Roe v Wade misinformation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The toxicity of abortifacient herbs has been widely studied as shown in Table 1 but there is little literature and few studies related to the topic of “herbal abortions” [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Most existing studies were done in countries where abortion was not legal until recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Abortion did not become legal in Uruguay until 2012 [63], for example, and a 2003 study found that the Montevideo Poison Centre had 86 cases of ingestion of herbal infusions with abortive intent from 1986 to 1999 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In the United States, misinformation surrounding “herbal abortions” in viral videos on TikTok has increased dramatically after legal abortion was overturned [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The consequences of these viral misinformation videos already brought several people to the emergency rooms seeking critical lifesaving treatment, making active prebunking and debunking by qualified health professionals imperative [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Herbal Abortifacients and Side Effects Common Name Side Effects Black Cohosh Hepatoxicity [34] Blue Cohosh Nicotinic toxicity: tachycardia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' hypertension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' headaches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' abdominal pain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' vomiting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' muscle weakness and fascicula- tions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' seizures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' and coma [84] Eastern Daisy Fleabane Insufficient evidence on the safety and effectiveness as an abortifacient agent [109] Mugwort Vomiting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' hypertension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' confusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' respiratory distress,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' coma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' and seizures [22] Parsley Abdominal pain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' vomiting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' genital hemorrhage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' anemia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' jaudice [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' internal bleeding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' convulsions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' and death [21] Pennyroyal Gastrointestinal upset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' fainting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' intestinal bleeding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' seizures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' hepatomegaly or injury,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' multiple organ failure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' coma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' cardiac arrest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' and death [41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 105] Rue Vomiting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' liver damage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' anemia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' tremors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' respiratory dis- tress,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' multiple organ failure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' and death [47] 4 MISINFORMATION AND ALTERNATIVE ABORTION NARRATIVES ON TIKTOK 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Research Questions The volume and output of abortion misinformation naturally prompted health experts, by them- selves, to dispel the inaccuracies related to herbal abortions, abortion pills, and abortion side-effects directly on social media [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This effort is by all means needed, but is likely not to be sufficient to prevent an “outbreak” of unsafe decisions about people’s reproductive health in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' An intuitive response, then, would be a systematic prebunking and debunking of abortion misinforma- tion in coordination with moderation of related content on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Based on past experiences 6 Abortion Misinformation on TikTok with health misinformation, however, such a response leaves little room for nuanced explorations of how people engage on their own with abortion misinformation in the first place [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As TikTok has been identified as a “hotbed of abortion misinformation” [39], such an exploration would be beneficial to the ongoing response of removing and moderating misleading content on TikTok [51] as it will provide knowledge on how users conceptualize, encounter, assess, and respond to abortion falsehoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' So far, such knowledge is scarce and only provides glimpses on how users conceptualize misinformation encountered on the traditional social media platforms [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' To address this knowledge gap, we set to conduct a study that aimed to answer the following research questions: (1) RQ1: Concept: How do social media conceptualize misinformation on TikTok (definition, origins, targets, and purpose)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' (2) RQ2: Encounters: What encounters with abortion misinformation users had so far on TikTok and how they dealt with it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' (3) RQ3: Response: What strategies users employ in assessing and responding to various abortion misinformation content on TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Dataset Preliminary, we set to collect a dataset of abortion misinformation on TikTok in the immediate period after the overturn of Roe vs Wade, up till the end of November 2022, as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We leveraged the unofficial TikTok-API python library [108] to scrape 8,226 videos, which we collected using a snowball sampling strategy, starting with scraping three initial hashtags, specifically #TikTokTaughtMe, #Healthcare, and #Abortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Due to the limitations of the API, each search returned no more than 300 videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' To continue collecting hashtags, we searched for each of the hashtags that were associated with each of the three seeding hashtags above, effectively performing a snowballing sampling of the TikTok’s base with abortion-related short-form videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok Abortion Hashtag Dataset Attribute Value Total Number of Posts 8,226 Number of Hashtags 77,880 Unique Hashtags 17,606 From here, we vectorized the hashtags using Scikit-Learn’s CountVectorizer [76] to create a dense boolean array of 1,754 tokens – character unigrams, bigrams, and trigrams – that appeared in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='01 - 99% of hashtag samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We then identified the closest hashtags to a given input hashtags using Minkowski distance, or ||𝑥||𝑝 where 𝑥 is the difference between an searched vector and the saved vectors from our hashtag dataset, and 𝑝 is a scalar that we selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' To identify 𝑝, we reviewed kernel density estimate plots of the distances to several searched hashtags and identified the most expected bimodal distribution, with a smaller left distribution of relevant hashtags, and a larger right distribution of less relevant hashtags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We settled on an ideal 𝑝 of 2, which is ||𝑥||2, or euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Using these hashtag representations, we were easily able to identify perturbations in hashtags that might otherwise be moderated by TikTok [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' For example, searching the representations for hashtags like #selfharm highlighted the existence of #sêlfhârm, and #abortion revealed #abotion and #anortion, as well as longer hashtags like #abortionishealthcare and #abortionishealthçare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A 7 Authors search with regular terms and hashtags like “#abortifacient” indeed does not present any videos tagged as such, but following our dataset analysis above, we discovered that a small change in the spelling – #ab0rtifacient, for example – unveils a lot of abortion videos that promote abortifacient solutions for miscarriage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Many of these videos were not necessarily were tagged with the exact search hashtag and may even be tagged with the original “#abortifacient.” One could argue that an ordinary users might not know what hashtags exist, but from the video posting functionality, a list of suggested hashtag completions provide additional variations, and the number of videos with each variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As such, even an incomplete, suggestive hashtag search on TikTok brings seemingly obscured tags for abortion misinformation, as exemplified in Figure 1 Using these built in features, we quickly identified dozens of videos that described methods for “at-home” abortions as candidates for misleading claims we wanted to test in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The images above demonstrate how, while a search for “#abortion #herbs” does not return any videos due to the TikTok’s guidelines for harmful content [51], a search for “#abotion #herbs” not only returns videos with similar typos, it also returns videos with the original “#abortion #herbs” spelled correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Additionally, when posting a video, TikTok suggests additional hashtags, any of which can be searched to find additional hashtags, like #ab0rtionishelathcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='3 Sample The analysis of the information in our dataset, given in section 7, helped us identify the main themes of abortion misinformation content on TikTok in the aftermath of Roe vs Wade decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As we were interested in better understanding how actual users deal with this content, we obtained approval from our Institutional Review Board (IRB) to conduct an exploratory survey (the questionnaire is provided in the Appendix) with a sample of TikTok users ages 18 and above in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We used Prolific for recruitment and after we consolidated the responses we obtained through Qualtrics, we ended with a sample of total of 60 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The responses were anonymous, and the survey allowed users to skip any question they were uncomfortable answering, taking 8 11:30 @45Gl92% Q #abortion #herbs X Top Users Videos Sounds LIVE Hashtags Noresultsfound Thisphrase maybeassociatedwithbehaviororcontent that violates our guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=" Promoting a safe and positive experience is TikTok's top priority." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content="For more information,we invite youto review our Community Guidelines11:30 Q45GEl 92% #abotion #herbs Q X Top Users Videos Sounds LIVE Hashtags All Unwatched Watched Recentlyuploaded Detailed Information Herbs that Induce Contained within Aborfion What method I've used to interrupt pregnancy at home naturally Perspective from a Herbalist 6/28 8/26 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='#abortion#herbs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='herbsOnHANDsothat #herbalmedicine you can even start taking t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' medical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='mamacita ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='569 boobygrow ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='157 SHOT Abortifacient Herbs proud of Texas!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' etthat babies will ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='.Sad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=" DON'T use these dangerous herbs O <11:35 Q45GEl 88% Post #abOrt Select cover #Hashtags @Mention OVideos #abOrt 467." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5Kviews #abOrtionsaveslives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='3Mviews #abOrto 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='8M views #abOrto 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='8M views #abOrtire Oviews #abOrtionishelathcare 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='6Kviews #abOrted 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='7K views #abOrtOlegal 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='4Kviews #abOrtionjustice 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='6Kviews #abOrtionrights 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2K views I= VAbortion Misinformation on TikTok around 25 minutes to complete it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants were offered a compensation rate of $5 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The demographic structure of our sample is given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Sample Demographic Distribution Gender Female 44 (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Male 15 (25%) Non-cisgender 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Age [18-20] 5 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) [21-30] 33 (55%) [31-40] 12 (20%) [41-50] 6 (10%) [51-60] 4 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) [61+] 0 (0%) Political leanings Left 38 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Moderate 14 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Right 5 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Apolitical 3 (5%) Highest Level of Education Completed High school 12 (20%) College 43 (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Graduate 5 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='4 Method and Analysis Participants were provided an open ended qualitative survey through Prolific that provided a list of questions and a predetermined set of TikTok videos we selected from our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We singled out seven videos in total from our dataset that contained abortion misinformation already debunked by the time of our study [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We used the input on general abortion misinformation from Table 1, information from authoritative verifiable sources [69], and verbatim misinformation terms from two fact-checking articles [23, 109] as a selection criteria for videos promoting the use of herbal abortifacients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We also chose to focus only on “at-home abortion remedies” as explicit health misinformation [101] and not alternative abortion narratives involving “religion” or “political contextualizaiton” to avoid bias and expressive responding [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We wanted to have as many varying modalities, formats, and creators in our selection as possible, therefore we selected two videos that contained only text and five videos featuring the creator of the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Six of the selected videos were created by women and one was created by an individual who identifies as transgender in their profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The creators of the videos were ethnically diverse, consisting of individuals who identify in their profile or other videos as White, Black, North American Indigenous, and Native Hawaiian or Pacific Islander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We must note that TikTok, in response to the increased scrutiny about their lax handling of health misinformation [51], claims to regularly remove misleading abortion content so there is a possibility that our dataset was considerably restricted for our particular selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants were asked to describe their experience with encountering misinformation on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Next, we asked participants to provide their opinions on where misinformation comes from, what purpose misinformation serves on social media, and who creates and benefits from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants were then asked to further elaborate how they determine a certain social media post is misinformation, and what tactics they employ when dealing with misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In reporting the results, we utilized as much as possible verbatim quotation of participants’ answers, emphasized in “italics” and with a reference to the participant as either PXYZ# or [PXYZ#], where P denotes participant, X denotes the number of the participant in the sample (ordered by the time of participation), Y denotes their gender identity (F - female, M - male, NC - 9 Authors non-cisgender), Z denotes their political identity (L - left-leaning, M - moderate, R - right-leaning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A - apolitical), and # denotes the upper bound of their age bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' For example, P16FL30 refers to participant 16, female, left-leaning, age bracket [21-30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 5 MISINFORMATION CONCEPTUALIZATION ON TIKTOK 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Definition First, we asked our participants to define misinformation in their own words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Exactly half the sample provided a definition that did not include any intention in the production or dissemination of questionable content, along the lines of the misinformation definitions outlined in [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' All of these participants conceptualized falsehoods through the inherently fallacious information mental model of misinformation on social media described in [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' For example, P31FL30 defined it as “untrue/unsubstantiated statements being presented as fact,” P13FR30 as “incorrect, skewed, or communicated incorrectly,” and P27MM20 as simply “false information.” In this half of the sample, 18 (60%) of the participants identified as left-leaning, 3 (10%) as right-leaning, 8 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) as moderate, and one (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) as apolitical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The other half of our sample expressed intentionality as an additional quality of misinformation, de facto referring to disinformation instead [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Using the folk models of misinformation on social media [96], more than half, 20 (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%), of the participants conceptualized misinformation as out-of-context narratives, for example, P36FL40 stated that misinformation is “is intentionally, either by using wrong information or leaving out context, misleading to the people reading it.” The next most popular folk model 6 (20%) was external propaganda and the participants pointed to “intentional spread of misleading information to stir an emotion or to further promote a system, product, or person” [P5FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The remaining 4 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) participants conceptualized misinformation as political (counter)argumentation pointing to cases where “a journalist or news source provides false information to persuade you in one political direction” [P47MR30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Here, the older participants were, the more they saw an intention in the spread of misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' For example, P50FL60 placed the intentionality where “videos get edited and changed, and convincing memes with cherry picked facts are created as part of misinformation that has been used extensively in politics and the pandemic.” 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Origins Three quarters or 45 of the participants in our sample felt that misinformation on TikTok came directly from a creator of the TikTok video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In the view of P20FL40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' misinformation on TikTok is brought by “people who are trying to gain clout,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' or get numerous views.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' P19FL30 went further and reckoned that “misinformation can come from the creators’ own consumption of misinformation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' or a creators’ misinterpretation of information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' or a creators’ attempt to sell something or an idea to influence others/gain attention.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The creators of Tiktok content,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' in the view of P27MM20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' “are people who doesn’t care about misinformation but more about views and attention”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The remaining 25% pointed to the “other” side of a polarized debate or issue i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' “people on both the left and right who want to increase views, as well as institutions and political groups with agendas to create misinformation” [P50FL60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' P2FL50, seeing misinformation as out-of-context narrative, felt that it “comes from a variety of places such as Republicans, Russia or China” and P12FL60 seconded the impression of external interference “directly from a bad-actor or a big-mouth source such as Fox News.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' P50FL60, using the political (counter)argumentation model of misinformation, directly accused the GOP for “catering misinformation to low information and low IQ people who will believe anything they get told because GOP knows they can’t fool the science/college crowd.” 10 Abortion Misinformation on TikTok 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='3 Targets Half of our sample felt that the targets of misinformation are “vulnerable people who do not know how to research and form their own opinions” [P7FM30], specifically “Younger people, older people, or more easily-influenced crowds, which are the people who are not likely to fact check a claim” [P40FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' P50FL60 expanded this list to include people “in areas that have low instances of college education and high poverty areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' who are lower income and very religious;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' who are already suffering themselves and see anyone who gets ahead as a threat to them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' and who have very little access to help so they resent people.” The other half felt that “anyone and everyone can be a target of misinformation” [P44FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' P5FL30 described the targets of misinformation on TikTok to be from “All ages, races, sexualities, and backgrounds are targets of misinformation because the algorithm brings it to your ‘For You page’.” 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='4 Purpose 20 (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) of our participants explicitly indicated that the purpose of misinformation on TikTok is for profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The profit was assigned either to “politicians and large corporations who either make money off what evolves from misinformation campaigns or who benefit politically and financially from legislation enacted when bad actors are elected to government offices” [P12FL60] or to content creators themselves as “they get paid from the views, and there’s probably some devout followers to these people which give them a recurring income from just watching the videos every time they post” [P15ML30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Implicit gains, such as “engagement boosts” [P1MR50] that ultimately lead to profit per the TikTok participation model [50], was the purpose that 14 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) of our participants identified behind the spread of misinformation on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' They identified “creators and influencers looking to gain followers and views” [P1MR50] and “people who doesn’t [sic] care about misinformation but more about views and attention” [P27MM20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation as a political ammunition was the purpose identified by 13 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) of our participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' P31FL30 indicated that the purpose of misinformation is “political influence feeding into distrust of science and government” and P59FL30 felt the misinformation on TikTok is brought “to divide people further and continue to build up the conservative party.” 5 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) of participants felt that misinformation was to “stir the pot” on TikTok, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' “foreign agency targeting the US or groups within the US that want superiority” [P41FM40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Videos created by “trolls make up a portion of deliberate misinformation” [P38FL30] to “gets a rise out of someone,” in the view of P5FL30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A subgroup of 8 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) participants, indicated that “no one” [P10MA40] benefits from misinformation on TikTok, both in a short-run and “ultimately, in the long-run” [P58FM30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 6 ABORTION MISINFORMATION ENCOUNTERS ON TIKTOK 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Encounters Exactly half of the sample indicated they have seen abortion misinformation on TikTok prior to the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The misleading content mostly consisted of “videos like these claiming at home abortion remedies” [P25FL20] but also included “misinformation rooted in religion – churches show videos of full term pregnancies being ripped apart by limbs from wombs” [P50FL60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants also indicate they were seeing politically contextualized abortion narratives “on both sides of the political spectrum” [P7FM30] to either ban or allow “birth control as well as contraceptives provided by the government” [P24FR20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants also indicated that they see “people who are Pro Life on TikTok that spread all kinds of rumors and lies about abortion all the time” [P5FL30] “mostly to cause fear” [P49Fl30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 11 Authors 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Response About half of the participants indicated that abortion misinformation invoked negative emotions in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants stated that the “videos in all just made me sad” [P6FA30], that they were “very disturbed by this abortion misinformation” [P13FR30], and “disappointed that people are making content like this” [P15ML30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Some of them said that their “first response was shock that someone would even think this” [P24FR20], that and it is “worrying that this type of misinformation is being shared because it can be dangerous” [P39-FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants in our sample felt “anger, resentment” [P42FL30] and “disgust that people will believe anything they see and try it” [P56FM50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The other half indicated they were mostly “intrigued and wanted to know if anything in these herbal videos was true or not” [P41FM40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In response to abortion misinformation content on TikTok, our participants said they “did not engage because it only further spreads the misinformation” [P8FL20] or “just ignored it” [P52MM40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Some participants indicated they took action on the video by doing “research on abortion and take what I gather on TikTok with a grain of salt unless it is information spread by an actual health professional” [P26FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' There were also participants that “blocked the creators that spread the misinformation” [P20FL40], “reported these videos for spreading false information” [P49Fl30], or “Liked comments pointing out the abortion falsehoods” [P31FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 7 RESPONSE TO ABORTION MISINFORMATION ON TIKTOK 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Post #1 The first post we presented to participants was labeled by the creator with the hashtags #roevwade, #abortion, #herb,s #knowyourherbs, #herbalist, #womensrights, #fightbackwithherbs, #herbalism, #homesteadinglife, #michigan, #crazyplantlady, and #farmlife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This post discusses the use of Eastern Daisy Fleabane root [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The use of fleabane as an abortifacient herbal tea was found misleading as it can “have unpredictable effects” and there is no evidence that this root can induce a miscarriage, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The screenshot of the post as it appeared in the standard TikTok app is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We broke down the results of the participants’ evaluation in two groups based on their baseline mental model of misinformation on TikTok we outlined in section 5 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The assessment results of the first TikTok video in our study are given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Out of the 30 participants who thought misinformation is disseminated on TikTok without intent, 14 were randomly selected to assess the first post with abortion misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Three of them thought the video is indeed misinformation, stating that “this is likely misinformation, as a claim such as this likely has very little evidence to substantiate it” [P46MM30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A surprising 50% of the participants in this group though the video was not misinformation, feeling that “it is true because she sound [sic] like she knows what she is talking about” [P8FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Four participants in this group were unsure if this video was misinformation, worried that “they state some historical context not verified in any way” [P42FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Out of the other 30 participants that saw misinformation being spread with intent on TikTok, 12 were randomly shown the first post (the random selection was done by the Qualtrics survey software we used, leading to slightly unbalanced group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Four of them confirmed the video contains falsehoods with quite a verbose justification: “This is misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' It is referencing clearing your liver which immediately points to something more like a Multi-Level Marketing (MLM) product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' She’s using the same terminology that essential oil salespeople use;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' These people can never name what toxins, etc you are eliminating because that is not actually happening” [P50FL60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Three participants thought otherwise, believing this post was not misinformation because “the content creator seems 12 Abortion Misinformation on TikTok Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok Post #1 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Is Post #1 Misinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 14 participants] Yes 3 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='43%) No 7 (50%) Unsure 4 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='57%) Disinformation (intent) [viewed: 12 participants] Yes 4 (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='34%) No 3 (25%) Unsure 5 (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) informed” [P10MA40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Five participants were unsure, because they had “no idea whether the claim in the video is based in truth or not” [P32NC50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants were also asked to describe what action they would take for each post, with their actions given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Of the participants that thought misinformation is disseminated on TikTok without intent, six (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='86%) said that they would “scroll past without interacting with the post” [P42FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants in this group were equally likely to “verify said information to see if it is accurate” [P20FL40] and “would like the post” [P8FL20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Only two (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='28%) participants in this group said they would “unfollow this person” [P56FM50] or “report this video for dangerous activities” [P24FR20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Of the participants who saw misinformation being spread with intent on TikTok, five (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) said they “would just scroll past it” [P10MA40] and five (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) said they would “look at the comments and then conduct my own personal research” [P58FM30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' There were also two (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) participants that said they “would block it” [P4MM50] or “would report this”[P50FL60] and no participants from this group said they would like the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 13 Following ForYou LIVE 229 herbs,savealife 38 thewheatwitch Fight back against the patriarchy by knowing what herbs J I sound - thewheatwitch - BethAuthors Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' What action would you take on Post #1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 14 participants] Ignore 6 (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='86%) Fact-check 3 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='43%) Block 1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='14%) Report 1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='14%) Like 3 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='43%) Disinformation (intent) [viewed: 12 participants] Ignore 5 (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Fact-check 5 (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Block 1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Report 1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Like 0 (0%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Post #2 The second post that we presented to the participants was labeled by the creator with the hashtags #roevwade, #women, #health, and #holistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This post discusses the use of an abortion tea containing multiple herbs, including rue, which has dangerous side effects noted in Table 1 and health advisories warn that it “can lead to death for both the mother and baby” [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The screenshot of the post as it appeared in the standard TikTok app is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok Post #2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The participants that thought that misinformation is disseminated on TikTok without intent were almost evenly split regarding this post, as shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Six (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='15%) said they “don’t believe this is misinformation” [P33FL30] and five (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='46%) said they “do think this post is misinformation” [P13FR30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The remaining two (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) participants said they “cannot confirm if this post is misinformation” [P17FR30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants that saw falsehoods as disinformation on TikTok did not see this post as inaccurate as only two (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) of them thought the post is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='“misinformation because it is suggesting that an herb blend is a safe and effective way to self administer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='LIVE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='FollowingFor You ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='Herbs: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1tspRue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='Crush/Grind: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1tspTansy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1tspFenugreekSeeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1/2tspWormwood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1tspAniseSeeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='9290 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1tspComfreyleaf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1tspAshwagandaroot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1tspSage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1tspDillSeed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1tspRosemary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='346 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1/2CinnamonStick ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2tsp Lemongrass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2tspNettleLeaf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='TikTo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='@loudm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='enia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='295 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='cleerly_clara ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='Period Tips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='#roevwade #women#health#holistic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='J1 -cleerly_clara-ClaraC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='origirAbortion Misinformation on TikTok ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='an abortion” [P36FL40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Five participants (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='46%) said they “don’t think this is misinformation because the post provides full context on the information it was trying to provide” [P40FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Six participants, or (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='15%) were unsure because they “don’t have enough knowledge to know if it’s misinformation” [P59FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Is Post #2 Misinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 13 participants] Yes 5 (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='46%) No 6 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='15%) Unsure 2 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) Disinformation (intent) [viewed: 13 participants] Yes 2 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) No 5 (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='46%) Unsure 6 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='15%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The 13 participants who thought of no intent behind misinformation on TikTok indicated they would perform a wide variety of activities for this post as shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Four (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='77%) of them said they would “potentially do some research into the other herbs that they are not familiar with” [P17FR30], three (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='08%) said they “would move past it“ [P8FL20], two (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) said they “would most likely block this account” [P13FR30], and two (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) said they “would probably like this post” [P31FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The two participants that said they would block this video indicated they felt very strongly about the content as it is “definitely misinformation because there is scientific evidence that home remedies for this almost never work” [P25FL20] and “it was extremely uncalled for in suggesting abortion tea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' If this was in my TikTok feed I would [also] report this video” [P24FR20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' None of the participants that saw misinformation being spread with intent on TikTok said they would block or report this post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The most popular responses included that six (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='15%) said they would “simply move past the video and not pay no mind to it” [P35FL30] and five (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='46%) said they would fact-check the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' P5FL30 included that to determine the post is not misinformation they would need to see “a Gynocologist [sic] [to] validate or debunk this information before I could trust it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This would include discussing the ingredients, linking relevant papers and studies, and reminding people to go see their doctor for personal medical information.” Two (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) of the participants indicated responses that they’ve “done research on natural remedies for several health issues and if it was on my feed I may comment or like it” [P6FA30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' What action would you take on Post #2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 13 participants] Ignore 3 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='08%) Fact-check 4 (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='77%) Block 2 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) Report 2 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) Like 2 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) Disinformation (intent) [viewed: 13 participants] Ignore 6 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='15%) Fact-check 5 (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='46%) Block 0 (0%) Report 0 (0%) Like 2 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='38%) 15 Authors 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='3 Post #3 The third post that we presented to the participants appears to be a ScienceDirect article indicating which herbs are abortifacients, but is actually a Google search result snippet with a caption stating “learn your herbs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This TikTok video was not labeled with any hashtags by the creator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Although the full ScienceDirect article purportedly referenced in this video provides warnings about the toxicity risks of the herbs [87], the screenprint also has directions on the dosage for pregnancy termination centrally positioned in the overall text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The screenshot, shown in Figure 4, includes the pennyroyal and mugwort herbs as abortifacients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok Post #3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Multiple participants in both groups observed that the TikTok post appeared to be from ScienceDirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A few noticed it was a Google search result or that the “source looks questionable” [P49Fl30] and reflected that in their responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In the misinformation group, as shown in Table 8, six (40%) of the participants said they “think this post maybe fake because the whole information is not given just the negative information that they want you to see” [P11ML40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Five (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) leaned towards it not being misinformation and weighing that “this post is presenting information from ScienceDirect, which I consider to be a reputable source of science-based and factual information” [P44FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Four (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) participants were unsure because “This post may contain some degree of accuracy given that the source is somewhat credible, but herbs are often not studied enough for these claims to be made with a high degree of accuracy” [P46MM30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Most of the participants in the disinformation group were unsure if this post was misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 10 (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5%) of them said they “honestly cannot tell if this is misinformation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' I see that the website credited is science based, but without going to the website, I am really unsure” [P41FM40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The next most common response was that “this tries to present as being medically accurate, but it’s just a screenshot of Google search results, so I would not trust it on its face” [P12FL60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Only P15ML30 16 d emmenaayowine hearyguinclude (Qt arenot limited to)tansy,thuja, safflower,scotchbroom,rue, angelica,mugwort,wormwood, yarrow,and essentialoil of pennyroyal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' MaternalConsiderations:Carboprostis ananalogof15-methylprostaglandin PGF2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='. DosageWithQualifiers:Pregnancy LINDIGEN termination-begin100mcgIMtestd then 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' DrugClass:Abortifacients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='Oxytocics Prostaglandins,Stimulants,uterine Indications:Pregnancytermination uterineatony E https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='com>topics Abortifacients-anoverview ScienceDirectTopics Share nikki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='kiya1 learn your herbs Aboutfeatured snippets Fee J No - Kreepa Oh No - KreepaAbortion Misinformation on TikTok said he did not “know if it’s misinformation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' I’d assume it isn’t because I’m not how someone could lie about a google search coming up.” Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Is Post #3 Misinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 15 participants] Yes 6 (40%) No 5 (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Unsure 4 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Disinformation (intent) [viewed: 16 participants] Yes 5 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='25%) No 1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='25%) Unsure 10 (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The participants in both groups, as indicated in Table 9, were mostly inclined to ignore this post, saying they “wouldn’t interact with such a post” [P3ML50] with P29FL60 noting that the post “was a standard google search so it may be true but again I would not respond and I would swipe on by.” The next most common response of the participants in the misinformation group was to “report this creator” [P16FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The remaining two (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) participants in this group said they would “likely look at the comments to see what others have said” [P19FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Six (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 5%) of the participants in the disinformation group said they “would probably do research on the specifics if it was something I desired to learn more about” [P40FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Only one participant in this group said they would block the post “because it does not state the risks of using these” [P34MM30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' What action would you take on Post #3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 15 participants] Ignore 9 (60%) Fact-check 2 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Block 0 (0%) Report 4 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Like 0 (0%) Disinformation (intent) [viewed: 16 participants] Ignore 9 (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='25%) Fact-check 6 (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5%) Block 1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='25%) Report 0 (0%) Like 0 (0%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='4 Post #4 The fourth post presented to the participants was labeled by the creator with the hashtags #fyp, #prochoice, #roevwade, and #herbodyherchoice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The text overlay in the video also includes penny- royal and mugwort under the heading “unfortunately it’s come down to this”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The screenshot of the post as it appeared in the standard TikTok app is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As shown in Table 10, six (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5%) participants in the misinformation group indicated that they “do think that this post is misinformation” [P13FR30], five (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='25%) said it is “not misinformation because it didn’t advise a particular viewpoint or way of thinking” [P54FL30], and five (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='25%) said “It’s not clear that the post is misinformation because it’s just a list of herbs, supplements, and foods” [P49Fl30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The disinformation group of participants felt more strongly that this post was misinformation, with nine (69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='23%) indicating “this one is completely lacking context or supporting information, so, yes, I would qualify it as misinformation” [P12FL60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Three (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='08%) were “not sure it’s misinformation but it may cause unsafe things to occur if not posted with caution” [P35FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Only P39FL30 said she “doesn’t really think it’s misinformation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 17 Authors Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok Post #4 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Is Post #4 Misinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 16 participants] Yes 6 (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5%) No 5 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='25%) Unsure 5 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='25%) Disinformation (intent) [viewed: 13 participants] Yes 9 (69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='23%) No 1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='69%) Unsure 3 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='08%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' When asked to describe the action they would take on this post, 8 (50%) participants in the misinformation group said they “would just ignore the video and move on” [P52MM40], as shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Three (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='75%) participants said they “would probably read through the comments and either search for similar videos on TikTok or Google it” [P33FL30], another three (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='75%) said they would “report this creator because their information is dangerous” [P16FL30], and the remaining two (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5%) said they would “most likely block this user” [P13FR30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Almost all of the participants in the disinformation group would ignore the fourth post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 11 (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='62%) said “would probably scroll past this without interacting because even if it is truth it isn’t helpful or informative” [P48FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The remaining two participants said they “would try to fact check as best as I could” [P6FA30] and “report this post” [P12FL60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' No participants in this group said they would block this post and no participants in either group said they would like the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 18 CIVE FollowingForYou Q unfortunatelytscome downto this chamomile cinnamon liver mugwort femaleginseng unripepapayaseed 12 vitaminC raweggs pennyroyal sesameseeds w/honey voidsnail Share stay safe outthere<3·#fyp#prochoice #roevwade #herbodyherchoice JNirvana Pennyroyal Tea-NirvAbortion Misinformation on TikTok Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' What action would you take on Post #4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 16 participants] Ignore 8 (50%) Fact-check 3 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='75%) Block 2 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5%) Report 3 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='75%) Like 0 (0%) Disinformation (intent) [viewed: 13 participants] Ignore 11 (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='62%) Fact-check 1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='69%) Block 0 (0%) Report 1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='69%) Like 0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='00%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5 Post #5 The fifth post that we presented was labeled with the hashtags #themoreyouknow, #parsley, #pes- saryinsertion, #pessary, #fertilityherbs, #fertilityeducation, #plantsheal, #herbalist, #apothecarycab- inet, #hippocraticoath, #ayurvedic, and #hippocratesfatherofmedicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This creator has a series of posts that contain potential “abortion inducers” and emmenagogues (herbs which purpotedly stimulate menstruation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The post explains how to insert parsley into the cervix or make it into a tea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In 2018, a women in Argentina died from attempting to induce a miscarriage while utilizing this method, which “stimulates blood flow in the uterus and can lead to massive internal bleeding and convulsions” [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The screenshot of the post as it appeared in the standard TikTok app is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok Post #5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As shown in Table 12, most participants from both groups felt this post was misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Twenty-two participants in total were randomly selected to assess this video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The groups were evenly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Six (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='55%) participants in the misinformation group stated 19 FollowingForYou Q LIVE 101 HERBS THINGSEVERYONE WITHA UTERUSSHOULDKNOW PARSLEY MILDEMMENAGOGUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='RELAXES CERVXCOMBNESWELLWITH VITAMINC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' PARSLEYPESSARY:2-4SPRIGSOF PARSLEYRINSED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='REMOVELARGER PARTOFSTEMJUSTBELOWFIRSTLEAF JOINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='RINSEAGAIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='PLACEINSIDE 282 INTERNALLYAGAINSTCERVIX CHANGEEVERY12HOURS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' *TIE STRINGAROUNDSTEMSFOREASIER REMOVAL TEA:BOILWATERINSMALLPOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='ADD2 65 HANDFOLLSOFCHOPPEDPARSLEY COVERTIGHTLY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='STEEP20-30MIN shenvalleyapothecary Reply to edits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='green Parsley is a mild emmenagogue but works well in conjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' See more J)aniRoccaMusicoriginal sounoAuthors they “feel this post is misinformation because it does not relay the effects of placing an unsterile object inside your body, which is what the video is promoting/suggesting” [P17FR30] and that “this post has no credible citations to support the claims it is making” [P14FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Three (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='27%) of the participants who thought misinformation is disseminated on TikTok without intent felt it was not misinformation because “It’s always the consumers [sic] responsibility to do their own research and make their own choice(s)” [P23FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Two (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='18%) participants in this group said they “have no idea” [P21FA40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Eight (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='73%) participants in the disinformation group thought “it is misinformation because the account is not stating what the benefits of parsley even are, nor where they gathered this information” [P7FM30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Three (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='27%) said they “suspect this is misinformation but do not know for sure]” [P36FL40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' No participants in the disinformation group thought the post wasn’t misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Is Post #5 Misinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 11 participants] Yes 6 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='55%) No 3 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='27%) Unsure 2 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='18%) Disinformation (intent) [viewed: 11 participants] Yes 8 (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='73%) No 0 (0%) Unsure 3 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='27%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The misinformation group participants, as shown in Table 13 primarily said they “would ignore this post” [P14FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Two (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='18%) participants said they would “report the video as unsafe due to the major consequences that doing what the video says to do could ensue on someone’s health” [P17FR30], one (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='09%) said they “would like to learn more and can always compare information on google” [P11ML40], and one (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='09%) said they “would most likely block this account” [P13FR30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participants in the disinformation group were mostly split on the post’s inaccuracies, saying they “wouldn’t respond and just keep scrolling” [P26FL30] and they “would look up the benefits of parsley to see if the post has an validation” [P6FA30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Two (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='18%) participants said they would “typically report this post“ [P7FM30] and P38FL30 said she would “block this account.” No participants in either group said they would like this video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' What action would you take on Post #5?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 11 participants] Ignore 7 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='74%) Fact-check 1 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='09%) Block 1 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='09%) Report 2 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='18%) Like 0 (0%) Disinformation (intent) [viewed: 11 participants] Ignore 4 (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='36%) Fact-check 4 (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='36%) Block 1 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='09%) Report 2 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='18%) Like 0 (0%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='6 Post #6 The sixth post that we presented participants was labeled with the hashtags #greenscreen, #womb, #roevwade, #pregnancyrelease, #abortion, #herbal, #safety, and #withlove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This post contains information on which herbs to use to perform a “pregnancy release”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The herbs in the video include 20 Abortion Misinformation on TikTok rue, pennyroyal, and mugwort, which were also included in prior videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The source cited in the video is indicated by the creator to be an article on herbal abortion from we.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='riseup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The screenshot of the post as it appeared in the standard TikTok app is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok Post #6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As shown in Table 14, six (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='86%) of the misinformation group participants indicated that this post was misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Participant P19FL30 explained that “the content creator in this video does tell viewers to do ‘proper research’ on the herbs she is promoting before using them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' However, she notes that if ‘something goes wrong’ and medical attention is needed that the viewer does not need to tell their medical provider that they have been taking these herbs - assuming that the creator is not a medical professional, I would consider this to be misinformation.” Five (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='71%) of the participants said they weren’t sure if this post is misinformation because “there is an article supporting the statements made by the speaker and the speaker seems to have genuine intent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' However the shared article is not in the form of a scientific verified and peer reviewed study” [P42FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The remaining three participants in this group said they “don’t believe this post is misinformation because she provides supporting facts that you can double check and they will confirm her point” [P54FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Six (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='29%) of the participants in the disinformation group felt “this is not misinformation because she is well spoken, shows where she is getting the information from, states what it is for and what it does and also mentions to seek medical help” [P51FL40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Six (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='29%) were unsure because they “can’t confirm if this is misinformation or not because I am not familiar with the scientific data that either backs up or refutes this info”[P1MR50] and five (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='41%) stated “Yes, this post is misinformation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' To start - the post screenshots a website: weriseup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='net;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' That website is not a reliable medical source of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' I would scroll past” [P41FM4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 15 indicates that more that half of the participants from both groups said they “would scroll past” [P41FM40], “would simply ignore this post” [P1MR50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Of the participants in the misinformation group, four (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='57%) said they would “encourage viewers to follow up with 21 Which HerbstoTeke LIVE Each herb has a unique action on the body and it is useiui toknow how each one works in order to select the right ones at the right time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='The herbs have been categorised by their properties and some herbs appear in more than one category as they affect the body in more than one way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Many herbs could cause abortion single-handedly but a combination ofcomplementaryherbs could provemore effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='Choosing herbs with different actions to each otherwould be complementary,rather than using several with the same action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='There is some evidence to suggest that using just one to four herbs at a time isbetterthan trying to combine a large numberofherbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' These categories could be seen as an oversimplification as whole plants have a variety of different actions combining to make cach one unique, but nevertheless these three basic groupings arean invaluableguidelineforgetting it right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=" ProgesteroneBlockingHerbs These are also referred to asimplantation inhibiting herbs'as they are the category of herbs that are used as cmergencycontraception after fertilisation,but they're effective after that period as well." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='They work by interfering with the normal production of the hormoneprogesterone,without whichthelining ofthewombbecomes unsupporti the fertilised egg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='Progesterone is crucial to the continued viability ofa pregnang egg has not yet implanted in the womb then it is prevented from doing so and is along with the lining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='If the egg has already implanted, it detaches from the unsupp womb liningandthe lining willbreakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=" Cotton Root Bark Wild CarrotSeed VitaminC (not a herb, but included here because it's readily non-toxic) Rue (this is not a progesterone blocking herb, is pla cause thewomb lining to break down." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' It does this by fblood capillaries in the womb) Uterine Contracting Herbs 6 These herbs cause contractions in the uteru ean ideal addition afteraprogesteroneblocking ofte these herbs can help to regulate and streng to the use of other abortifacients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' ue Angelicaroot DongQuai (Chinese angelica) Green Screen earthnmaya linkin mybio for full article#greenscreen#womb #roevwade #pregnancyrelease #abortion #heFee more JJoriginal sound-earthnmaya-MAuthors Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Is Post #6 Misinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 14 participants] Yes 6 (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='86%) No 3 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='43%) Unsure 5 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='71%) Disinformation (intent) [viewed: 17 participants] Yes 5 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='41%) No 6 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='29%) Unsure 6 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='29%) doing their own research and a notice that every woman’s body still responds differently to methods” [P54FL30], two (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='29%) said they “would simply block the user and move on” [P52MM40], and one (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='14%) said they “would report this post” [P25FL20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The participants in the disinformation group did not say they would block this post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Three (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='65%) of them indicated they would “read the full article this video links, and find other videos to make a better judgement on the trueness of this post” [P5FL30] and two (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='76%) said “there are natural remedies that are legitimate but telling people they can take an herb for contraception is dangerous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' I would report this one[P50FL60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' No participants in either group said they would like this post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' What action would you take on Post #6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 14 participants] Ignore 7 (50%) Fact-check 4 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='57%) Block 2 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='29%) Report 1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='14%) Like 0 (0%) Disinformation (intent) [viewed: 17 participants] Ignore 12 (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='59%) Fact-check 3 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='65%) Block 0 (0%) Report 2 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='76%) Like 0 (0%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='7 Post #7 We also selected a post that was soft moderated by TikTok [51] and contained a trustworthiness tag with stating “Participating in this activity could result in you or others getting hurt.” Post seven was labeled by the creator with the hashtages #greenscreen and #roevwade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As this is the only explicitly moderated post, we presented it to all 60 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The post, shown in Figure 8, has overlay text that states “they may be able to ban abortion but they can’t ban these” and has slides of images that include pennyroyal, juniper berries, vitamin C, mugwort, aspirin, and a wire hanger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As shown in Table 16, 14 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) of the 30 participants in the misinformation group said the post was misinformation “considering this person is unqualified to be spreading this information” [P16FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Eleven (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) said the post was not misinformation because they “don’t agree with using some of those products for achieving the goal that the video is promoting, but all of those products shown, used in the correct way/dosage can result in abortion” [P17FR30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The remaining five (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) participants indicated they would “have to research before deciding if it was misinformation or not” [P44FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In the disinformation group, 13 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) of participants indicated that “this is misinformation as there is no evidence this is safe and effective means of self administering an abortion” [P36FL40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Nine (30%) of the participants said “It is not helpful or informative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' And the hanger is such a danger suggestion to put out there;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' I wouldn’t consider this misinformation but it isn’t helpful information” 22 Abortion Misinformation on TikTok Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok Post #7 [P48FL30] and 8 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) stated “I don’t have the scientific data to comment on whether or not this is misinformation” [P1MR50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Is Post #7 Misinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 30 participants] Yes 14 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) No 11 (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Unsure 5 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Disinformation (intent) [viewed: 30 participants] Yes 13 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) No 9 (30%) Unsure 8 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' As indicated in Table 17, 13 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) of the misinformation group and 18 (60%) of the disinformation group said they “wouldn’t do anything with the post” [P6FA30] and they “would not respond and just keep scrolling” [P26FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Eleven participants of the misinformation group (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) and 8 of the disinformation group (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) said they “would even consider reporting it as harmful because some people are viewing it and are not aware of how to properly use the products” [P17FR30] and “because it’s dangerous” [P16FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Three (10%) participants from each group said they would “have to do more research” [P2FL50] and they “would read the comments, look at similar videos, or Google it” [P33FL30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Two (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) of the participants from the misinformation group said they would like the post as they “agree with this post and have heard these methods before” [P8FL20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=" One of participants from each group said they would block the post because “it’s not humorous joking around about women feeling they are being pushed to use alternative and possibly 23 Following Q LIVE For You 286 they may be able to bal abortion but they can't ban these 29." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='5K 215 GreenScreen 2753 13kasettetapes #greenscreenthelast oneisajoke(kind)#roevwade 171 Dalsound-speedaudiios_-Nick Participating in this activity could result inyou or others gettinghurt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='Authors dangerous medicine” [P60ML40] and “it is very disturbing and should not be allowed to be showed to young women in particular” [P13FR30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Table 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' What action would you take on Post #7?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation (no intent) [viewed: 30 participants] Ignore 13 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Fact-check 3 (10%) Block 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Report 11 (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Like 2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Disinformation (intent) [viewed: 30 participants] Ignore 18 (60%) Fact-check 3 (10%) Block 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='33%) Report 8 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='67%) Like 0 (0%) 8 DISCUSSION Our first research question aimed to uncover how social media users conceptualize misinformation on TikTok, where does it originate, who are the targets, and what’s its purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Misinformation, even within such a small sample as ours, invokes nuanced interpretations as people do not always stick to the popular “fake news” association [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' According to the participants in our sample, misinformation on TikTok might involve falsehoods only, but falsehoods could be interspersed with biased interpretations, facts taken out of context, or emotion-provoking narratives [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The amalgamation of factual and inaccurate content was mostly a craft assigned to individual content creators, often for personal gain, but the “usual suspects” – political parties and foreign unfriendly countries to the United States – were not spared in disseminating misinformation on Tiktok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This finding suggests that while TikTok’s affordances are fit for selling products or participate in challenges [50], they are not prohibitive for political and foreign actors to adapt their agenda setting and information operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In the past, misleading narratives and content were migrating from fringe and alt-communities to the mainstream platforms [124] and the evidence from our study suggest that TikTok could be a future candidate destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Our participants’ view that many vulnerable people and easily-influenced crowds would unlikely step out from the “For You” page to check a claim adds to this impression, confirming previous studies that identified such users as the most sought out targets of misinformation [19, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The TikTok participation model adds additional incentive to engage with misinformation as it provides immediate profit and long-term gains, according both to the participants in our sample and studies exploring political and conspirative influence on TikTok [14, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Our second research question narrowed our exploration on misleading abortion narratives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Given that we conducted our study after Roe vs Wade was overturned, health misinformation regarding herbal abortifacients dominated the TikTok’s ‘For You’ pages of our participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' While some of them were simply intrigued about these “at-home remedies,” there were participants that were disturbed, worried, angered, and disgusted that dangerously misleading content like this finds its way on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This emotion-provoking response, though concentrated on a topic of limited polarization, is worrisome because this is precisely the response that foreign actors sought to incite in past polarizing debates on social media [90, 125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' These debates always involved politicization of the discourse, so the evidence that our participants also saw politically contextualized abortion narratives further suggest that TikTok might be the next health/political misinformation battleground [36], despite the impression of mostly apolitical participation on the platform so far [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 24 Abortion Misinformation on TikTok But to substantiate such conjectures, one needs evidence on how actual users engage with essential, health-related abortion misinformation in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Uncovering this evidence was central to our study and we attempted to gain as much as possible a nuanced insight into how users actually deal with various abortion misinformation content on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Our findings indicate that a significant number of TikTok users do not see videos promoting herbal abortifacients as abortion misinformation, despite the scientific evidence to the contrary [9, 37, 41, 74, 89, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Regardless of whether the video included the creator itself, contained explicit tags to the herbal abortifacients listed in Table 1, or was simply a textual post promoting these at-home remedies, many participants were unconvinced the posts were misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The majority of them conceptualized misinformation as spreading falsehoods without intent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' those that saw intent behind the spread of falsehoods on TikTok, in general, were far more skeptical of the posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A deeper look into the responses of the participants that did not see misinformation in the selected posts reveals that it was sufficient for the creators to appear “like they know what they are talking about,” “provide context,” “seems informed,” and are “well spoken” to make the misinformation believable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' There was no particular demographic trend among these participants as the posts were equally convincing to all gender, political, and age groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' The group of participants that called out the misinformation indicated both analytical and heuristic cues that could be helpful for the wider audience on TikTok to employ in dealing with misleading abortion content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' For example, videos that “look like MLM promotions”, “omit any side-effects” of a proposed treatment, “do not include credible citations”, and the “creator is unqualified for making any health-related claims” should be dismissed as abortion misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' All of these strategies have been proven to work against health misinformation in the past [80], so our findings strengthen the case for continuing the “inoculation” against false and misleading abortion narratives [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This is especially important in situations where many users, like the ones in our study, remain unsure whether these misleading videos are abortion misinformation or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Our results suggest that these users lack knowledge on the safety and the effectiveness of herbal abortifacients, which in turn precludes them to make a decisive dismissal of the misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Educational interventions on TikTok regarding abortions have already be proposed [30], but question is if the algorithmic curation of the ‘For You’ page for TikTok users “interested” in alternatives includes them too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' According to our findings, the undecided participants would be easily “nudged” with accurate scientific information, which is a strategy that worked for other types of misleading content [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' It is reassuring to see a fact-checking trend among even a small sample like ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Perhaps affordances of TikTok probably encourage majority of the users to simply ignore many misleading posts, as many of the participants in our study did, and avoid to critically discern the content [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' But recalling that familiarity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' a repeated exposure to such videos makes the content look truthful [80], this fact-checking trend might need to be appended with better debunking and soft moderation efforts suggested to work for other health-related misinformation [95, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Such a need is further corroborated by our findings suggesting that even in the presence of the current TikTok soft moderation labels on post #7, at least 30% of the entire sample still believed that the content is not misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Equally important for consideration is that our results suggest a number of users are unafraid to block or report abortion misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Prior evidence on flagging and reporting misleading content suggests that social media users might not opt for such actions as to preserve interpersonal relationships and avoid social arguments, especially for issue that do not matter much to them [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' However, reporting misinformation becomes a proactive endeavour when health-related misinformation is in question [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Our results confirm that this effect also takes place for abortion- related misinformation and occurs across all demographics, making a further case for an actionable framework of misinformation sharing and correction sharing on TikTok [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 25 Authors 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1 Ethical Considerations The purpose of our study was not to generalize to a population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' rather, to explore the phenomenon of individual dealing with abortion misinformation from a regular user perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' While we added the participants’ self-reported age, gender, and political leanings for more detailed reporting, we avoided providing definitive numbers accompanying the assessments and response strategies for each of the seven intervention posts with these demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Instead, we supported our findings with descriptive quotations from participants that convey the way ordinary users deal with misleading videos on TikTok in a hope that the results can help to elevate the study of abortion misinformation on social media as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We acknowledge that there might be a potential risk of repeated exposure to abortion misinfor- mation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' an “implied truth effect” [80], as each participant saw four of the short-form videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' To mitigate this risk we explicitly pointed to the debunked information for the related “at-home” reme- dies and their associated harms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' There is also a risk from oversimplification of our results where the participants’ conceptualization, assessment, and response to abortion misinformation, expressed on their behalf, might not represent the entirety of the strategies used to deal with misleading videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We, of course, agree that users do employ others ways and means to deal with misleading videos and we welcome every work that brings them to the fore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' This will facilitate a scientific work on abortion misinformation beyond simple oppositional responses [101], a corollary we also want to avoid when contextualizing our results for future anti-misinformation interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='2 Limitations Our research was limited was limited in its scope to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok media users and the state of abortion misinformation regarding at-home remedies that existed on TikTok in the period immediately after the overturn of Roe vs Wade by the Supreme Court of the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' In as much as we attempted to have balanced, diverse, and inclusive set of short-form videos regarding herbal abortifacients, there certainly are, and will be, other similar content which our participants might assess and respond to differently than they did in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' While the content of these videos was scientifically debunked and constituted misinformation during our study, we acknowledge that this might change in light of new scientific evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We are also limited by the state of the content moderation policies on TikTok that, together with public policy changes and new Supreme Court rulings, change and impact the relevance of our results, therefore we exercise caution in considering these results in the narrow post-Roe vs Wade context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' A limitation also comes from the sampling method and the use of a survey provider [85], as other users and other samples might provide results that differ from the once we obtained as there is little insight into general sampling and sample-related differences when users are broadly queried about abortion misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' By asking users directly about how they interact with abortion misinformation and misleading video content on TikTok, we got a wide variety of insights from a broad range of perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' We did not measure the efficacy of users’ assessment approaches and response strategies for explicitly politicized abortion misinformation content, nor did we ask how users dealt with other abortion misinformation on other social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Short-form videos are a relatively new way of persuasive communication appealing to younger users, but traditional social media text, memes, and visceral images [5] might provoke different responses for a wider population of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Therefore, we are careful to avoid any predictive use of our findings because TikTok’s affordances might change in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 26 Abortion Misinformation on TikTok 9 CONCLUSION Abortion misinformation undoubtedly shapes the way people make reproductive decisions and TikTok, a very popular social media platform, allows misleading content regarding “at-home” abortion remedies to reach wide audiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Users possess the critical ability to assess, discern, and reject misleading and scientifically debunked abortion claims, but a worrying number of them are not ready to dismiss these alternatives for self-induced terminations of unwanted pregnancies in a post-Roe v Wade America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Time will tell whether this proclivity for abortion misinformation will remain, but meanwhile we hope that TikTok, the scientific community, and health authorities take our results as actionable insight that can prevent harmful outcomes of abortion misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' REFERENCES [1] Crystal Abidin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Mapping internet celebrity on TikTok: Exploring attention economies and visibility labours.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='com/topics/nursing- and-health-professions/abortive-agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 30 Abortion Misinformation on TikTok [88] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Rubin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Abbasi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Suran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2022.' metadata={'source': 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Communication 33, 3 (2016), 460–480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [111] TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok Safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='tiktok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='com/safety/en-us/topics/ [112] Nina Totenberg and Sarah McMannon.' metadata={'source': 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Shenanigans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Data and Information Quality 11, 3, Article 10 (may 2019), 37 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='1145/3309699 [127] Marco Zenone, Nikki Ow, and Skye Barbic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' TikTok and public health: a proposed research agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' BMJ Global Health 6, 11 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [128] Jingwen Zhang, Jieyu Ding Featherstone, Christopher Calabrese, and Magdalena Wojcieszak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Effects of fact-checking social media vaccine misinformation on attitudes toward vaccines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Preventive Medicine 145 (2021), 106408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [129] David X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Zheng, Anne Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Ning, Melissa A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Levoska, Laura Xiang, Christina Wong, and Jeffrey F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Acne and social media: A cross-sectional study of content quality on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Pediatric Dermatology 38, 1 (2021), 336–338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [130] Chris Zielinski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Infodemics and infodemiology: a short history, a long future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Rev Panam Salud Publica 45 (2021), e40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' STUDY QUESTIONNAIRE Exposure and Preconceptions (1) Can you please define “misinformation” in your own words (please be verbose): [Open Ended] 32 Abortion Misinformation on TikTok (2) Have you encountered misinformation on TikTok and in what form?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Please provide examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] (3) Where does misinformation on TikTok come from, in your opinion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] (4) Who is the target of misinformation on TikTok, in your opinion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] (5) Who benefits from misinformation on TikTok, in your opinion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] Engagement Strategies (1) How do you suspect or know that a certain TikTok post is a misinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Please elaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] (2) What is your strategy for dealing with misinformation posts on TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Please elaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] (3) Have you used any engagement features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' share, comment, like, follow) for misinformation posts on TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' If so, in what circumstances?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] (4) Have you used any action features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' block, mute, report, unfollow) for misinformation posts on TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' If so, in what circumstances?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] (5) Have you talked about a particular TikTok misinformation post outside social media?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' If so, in what circumstances?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] Abortion Misinformation Exposure (1) On what other occasions you have encountered misinformation regarding abortion on TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Please elaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] (2) What was your response to this particular abortion misinformation on TikTok?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Please elaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] (3) Where else does abortion misinformation exists outside of TikTok, in your opinion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' Please elaborate and share any experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} +page_content=' [Open Ended] 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQfhw2D/content/2301.05128v1.pdf'} diff --git a/K9AzT4oBgHgl3EQfkP22/content/2301.01529v1.pdf b/K9AzT4oBgHgl3EQfkP22/content/2301.01529v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b351e5962a028d7702d1eab54140b0a17404e225 --- /dev/null +++ b/K9AzT4oBgHgl3EQfkP22/content/2301.01529v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:26cbdd22f401666ca24201dcedf9a46ae57a11f427a668df711248cd9e49799e +size 445794 diff --git a/K9AzT4oBgHgl3EQfkP22/vector_store/index.pkl 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Gerrard St E, Toronto, M5B 1G3, Ontario, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2 University of Toronto, 5 King’s College Rd, Toronto, M5S 3G8, Ontario, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3 Bogazici University, Bebek, Istanbul, 34342, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' E-mail(s): mcevik@torontomu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='ca;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Abstract Wind power forecasting helps with the planning for the power systems by contributing to having a higher level of certainty in decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Due to the randomness inherent to meteorological events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', wind speeds), making highly accurate long-term predictions for wind power can be extremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' One approach to remedy this challenge is to utilize weather information from multiple points across a geographical grid to obtain a holistic view of the wind patterns, along with temporal information from the previous power outputs of the wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our pro- posed CNN-RNN architecture combines convolutional neural networks (CNNs) and recurrent neural networks (RNNs) to extract spatial and temporal information from multi-dimensional input data to make day- ahead predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In this regard, our method incorporates an ultra-wide learning view, combining data from multiple numerical weather predic- tion models, wind farms, and geographical locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Additionally, we experiment with global forecasting approaches to understand the impact of training the same model over the datasets obtained from multiple dif- ferent wind farms, and we employ a method where spatial information extracted from convolutional layers is passed to a tree ensemble (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', Light Gradient Boosting Machine (LGBM)) instead of fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The results show that our proposed CNN-RNN architecture out- performs other models such as LGBM, Extra Tree regressor and linear regression when trained globally, but fails to replicate such performance 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='00819v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='LG] 2 Jan 2023 2 Multi-Step Wind Power Forecasting when trained individually on each farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We also observe that passing the spatial information from CNN to LGBM improves its performance, pro- viding further evidence of CNN’s spatial feature extraction capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Keywords: Time series forecasting, CNNs, RNNs, Machine learning, Regression 1 Introduction Rapid economic development and the continuous rise of living standards have raised the need for electric power production in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The most common form of energy extraction is from fossil fuels, such as coal, oil, and natural gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' However, using fossil fuels comes with serious consequences such as air pollution, ozone depletion and global warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Furthermore, due to the non- renewable nature and limited reserves, unrestrained exploitation of fossil fuels might lead to energy resource depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' According to the Paris agreement, to achieve the goal of limiting the global temperature rise below 2 ℃, renewable energies have to supply two-thirds of the global energy demand up to the year 2050 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The need for a pollution-free and environmentally friendly form of electricity generation has attracted increasing attention over the years and has brought significant focus on renewable sources of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Renewable energy sources such as solar photovoltaic, tidal and modern bioenergy play a crucial role in reducing global carbon footprint by acting as clean alternatives to fossil fuels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Wind power generation has witnessed rapid growth over the years for its abundance of availability, low land-based utility, and economic feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The wind is a significant and valuable source with the potential to produce energy continuously and sustainably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' It has the potential to generate electricity for each hour of the day, unlike for example solar energy, which cannot operate at night, and is suitable for systems that require energy continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Additionally, wind turbines can be built without occupying large areas of land, preventing the loss of agricultural areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Accordingly, wind power systems have developed rapidly around the world as a promising avenue for renewable energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' They have become an important component of the smart grid, smart microgrids, smart buildings and smart homes, playing a big role in providing electric power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The use of wind energy has several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Due to the intermittent nature of wind and its corresponding environmental factors, wind power pro- duction becomes inherently stochastic, which makes grid distribution planning and resource scheduling extremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Additionally, sudden dramatic fluctuations in the wind speed cause the turbines to rotate at a much faster rate than usual, causing sharp increases in electricity production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Such an event is referred to as a ramp event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' On the other hand, when wind speed is too low, the wind turbines do not rotate as fast, leading to a sharp decrease in electricity production, leading to what is known as a down ramp event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This Multi-Step Wind Power Forecasting 3 often contributes to equipment damage, transmission and distribution losses, and capital loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' However, this can effectively be dealt with by employing accu- rate wind power and ramp event forecasting, which can enable informed and reliable decision making, allowing for better planning, improved efficiency and reduced risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' As such, accurate forecasting can play an important role in reduc- ing operating costs and enhancing the competitiveness of wind power systems in the energy industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Wind power forecasting strategies often rely on temporal information extracted from the past production outputs of a wind farm, as well as spatial information derived from meteorological readings at various locations across a geographical grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Various wind power forecasting strategies make use of learn- ing techniques to generate accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' While the data for production outputs is specific to the power curve of each wind farm, meteorological data for a given geographical location can be found using Numerical Weather Pre- diction (NWP) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' NWP models provide a complete forecast of the state of the atmosphere at a given time, and a geographical location based on its lat- itude and longitude coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Most wind power forecasting works in recent literature make concurrent use of historic outputs and NWP data and apply learning methods to make reliable short-term and long-term predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' With the advances in artificial intelligence and machine learning (ML) technologies, a large number of deep learning-based models have been considered for wind speed and wind power forecasting due to their superior ability to deal with complex nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Wind power forecasting models can be categorized according to their fore- cast horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' These include ultra short-term forecasts, ranging from a few seconds to 30 minutes ahead, which are useful for turbine control and power load tracking in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Short-term forecasts, ranging from 30 minutes to 6 hours ahead, are often used for load dispatch planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Medium-term forecasts, ranging from 6 hours to 1 day, are utilized for energy trading and power system management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Lastly, long-term forecasts, from 1 day to 1 week or more ahead, allow for optimal maintenance scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our study focuses on medium-term forecasting, with 1-day ahead predictions for both wind power forecasting and ramp detection, and our proposed models take into consideration long-term historical trends (up to 48 hours) as the lookback window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Research objectives and contributions Our main research objective is to design novel ML models to achieve highly accurate wind power forecasts based on meteorological data and past wind power production output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We propose a novel multi-head, multi-layer, deep architecture, which combines Recurrent Neural Network (RNN) and Convolu- tional Neural Network (CNN) structures in parallel to extract spatio-temporal information from meteorological NWP data, and sequential information from historic wind power data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We evaluate our model on data collected from seven unique wind farms and compare its performance when trained on each farm 4 Multi-Step Wind Power Forecasting independently, against when the model is trained on a combined wind farm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The contributions of this study can be summarized as follows: We propose a novel CNN-RNN architecture which extracts spatial and temporal information in parallel for improved learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our numerical analysis shows that the proposed model is able to outperform competing ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our method incorporates an ultra-wide learning view, combining data from multiple NWP models, wind farms, geographical locations and atmo- spheric levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our analysis points to the benefits of global learning for wind power forecasting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We design a mechanism to employ CNN layers to extract features from the spatial data, and feed those into another machine learning model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', a tree-based ensemble such as Light Gradient Boosting Machine (LGBM)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This way, we examine the effectiveness of interdependent learning using spatial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Organization of the paper The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Section 2 provides an overview of the relevant studies on wind power forecasting in the literature and their applications across various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Section 3 introduces our pro- posed CNN-RNN architecture and the combined CNN and LGBM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Section 4 provides the experimental setup in terms of evaluation techniques, metrics and model parameters, followed by numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Lastly, Section 5 concludes the paper with a summary of our findings and a discussion on future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2 Literature Review Time series forecasting has been a prominent research field with applications in various domains and it has undergone major methodological advancements over the recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Earlier studies focused on linear statistical models such as auto-regressive (AR), moving average (MA) and auto-regressive integrated moving average (ARIMA), which account for linear correlations between past data points to make future predictions [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' With the growing availability of exogenous variables, ML models such as Random Forests (RF), Support Vector Machines (SVM) and eXtreme Gradient Boosting (XGB) grew in popularity for their effectiveness in dealing with cross-sectional feature spaces [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' More recently, RNNs such as Long Short Term Memory (LSTM) [20] and Gated Recurrent Unit (GRU) [10] architectures have been frequently employed for forecasting tasks due to their ability to extract long-term dependencies between temporal sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Similarly, CNN-based architectures have been used for time series forecasting due to their ability to capture information along spatial and time coordinates [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Various studies pointed to improved forecasting performance when com- bining multiple methods, which allows for better distinguishing patterns from Multi-Step Wind Power Forecasting 5 noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Commonly used ensemble techniques include stacking, bagging and boosting, which have been applied to obtain more accurate forecasting perfor- mance than the ones that constitute the ensemble [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Galicia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [14] used decision trees, gradient boosted trees and random forest models for forecasting big data time series, such that the predictions for each ensemble member are obtained by dividing the forecasting problem into forecasting sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Makridakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [32] conducted a study on the M4 forecasting competition which involves 100,000 time series and 61 forecasting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' They observed improved results when multiple methods were combined to obtain the fore- casts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' They noted that using a single model might lead to the difficulty of separating the pattern from the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Custom boosting algorithms were also shown to achieve high performance for time series modeling [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For instance, Ilic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [24] proposed explainable boosted linear regression, a method which involves training a generic forecasting model to obtain the initial forecasts and then exploring the residuals of the existing model using a regression tree which is trained on all available features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Recent studies have primarily focused on deep learning models for time series forecasting, with performance improvements over standard approaches for large datasets consisting of a large number of time series [9, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Some of these studies adopt a global learning approach to forecasting where training is performed over multiple related time series together in order to capture the seasonal behaviors and dependencies across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Hewamalage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [19] demonstrated that no matter how heterogeneous the data may be, a global forecasting model, that can perform equally well, or even better than a collection of independent models always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Complex structures such as DeepAR [39], temporal fusion transformer [30], Spacetimeformer [16], and N-BEATS [35] are examples of models which effectively make use of global learning when provided with large enough samples of related time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' These complex deep learning architectures also provide probabilistic forecasting capa- bilities, for which the typical objective is to predict the parameters of the underlying probability distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', mean and variance) for the target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Alexandrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [2] provided implementations for different probabilistic time series models and created an extensive Python library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Time-series forecasting for power generation from renewable energy sources is considered to be challenging due to the uncertainties associated with natu- ral events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Most energy problems take advantage of long historical patterns of production output along with domain-specific and seasonality-based features to make future predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Many studies employ ML techniques to effectively forecast for long-term and short-term power generation in order to ensure smooth operational planning and efficient distribution of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Ozoegwu [36] created a hybrid method based on a combination of nonlinear autoregres- sive and structural artificial neural networks to forecast monthly mean global solar energy production on a daily basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Similarly Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [15] used LSTM networks to make day-ahead solar power generation predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Dehghani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [11] applied the Grey Wolf optimization method [33] coupled with an 6 Multi-Step Wind Power Forecasting adaptive neuro-fuzzy inference to forecast the monthly hydropower genera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Other ML models deployed in the energy forecasting domain include SVMs [40], ANNs [43], and CNNs [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Wind power forecasting is arguably the most challenging form of energy forecasting due to the random fluctuations inherent to wind speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Previous studies used information from meteorological factors, including wind speed, wind direction, humidity and temperature, recorded at various locations and atmospheric levels across a wind farm, to obtain a diverse set of features for the prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' As such, wind power datasets involve dense spatial attributes, making them a natural fit for CNN-based architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Below, we discuss pre- vious works within the wind power forecasting domain, which effectively utilize CNNs to extract spatio-temporal information from nonlinear meteorological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Yildiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [48] proposed a novel residual-based CNN, where historical wind patterns across 54 different wind turbines are concatenated to be repre- sented as 2D RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The architecture is able to utilize spatial attention and extract daily and hourly correlations of input data more effectively than other state of the art deep networks including AlexNet [3], SqueezeNet [22], ResNet-18 [7], VGG-16 [3], and GoogLeNet [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Kazutoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [27] proposed a similar 3D-CNN architecture, where wind data from a 50 × 50 grid is given a video-like representation to account for spatial and temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [26] extracted spatio-temporal information from CNN layers and fed the flattened output to a LGBM model [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' They noted that lack of ade- quate training data may cause nonlinear convolutional output to fall into a local optimum, which can be avoided by replacing the fully connected layer with a stronger classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' While they feed CNN output to the LGBM model exclusively, our work involves feeding CNN output on top of the original input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Several studies combined the CNN and RNN structures to enhance the prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' ConvLSTM [41] is an example of such combined struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' It is an extension of standard LSTM networks which replaces matrix multiplication with convolution operation at each gate in the LSTM cells to capture the underlying spatial features present in multi-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [8] and Agga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [1] compared 1D and 2D variants of the Con- vLSTM network against standalone LSTM and CNN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [44] proposed a combined CNN-RNN architecture, where spatio-temporal informa- tion from multiple meteorological factors of previous timesteps is extracted using CNN and then fed into LSTM to extract long-term historical temporal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Alternatively, Zhen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [49] proposed BiLSTM-CNN, where temporal information for each meteorological factor is first extracted using a BiLSTM model, then fed into CNN to extract spatial dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' They noted that extracting temporal characteristics of input historical sequences first and then feeding them into CNN results in higher prediction accuracy than doing the vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Multi-Step Wind Power Forecasting 7 Different from these approaches, the AMC-LSTM architecture by Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [46] extracts spatial and temporal features in parallel using CNN and LSTM, respectively, before fusing them together to make final predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Such parallel structures are computationally inexpensive [47] and they are included in our study as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Additionally, their architecture uses attention mechanism to effectively assign feature weights based on influence factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [45] also incorporated attention mechanism within their SATCN- LSTM architecture and they employed a model validation strategy in order to select the best performing version of their model based on validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' While our work does not include an attention-based mechanism, it does incorporate a similar model validation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Different from many of the previous works, our analysis is based on a dataset from a massive grid consisting of seven wind farms, each with 48 tur- bine locations, and 20 unique atmospheric levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Since our dataset is extremely dense, instead of taking into consideration historic wind speed patterns from across all sources, we extract temporal information only from the power curve, as it is a function of all meteorological features, and extract corresponding spatial information at each time step independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' A comparative summary of relevant studies is provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Table 1: Summary of relevant papers in the wind power forecasting domain Paper Architecture Methodology Forecast hori- zon Temporal resolu- tion Data instances # loca- tions # wind farms # atmo- spheric levels Yildiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [48] ResCNN 2D-CNN to extract spatio-temporal information 1, 2, 3 step 1 hr 70080 54 1 2 Kazutoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [27] 3D-CNN 3D-CNN to extract spatio-temporal information 96 step 30 min 40320 50x50 grid 1 2 Jiajun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [25] WT-DBN- LGBM Features extracted using DBN fed to LGBM 1, 2, 3 step 10 min 800000 4 1 1 Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [26] CNN- LGBM Spatio-temporal information extracted using CNN fed to LGBM 1 step 5 min 5 1 1 Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [8] ConvLSTM1D ConvLSTM1D to extract temporal information from univariate time series 1 step 15 min 6000 3 1 1 Agga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [1] ConvLSTM2D ConvLSTM2D to extract spatio-temporal information from multivariate time series 1, 3, 5, 7 step 24 hr 1 1 Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [44] STCM Spatio-temporal information from previous timesteps extracted using CNN and then fed into LSTM 12 step 5 min 104800 33 1 1 Zhen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [49] BiLSTM- CNN Temporal information from previous timesteps extracted using BiLSTM and then fed into CNN 1 step 5 min 4896 1 4 Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [45] SATCN- LSTM Spatio-temporal information from previous timesteps extracted using CNN and then fed into LSTM using attention mechanism 16 step 5 min 10468 16 2 1 Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [46] AMC-LSTM Spatio-temporal information from previous timesteps extracted using CNN fed into LSTM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' temporal information from previous wind power timesteps extracted using LSTM 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 5 step 3 min 13440 1 1 1 Our study CNN-RNN Spatial information for each future timestep extracted using CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' temporal information from previous wind power timesteps extracted using LSTM 24 step 1 hr 12000 48 7 20 8 Multi-Step Wind Power Forecasting 3 Methodology In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' we discuss the various strategies used for time series forecast- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We first provide details on our dataset, including the distribution of the production outputs as well as general data characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Then, we elabo- rate on the models and architectures employed in our analysis and assess their strengths and drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Finally, we provide our proposed architectures for time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 Dataset In our analysis, we use meteorological data from seven wind farms located in Turkey, which was extracted using the Global Forecast System (GFS), and the Action de Recherche pour la Petite Echelle et la Grande Echelle (ARPEGE) NWP model, with meteorological features compromising Pressure (Pa), Tem- perature (K), Relative Humidity (%), and vertical (VGRD) and horizontal (UGRD) components of wind speed (m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our analysis encompasses only a normalized vector of UGRD and VGRD as the unique meteorological feature, which we refer to as wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Data for each farm was taken from early 2020, up until March 2022, with the sample size averaging around 12,000 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' A summary of further data characteristics of GFS and ARPEGE datasets is provided in Table 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Table 2: Comparison of GFS and ARPEGE data characteristics GFS ARPEGE Temporal Resolution 3-hourly hourly Height Levels 24 27 Latitudes 4 5 Longitudes 4 5 After carefully analyzing the correlations between the different atmospheric levels and wind power, we selected atmospheric level features that were most relevant to power prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This allowed us to select 11 unique levels from ARPEGE data and 9 unique levels from GFS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Since the data resolution of GFS 3-hourly instead of hourly, we use the mean of the one-step lag and one- step ahead values to augment the missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We normalize the wind power production outputs using min-max normalization as shown in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In addition to meteorological features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' we use cyclic month-of the year (moy) and hour-of-day (hod) features [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' which are incorporated using sinusoidal and cosinusoidal transformations as follows: ˆxi = xi − xmin xmax − xmin (1) Multi-Step Wind Power Forecasting 9 month of yearsin = sin(moy × 2π 7 ) month of yearcos = cos(moy × 2π 7 ) (2) hour of daysin = sin(hod × 2π 24 ) hour of daycos = cos(hod × 2π 24 ) (3) Figure 1 shows sample normalized wind power output time series from the years 2018,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2019 and 2020 to illustrate the evolution of the wind power data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Figure 2 shows hourly and monthly values for the wind power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Through the years 2018, 2019 and 2020, the power output distribution is very similar and no substantial yearly changes are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The hour of the day trend (Figure 2a) reveals that power output is maximized after midnight, and the lowest values are observed between noon to around 4 PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Monthly trends (Figure 2b) show that production is low during the summer months, with the lowest production in June, and peaks are achieved during winter, especially in January and February.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2018-08-20,15 2018-09-04,15 2018-09-19,15 2018-10-04,15 2018-10-28,14 2018-11-15,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 Normalized Power Output (a) Sample wind power output from 2018 2019-01-08,15 2019-01-23,15 2019-02-07,15 2019-02-22,15 2019-03-10,14 2019-03-25,14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 Normalized Power Output (b) Sample wind power output from 2019 2020-01-19,14 2020-02-03,14 2020-02-18,14 2020-03-04,14 2020-03-19,14 2020-04-03,14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='6 Normalized Power Output (c) Sample wind power output from 2020 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 1: Power production output samples from 2018, 2019 and 2020 Figure 3 demonstrates the normalized distribution of the production out- put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We note that the output is skewed towards the left, illustrating that most 10 Multi-Step Wind Power Forecasting 0 5 10 15 20 Hour of Day 36 37 38 39 40 41 42 Power Output (kW) (a) Average hourly values 2 4 6 8 10 12 Month of Year 15 20 25 30 35 40 45 50 Power Output (kW) (b) Average monthly values Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2: Average hourly and monthly trends of power production output of the production outputs lie within the 10th percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In terms of the ramp events present in the dataset, we note that the ratio of a ramp event against a no-ramp event is 1:5,673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This indicates a high class imbalance issue for the ramp detection task, which requires data balancing for model training, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', assigning more weight to the class that occurs less frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' On the other hand, undersampling and oversampling cannot be applied to cure the data imbalance due to the nature of time series datasets, removing or adding any additional points to the data disrupts the overall pattern and trend of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 Normalized Production Output 0 1000 2000 3000 4000 5000 6000 Frequency Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3: Distribution of the production output across all wind farms When combining data for the seven wind farms for global training pur- poses, we concatenate the independent datasets along the zeroth axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', concatenating the rows) and label the farms with one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Multi-Step Wind Power Forecasting 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 Review of Standard ML Methods for Forecasting While several different ML models have been previously used for time series forecasting tasks, in our analysis, we use linear regression as a baseline model, and tree ensembles such as LGBM and extra-trees (ET) as strong baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Note that we choose these models as they reportedly show high performance for various time series forecasting tasks [24, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' ML models require careful fea- ture extraction for training high-performance forecasting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Commonly extracted features for time series forecasting include features obtained from timestamps, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', time of the day, day of the week, and month of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' One- hot encoding or sine/cosine transformation can also be considered for these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For our dataset, since we have a linear representation of the features, where at each time step there exists a wind speed value extracted from various locations and height levels, we column-wise concatenate all the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Note that the lag values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', the values associated with previous time steps) can also be included in the feature space in a similar manner to allow for a cross- sectional feature matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' However, the main downside of using standard ML models for time series forecasting is that they do not consider the sequential information present in historical patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In general, forecasting can be done for making one-step or multi-step ahead predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For one-step-ahead predictions, a base model is trained such that the lag values up to time t, along with any exogenous features are used to pre- dict the value for time t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For multi-step ahead forecasting, two strategies are commonly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The first strategy is Direct Multi-step Forecast Strategy, which uses a base model to forecast for every time step in a time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For exam- ple, in a scenario which requires making n-step ahead predictions, a different base model is trained for each of those nth step predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The advantage of this method is that since it uses lags on the same time instance in a particular data series, it is easy to implement and experiment with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' However, a downside to this approach would be the high computational cost of training each of the separate base models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The second approach is to use a multi-output prediction model that is capable of generating multiple predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' These models can learn the dependencies between inputs and outputs as well as those between outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Deep learning models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', RNN- and CNN-based architectures) are typically designed to generate multi-output predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 Proposed Architectures RNNs and CNNs have been commonly employed for time series modeling [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' RNNs process given information incrementally, while maintaining an internal model of what is being processed based on the past information, and con- stantly updating its state as new information is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' As such, they are suitable for problems where the sequence of the data matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Specific RNN architectures such as Long Short-Term Memory (LSTM) and Gated Recur- rent Unit (GRU) networks are designed to model temporal sequences and their long-range dependencies more accurately than conventional RNNs, and they 12 Multi-Step Wind Power Forecasting are frequently employed to capture temporal information in complex neural networks used for time series forecasting [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' On the other hand, CNNs are known for their feature extraction capabilities from large datasets [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For time series modeling, a CNN can be seen as applying and sliding a filter over the sequential data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Unlike RNNs, the same convolution is used to find the relevant values for all the time stamps, which is a powerful property of the CNNs, referred to as weight sharing, as it enables learning filters that are invariant across the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In this study, we propose a novel CNN-RNN architecture for the wind power forecasting task, and compare it against standard ML models and vanilla CNN architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In addition, we provide a method, Conv2D + LGBM, which use 2D CNNs as a feature extractor for other ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In this method, we consider LGBM as a representative ML model, however, other ML models such as XGB and Random Forests can be employed to utilize the extracted features from CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 CNN-RNN Our proposed CNN-RNN architecture combines CNN and RNN architectures as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Two CNN models, one for GFS and the other for the ARPEGE dataset, extract spatio-meteorological information at each time step of the forecast horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Then, final predictions are obtained by combining CNN out- puts with the output from an RNN encoder for that time step, which stores temporal information of the previous wind power values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The idea behind relying solely on the previous wind power values to extract temporal informa- tion, while disregarding the historical meteorological features, is that since the wind power output is a function of those features, it summarizes the relevant information contained within the historic meteorological patterns [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Note that this approach is computationally efficient and it prevents high memory consumption caused by storing information of the previous n time steps, con- sisting of dense meteorological features accumulated from numerous different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Additionally, we only select wind speed as the meteorological feature for our analysis, since the goal for our model is to only capture location-wise interdependencies, and not waste computational resources by accounting for interdependencies between the meteorological features and the values at differ- ent time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Since our architecture does not store any previous exogenous features, we do not require using an attention mechanism to pay attention to important segments in a long historical sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In our proposed architecture, historic wind power data from time t−n to t−1 is fed into an RNN encoder, which encapsulates the sequential information of the input vector within its internal states ht (hidden state) and ct (cell state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The internal states are then passed along the forecasting horizon in the form of a context vector, where at each time step t+1, RNN cell outputs are combined with the flattened output of CNN, generated based on wind speed inputs corresponding to that same particular time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' At each time step, CNN models perform two-layered convolution with a filter bank to produce Multi-Step Wind Power Forecasting 13 a set of feature maps for the input data, which are then batch normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Then an element-wise ReLU non-linearity, max(0, x), is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Following that, max-pooling with a 2×2 window and stride 2 is performed, and a dropout layer is applied to the resulting output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Max-pooling is used to achieve pattern invariance over small spatial shifts in the 2D feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The combined CNN and RNN output are then passed to a fully connected network involving dense layers, before undergoing a linear activation function to generate the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The proposed approach is summarized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' RNN Encoder Historic Wind Power Spatial Information Temporal Information Wind Speed CNN CNN CNN Wind Speed Wind Speed Predicted output t = 0 t = 1 t = 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 4: CNN-RNN architecture illustrating a representative three-step ahead prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Since our proposed method only considers exogenous features for their respective time step, it involves preparing data so that data for all future prediction time steps gets stored in an array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This leads to three input arrays in total;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' one consists of the lag values of length of the considered historical period, and the other two compromising meteorological features of lengths equivalent to the forecasting horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Hence, the shape of the input tensors for CNN is (samples, timesteps, height, width, channels), while for the RNN encoder the input is of the form (samples, timesteps, feature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 Conv2D + LGBM CNNs are well-known for their ability to automatically extract important fea- tures from large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In this regard, the features extracted from CNNs can be fed into another ML model e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', Random Forests and LGBM to benefit from the strengths of another model to enhance the prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For our analysis, we employ LGBM [28] as the representative ML model, since it consistently provides high performance for our forecasting tasks (as observed in our preliminary analysis) and train LGBM using both the original input features and the extracted features from 2D CNNs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', Conv2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our Conv2D + LGBM method makes use of the spatial feature handling capabilities of the initial Conv2D layers within a CNN architecture, while tanh aa a tanh14 Multi-Step Wind Power Forecasting replacing the fully connected layers with a strong LGBM regressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We use two Conv2D layers, followed by a max pooling layer to extract spatial information from the input data, and then pass the flattened output to an LGBM model, which combines this information, along with the original set of input features to train itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The process of retrieving the flattened output is demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This approach might benefit the conventional CNN since replacing the fully connected layers with LGBM can help avoid falling into a local optima due to the amount or quality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Similarly, the performance of a traditional LGBM model can be enhanced by this approach, as it provides the LGBM with nonlinear feature interactions that it might not be able to learn otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Longitude Height Latitude 2D spatial representation Structured data (a) 2D spatial data Spatial data Max Pooling Flatten Conv2D Conv2D (b) Feature extraction by CNN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 5: Visual representation of the 2D spatial data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' and proposed CNN architecture to extract information from the 2D spatial data 4 Numerical Study In this chapter, we investigate the effectiveness of different machine learn- ing methods for our wind power forecasting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Below, we first provide the details of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Then, we present the results from our detailed numerical study and discuss the performance improvements that can be attributed to our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 Experimental Setup We first provide the details of our experimental settings including the metrics to evaluate forecasting models, hyperparameters for these models and train- test split of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Specifically, in our numerical study, we include four different forecasting methods and seven distinct wind farm datasets to ensure valid research outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We conduct the numerical experiments using Scikit- learn version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 and TensorFlow version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2, on a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='7 GHz dual-core i5 processor with 8GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' All the implementations are done in the Python programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Multi-Step Wind Power Forecasting 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 Evaluation metrics We consider two performance evaluation metrics, Normalized Deviation (ND) and Normalized Root Mean Squared Error (NRMSE), to compare the perfor- mances of forecasting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' ND(y, ˆy) = N � i=1 | ˆyi − yi| N � i=1 |yi| , NRMSE(y, ˆy) = � � � � 1 N N � i=1 ( ˆyi − yi)2 1 N N � i=1 |yi| where y = [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' , yN] and ˆy = [ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' , ˆyN] represent ground truth and pre- dicted values over a prediction horizon N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' RMSE is a popular metric for assessing the performance of regression models and it is typically the preferred method when the model errors follow a Gaussian distribution [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' ND, likewise, increases linearly with an increase in deviations from the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' These metrics are applied to each batch in the test set indepen- dently, and the average across the batches is reported as the final performance value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Statistical significance of the results is measured using the two-sided paired t-test [21] at 95% as the significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For ramp classification, we use “precision” (a measure of the percentage of instances predicted as ramp actually belongs to the same class), “recall” a measure of the percentage of instances detected as ramp are identified correctly), and “F1-score”, which is the harmonic mean of precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 Model settings Our proposed architecture is compared against three baseline models, namely linear regression (LR), extra tree regressor (ET), and LGBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our experiments are conducted using random seed and random state values to mitigate the stochasticity involved with ML model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We perform extensive hyperpa- rameter tuning for all the forecasting models, where, for our deep architectures, we experiment with different combinations of stacked layers and hidden units, along with other parameters including optimizer and batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For other ML models, we apply grid search to identify the best-performing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The final set of hyperparameters used for each model is provided in Table 3, while the search space compromising all different parameter combinations used in our hyperparameter tuning experiments is provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 Performance evaluation For each wind farm, we select the last 120 days as our testing period, which corresponds to 120 unique test samples consisting of 24 time steps each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The rest of the dataset is used for model training and we use the last 10% of the training data as our validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' After identifying the parameters that leads 16 Multi-Step Wind Power Forecasting Table 3: The hyperparameter settings used in the experiments for the employed models Model Final Parameters CNN hidden units: {264,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 128},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' kernel size: {4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' strides : {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' optimizer: Adam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' loss: mse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' batch size: 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' lookback: 48 LSTM hidden units: {128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 64},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' optimizer: Adam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' loss: mse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' batch size: 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' lookback: 48 ET # of trees: 120,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' n estimators: 100 splitting criterion: mse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' max depth: ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' lookback: 24 LGBM # of leaves: 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' n estimators: 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' learning rate: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='07,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' max depth: ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' n estimators:100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' min child samples: 22 Table 4: Hyperparameter search space for forecasting models (a) Deep learning models Parameter Search space hidden layers [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 4] hidden units [32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 264,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 518] kernel size [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 9] strides [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 4] optimizers [Adam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Adamax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' SGD] batch size [32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 264,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 518] (b) Tree-based models Parameter Search space # of trees [80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 120,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 140] n estimators [80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 120,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 140] # of leaves [30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 60,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 120] max depth [50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' ∞] splitting criterion [mae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' mse] min child samples [10:50] to the best forecasting performance using the validation set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' we merge the validation set back to the training set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' and retrain the models on this merged dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Note that we use the same test sets for global forecasting models as well, therefore, the predictions from individual and global forecasting models are directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We employed a two-sided paired t-test for pairwise comparison of forecasting models and understand the statistical significance of the performance improvements attributed to each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 Results In this section, we provide results from our numerical study, and discuss our findings for the wind power forecasting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' First, we compare the CNN archi- tectures against other well-known time series forecasting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Then, we examine the impact of incorporating sequential data to the forecasting models, and we assess the effectiveness of Conv2D as feature extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 Comparison of CNNs against other forecasting methods Table 5 summarizes the average ND and NRMSE values of the time series forecasting algorithms trained on spatial data alone (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', temporal informa- tion is excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The global forecasting approach (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', training a model using Multi-Step Wind Power Forecasting 17 the combined wind farm dataset) benefits the CNN model the most, reducing average ND values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='281 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We note that, when trained individ- ually for each wind farm, CNN performance is worse than other models such as LGBM and ET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This could be attributed to the fact that complex CNN models often require large training samples to fully capture the nonlinearities within the data [4], and when trained individually for each farm, there might not be enough data instances to generalize the learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' On the other hand, baseline forecasting models either do not benefit from the global forecasting approach or experience a slight decline in average ND and NRMSE values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' That is, these models are not able to make use of transfer learning as effec- tively and they are not able to combine information coming in from multiple farms to extract meaningful information in the combined dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In terms of average ND and NRMSE values, LGBM and ET perform similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We observe that out of the seven wind farms, CNN is able to outperform other models on farms 1, 3, 5 and 7 in terms of average ND values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This result indicates that each wind farm exhibits different behaviour and possesses different data characteristics, implying that no single model is best suited for universal wind power forecasting for our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Table 5: Performance comparison of models trained on spatial data Individual Global CNN LGBM ET LR CNN LGBM ET LR (a) ND WF1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='387 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='345 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='398 †: Significant at the 95% level (2-sided paired t-test) To determine whether the performance difference between these models is statistically significant, we conduct a two-sided paired t-test and adopt a similar approach to Oreshkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' That is, we employ a statistical proce- dure to identify whether the mean difference between two sets of results that 18 Multi-Step Wind Power Forecasting are compared against each other is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Figure 6 shows the distributions of ND and NRMSE errors, indicating that these results are suitable for conduct- ing two-sided paired t-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We compare the performance of globally trained CNN against ET model trained over individual wind farms, as these two mod- els show a similar level of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The null (H0) and alternative (H1) hypotheses are characterized as follows: H0: µET = µCNN H1: µET ̸= µCNN Here H0 signifies that the mean ET and CNN scores are equal, while H1 signifies that the mean ET and CNN scores are not equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The p-values for ND are smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='05 for wind farms 2 and 5, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='038 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='037, respectively, and for NRMSE, for wind farms 5 and 7, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='011 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='045, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Therefore only these results can be considered statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 ND 0 2 4 6 8 10 12 14 Density (a) ND 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 NRMSE 0 2 4 6 8 10 12 Density (b) NRMSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 6: Plots showing normal distributions of ND and NRMSE errors making them suitable for two-sided paired t-test 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 Impact of incorporating sequential data Table 6 summarizes performance of models when trained on spatial data and lag values of wind power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We observe that while the performance of all the models improve with the inclusion of temporal information, the most significant improvement is witnessed for the combined CNN-RNN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This could be due to LSTM’s ability to preserve long sequential information of the lag inputs, which is not possible for the baseline forecasting models such as LGBM and ET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We note that CNN-RNN improves the performance over the CNN model, with average ND values improving from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='268 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='249 for the global forecasting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We consider the following hypothesis tests to assess the statistical signifi- cance of the performance differences between CNN-RNN and ET models: H0: µET = µCNN-RNN H1: µET ̸= µCNN-RNN Multi-Step Wind Power Forecasting 19 Table 6: Performance comparison of models trained on spatial and temporal data Individual Global CNN-RNN LGBM ET LR CNN-RNN LGBM ET LR (a) ND WF1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='260 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='327 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='370 †: Significant at the 95% level (2-sided paired t-test) where H0 signifies that the mean performance values for ET and CNN- RNN scores are equal, while H1 characterizes the alternative hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The CNN-RNN architecture is able to outperform other methods overall at 95% sta- tistical significance level, with p-values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 for ND and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='047 for NRMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For the ND performance over individual wind farms, farms 3, 5 and 7 indicate statistically significant improvements for CNN-RNN with p-values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='024, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='047 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='002, respectively, while, for NRMSE, only the performance for farm 7 is statistically significant with p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Figure 7 demonstrates model predictions across six unique sample test batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We observe that while none of the models provide highly confor- mal predictions, the forecasts largely follow the trends for the ground truth (Actual) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Figure 8 illustrates the average ND and NRMSE errors over all the trained forecasting models across the test batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' That is, errors from all the different models are averaged for each test batch, with the goal of understanding which test batches are more difficult to predict, and whether there are any visible outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We find that while the ND error is below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0, and NRMSE is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='5 for the first 50 test batch samples, the errors drastically increase towards the last set of test batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This shift in performance is typically expected for time series forecasting tasks, as the further the test predictions are from the last training batch set, the more difficult it is to obtain accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Figure 8 also shows that the performance across the different batches inherits 20 Multi-Step Wind Power Forecasting 2020-10-10,12 2020-10-11,12 Forecast Horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 Normalized Production Output Actual CNN-RNN ET LR (d) Test sample 4 2020-10-14,12 2020-10-15,12 Forecast Horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 Normalized Production Output Actual CNN-RNN ET LR (e) Test sample 5 2020-10-15,12 2020-10-16,12 Forecast Horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='9 Normalized Production Output Actual CNN-RNN ET LR (f) Test sample 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 7: Visual comparison of model predictions across six 24-hr long test samples high variance, with the spikes indicating high noise in the testing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This observation indicates that our datasets are highly complicated and they contain a significant amount of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Multi-Step Wind Power Forecasting 21 0 20 40 60 80 Testing Batch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='5 ND (a) ND 0 20 40 60 80 Testing Batch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='5 NRMSE (b) NRMSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 8: Average ND and NRMSE across all test batches for all the models 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 Using Conv2D as spatial feature extractor We next examine whether spatial features extracted from Conv2D layers ben- efit the performance of LGBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Specifically, when added to LGBM, we expect the extracted features to enhance the prediction performance because the orig- inal features are kept in the LGBM training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' On the other hand, replacing the fully connected layers of a CNN with LGBM help might achieve better results than a traditional CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For this experiment, we use the global fore- casting approach and convolutional 2D layers, while LGBM is trained for each wind farm individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Table 7 shows the absolute change in performance for Conv2D + LGBM versus LGBM and CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The downward and upward arrows represent a drop or increase in error, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We conduct two hypothesis tests to measure the significance of performance improvements attributed to using Conv2D as spatial feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The null and alternative hypothesis for the first hypothesis test is as follows: H0: µLGBM = µConv2D + LGBM H1: µLGBM ̸= µConv2D + LGBM where H0 signifies that the mean LGBM and Conv2D + LGBM performance values are equal, while H1 corresponds to the alternative hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We observe that Conv2D + LGBM reduces average ND error by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='004, and aver- age NRMSE error by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='006 when compared against LGBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We find that the p-values for this test for ND and NRMSE are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='004 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='002, respectively, indicating that the performance improvements are statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For individual wind farms, Conv2D + LGBM is able to outperform LGBM on 22 Multi-Step Wind Power Forecasting Table 7: Performance improvements for Conv2D + LGBM (a) ND WF1 WF2 WF3 WF4 WF5 WF6 WF7 Average Conv2D + LGBM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='260 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='267 vs LGBM ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='006 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='005 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='012† ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='010† ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='004† vs CNN ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='004 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='006 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='005 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='002 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='012† ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='018† ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 †: Significant at the 95% level (2-sided paired t-test) (b) NRMSE WF1 WF2 WF3 WF4 WF5 WF6 WF7 Average Conv2D + LGBM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='335 vs LGBM ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='009 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='000 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='013† ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='014† ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='006† vs CNN ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='004 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='013 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='002 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='015† ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='021† ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='003 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='002 †: Significant at the 95% level (2-sided paired t-test) wind farms 6 and 7 with p-values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='045 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='008, respectively, for ND and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='043 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='003, respectively, for NRMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The null and alternative hypothesis for the second hypothesis test is as follows: H0: µCNN = µConv2D + LGBM H1: µCNN ̸= µConv2D + LGBM where H0 signifies that the mean CNN and Conv2D + LGBM scores are equal, while H1 corresponds to the alternative hyppothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We observe smaller improvements for Conv2D + LGBM over CNN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='001 and 0002, for ND and NRMSE, respectively), and with p-values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='99 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='83 for ND and NRMSE, respectively, the improvements were not found to be significant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', fail to reject the null hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We also find statistically significant improvements attributed to Conv2D + LGBM over CNN for wind farms 5 and 6 with p-values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='023 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='018 for ND, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='012 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='010 for NRMSE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' These results show that, in most cases, it is possible to utilize convo- lutional layers to extract spatial information from location-based features and use that additional information to further improve the performance of a strong ensemble-based regressor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', LGBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Additionally, for some cases, by replacing the fully connected dense network of a CNN architecture with a strong ensemble-based regressor, we are able to achieve better performance than CNN baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Figure 9 shows the distribution of the difference in ND and NRMSE errors across the seven wind farms, for LGBM and CNN, when compared against Conv2D + LGBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 5 Conclusions and Discussions Wind power forecasting and ramp event prediction have garnered significant interest as they can be used for various purposes in practice such as taking preventative actions to reduce equipment damage and improving operational efficiency within the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This work focuses on a complex combined Multi-Step Wind Power Forecasting 23 WF1 WF2 WF3 WF4 WF5 WF6 WF7 Farm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 ND Difference vs LGBM vs CNN (a) ND WF1 WF2 WF3 WF4 WF5 WF6 WF7 Farm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='3 NRMSE Difference vs LGBM vs CNN (b) NRMSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' 9: Statistical distribution of performance differences for Conv2D + LGBM vs LGBM and CNN architecture involving CNNs and RNNs to extract useful information from an ultra-wide input matrix that consists of entries from multiple NWP models, wind farms, geographical locations and atmospheric levels, to make a day- ahead wind power and ramp event predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our analysis highlights the capabilities of CNN architectures in extracting spatial information by learning the underlying interdependencies of input data across various locations and RNN’s capability towards extracting long-term temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We also make use of CNN’s spatial feature extraction ability by feeding the extracted features to a tree ensemble-based regressor, namely LGBM, to further boost its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We conduct numerical studies to assess the impact of global learning on wind power prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We evaluate the performance of our models using the ND and NRMSE metrics and employ the two-sided pairwise t-test to assess the significance of performance improvements attributed to the proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our results show that CNN and RNN make effective use of the spatial and temporal infor- mation in the dataset and, when combined together to form a complex neural structure, they are able to outperform other ML models including LR, ET and LGBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our numerical results also show that the Conv2D + LGBM method obtained by feeding the features extracted from CNN to LGBM performs bet- ter than standalone usage of LGBM, highlighting the importance of extracting spatial information from location-based features, and in some cases better than CNN, signifying that, for certain datasets, a simpler regression model may better learn from the available data compared to the complex neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' We observe that global learning contributes significantly to the performance of CNN and CNN-RNN, as these complex neural networks can leverage increased training set sizes from the combined dataset of multiple wind farms, and they are better able to extract the interdependencies between the related data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' There have been several challenges and limitations to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The data sets we incorporated in this study were extremely noisy, with a high variance 24 Multi-Step Wind Power Forecasting within the testing batches, making it difficult for us to achieve highly accu- rate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This issue is very common in medium and long-term wind power forecasting problems, and a significant amount of research is dedicated to improving the forecasting performance with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' For multi-step ahead ramp classification, we have dealt with a highly imbalanced class distribution, with the ratio of ramp event to no-ramp event being 1:5,673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' While we adjust the weights of the classes in the ML model training, it is still difficult to achieve satisfactory performance for the ramp detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Due to the massive scale of our input data combined from multiple sources, it is difficult to perform extensive hyperparameter tuning and experiment with more complex models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', by combining ConvLSTM with our CNN-RNN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Several research directions can be considered to extend our work in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' The proposed CNN-RNN architecture can be further enhanced by adding a ConvLSTM component to it for extracting spatio-temporal infor- mation from the historical meteorological features and using TCN along with RNN for temporal feature extraction from the historic wind power outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' This was not possible at present due to the high computational cost involved with this modified architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Our experiments involving data from seven distinct wind farms show that each wind farm has its own unique characteris- tics, which may be due to its location or the differences in the wind turbines installed, and that no single model is best suited to make predictions across all the wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' In this regard, an ensemble method can be considered for our forecasting task where final predictions are a combined result of individu- ally and globally trained models with adjusted weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' As more data becomes available, a continuous latitude and longitude grid can be employed, similar to Kazutoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' A hybrid deep learning model and comparison for wind power forecasting considering temporal-spatial feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} +page_content=' Sustainability 12, 9490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQf6Pqt/content/2301.00819v1.pdf'} diff --git a/LNAyT4oBgHgl3EQf6fpN/vector_store/index.faiss b/LNAyT4oBgHgl3EQf6fpN/vector_store/index.faiss new file mode 100644 index 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Aofi Al-Akbi and S. Hadi Jafari +Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran +January 31, 2023 +Abstract +There has been a great importance in understanding the nilpotent multipliers of finite groups in recent past. +Let a group G be presented as the quotient of a free group F by a normal subgroup R. Given a positive integer +c, the c-nilpotent multiplier of the group G is the abelian group M(c)(G) = (R ∩ γc+1(F))/γc+1(R, F), where +γ1(R, F) = R, γc+1(R, F) = [γc(R, F), F], and γc+1(F) = γc+1(F, F). In particular, M(1)(G) is the Schur +multiplier of G. The crucial aspect of the research in to the c-nilpotent multipliers of groups includes either +establishing their structures, or estimating their sizes and exponents. One reason for studying the c-nilpotent +multiplier is its relevance to the isologism theory of P. Hall. The study of Schur multiplier of finite metacyclic +groups goes back to the paper by F. R. Beyl in 1973. +In this article, we study the 2-nilpotent multiplier +of finite split metacyclic groups with the help of their nonabelian tensor squares. +In particular, we give a +complete description of the triple tensor product, the triple exterior product, and the 2-nilpotent multiplier of +such groups. +Keywords Metacyclic group . Group action . Nonabelian tensor product . Nilpotent multiplier +Mathematics Subject Classification 20F05 . 20C25 . 20J99 +1 +Definitions And Motivation +A group G is said to be metacyclic if it possesses a cyclic normal subgroup N such that G/N is also cyclic. It is clear +that a metacyclic group G can be written G = HN with H ⩽ G, N ✂ G and both H and N cyclic. If H ∩ N = 1, +then G is called the split metacyclic group. It was already known to H¨older that finite metacyclic groups can be +presented on two generators and three defining relations. When G = HN is split, by taking generators x of H +and y of N, it is well-known that G has a presentation of the form +⟨x, y | xm = yn = 1, xyx−1 = yl⟩, +(1) +1 + +Finite split metacyclic... +2 +where the integers m, n, and l satisfying lm ≡ 1 (mod n). In the rest of the paper we will fix m, n, and l as the +integers of the above presentation. The purpose of this paper is to show that the 2-nilpotent multiplier M(2)(G) is +the direct product of two cyclic groups of orders (n, l − 1, 1 + l + ... + lm−1) and 1 +n(n, 1 + l + ... + lm−1)(n, (l − 1)2), +where by (r, s) we mean the greatest common divisor of the integers r and s. +Given a positive integer c, recall that the c-nilpotent multiplier of a group G which is presented as the quotient +of a free group F by a normal subgroup R, is the abelian group M(c)(G) = (R ∩ γc+1(F))/γc+1(R, F), where +γ1(R, F) = R, γc+1(R, F) = [γc(R, F), F] and γc+1(F) = γc+1(F, F). It was shown by R. Baer [1] that these +invariants are, up to group isomorphism, independent of the choice of free presentation of G. The group M(G) = +M(1)(G) is more known as the Schur multiplier of G. +One motivation to study the c-nilpotent multipliers is its relevance to the isologism theory of P. Hall [8, 9] +in which groups can be classified into isologism classes (see [5, p. 406] for more details). Another important +reason is that, determining the structures of c-nilpotent multipliers is essential in studying generalized capability +and covering groups, and would be generally useful in developing such a framework. Furthermore, although the +above formula of M(c)(G) looks simple enough; it is mostly a hard but interesting problem to compute it. In +particular, according to the method proposed in [5], which is based on the notion of triple exterior product, we +hope our different approach of studying 2-nilpotent multipliers can be extended to more general cases of c-nilpotent +multipliers and provides some new insights. +Let’s first recall the definition of the nonabelian tensor product of two groups G and H, by supposing that +they act on each other satisfying certain compatibility conditions [3, pp. 178-179], and act on themselves by +conjugation. The nonabelian tensor product G ⊗ H, was introduced by Brown and Loday in [4], is defined as a +group generated by the symbols g ⊗ h, where g ∈ G and h ∈ H, with defining relations +gg′ ⊗ h = (gg′ ⊗ gh)(g ⊗ h) +and +g ⊗ hh′ = (g ⊗ h)(hg ⊗ h′h), +for all g, g′ ∈ G and h, h′ ∈ H, where gg′ = gg′g−1. It was shown in [6] with homological methods, involving the +original approach in [3, 4], that G ⊗ H is a finite group, when G and H are finite groups. +Since the compatible actions are satisfied when a group acts on itself by conjugation, it leads to define the +nonabelian tensor square G⊗G. The nonabelian exterior square G∧G is the quotient of G⊗G modulo the central +subgroup ∇(G) generated by the elements g ⊗g for all g ∈ G. The image of the generator g ⊗g′ in G∧G is written +g ∧ g′. The commutator map induces a homomorphism [ , ] : G ∧ G −→ G mapping g ∧ g′ to [g, g′] = gg′g−1g′−1, +in which its kernel is isomorphic to M(G) [12]. +The first attempt at determining the tensor square of a finite split metacyclic group (1) was made by Brown, +Johnson, and Robertson [3] where they achieved the favorable special case when n is odd. + +Finite split metacyclic... +3 +Proposition 1.1. ([3, Proposition 15]) Let G be a metacyclic group with presentation (1) and n is odd. Then +G ⊗ G is the direct product of four cyclic groups with generators x ⊗ x, y ⊗ y, (x ⊗ y)(y ⊗ x), x ⊗ y, of orders +m, (n, l − 1), (n, l − 1, 1 + l + ... + lm−1), and (n, 1 + l + ... + lm−1), respectively. Furthermore, ∇(G)=⟨x ⊗ +x, y ⊗ y, (x ⊗ y)(y ⊗ x)⟩ and M(G) is cyclic of order 1 +n(n, l − 1)(n, 1 + l + ... + lm−1). +Next Johnson [11] completed such a computations be evaluating G ⊗ G for even n. +Proposition 1.2. ([11, Proposition 16]) Let G be a metacyclic group with presentation (1) and n is even. Then +G⊗G is the abelian group with generators X = x⊗x, Y = y⊗y, Z = (x⊗y)(y⊗x), T = x⊗y, and A = (y⊗y)l−1, +and relations +Xm = Y n = Zl−1 = Zn = Z1+l+...+lm−1 = T n = A2 = Am = 1, Y l−1 = A, T 1+l+...+lm−1 = A(l−1)(m−1)m/4. +Furthermore, M(G) is cyclic of order 1 +n(n, l − 1)(n, 1 + l + ... + lm−1). +With the above assumptions, one could easily observe that ∇(G) ∼= Cm × C(n,2(l−1)) × C(n,l−1,1+l+...+lm−1). +Here we need to give a brief recall of the notion of triple tensor (exterior) product of groups. The notation +and terminology are the same as those in [10]. Since conjugation yields compatible actions, there is a diagonal +action of G on G ⊗ G such that g(g1 ⊗ g2) = +gg1 ⊗ gg2. Also the tensor square G ⊗ G acts on G by conjugation in +G via the homomorphism [ , ] : G ⊗ G −→ [G, G] induced by the commutator map. Evidently these actions are +compatible and we can thus construct the triple tensor product ⊗3G = (G ⊗ G) ⊗ G. Also by [4, Definition 2.11] +the triple exterior product ∧3G = (G ∧ G) ∧ G is obtained from the tensor product (G ∧ G) ⊗ G by imposing the +additional relations (g ∧ g′) ⊗ [g, g′] = 1 for all g, g′ ∈ G (see [10] for more details). Recently, the second author +in joint work with some others [7, 10] established the explicit structures of the triple tensor (exterior) products of +certain groups. +Like the above results, we’ll see below that although the structures of ∧3G and M(2)(G) remain invariant for +any integer n, but the evaluation of ⊗3G for even n is somewhat different from the odd n. Because for even case +the computations depend on the relation T 1+l+...+lm−1 = A(l−1)(m−1)m/4 which appeared above. The main result +of this paper is the following theorem. +Main Theorem. Let G be a finite split metacyclic group with presentation (1). +(i) If n is odd, then +⊗3G ∼= Cm × C(n,l−1) × C2 +(m,n,l−1) × C(n,1+l+...+lm−1) × C2 +(n,l−1,1+l+...+lm−1) × C(m,n,l−1,1+l+...+lm−1), +(ii) If n is even, then under the assumptions of Proposition 1.2, +⊗3G ∼= (A ⊗ Gab) × (B ⊗ G) in which A = ⟨X, Z⟩ and B = ⟨Y, T ⟩ are subgroups of G ⊗ G such that +A ⊗ Gab ∼= Cm × C(m,n,l−1) × C(n,l−1,1+l+...+lm−1) × C(m,n,l−1,1+l+...+lm−1), and B ⊗ G is the abelian group + +Finite split metacyclic... +4 +generated by Yx = Y ⊗ x, Yy = Y ⊗ y, Tx = T ⊗ x, and Ty = T ⊗ y, subject to the relations +Y (m,n,2(l−1)) +x += Y (n,l−1) +y += T (n,4(l−1),1+l+...+lm−1) +y += 1, Y l−1 +x += T l2−1 +y +, +and either +� +� +� +� +� +� +� +T (n,1+l+...+lm−1) +x += 1, +if +m ≡ 0 or 1 (mod 4) +T n +x = 1, T 1+l+...+lm−1 +x += T −(l−1)2 +y +, +if +m ≡ 2 or 3 (mod 4) +when T 1+l+...+lm−1 = 1, or T (n,1+l+...+lm−1) +x += 1 when T 1+l+...+lm−1 ̸= 1, +(iii) ∧3G ∼= C(n,l−1,1+l+...+lm−1) × C(n,1+l+...+lm−1), +(iv) M(2)(G) ∼= C(n,l−1,1+l+...+lm−1) × C(n,1+l+...+lm−1)(n,(l−1)2)/n. +It should be mentioned that the HAP package (Ellis, 2008) of GAP [13] has methods for checking the above +results in the case when G is nilpotent. We applied them for several of metacyclic groups by performing the +relevant computations (for more details see [10, Section 4]). The paper is organized as follows. In Section 2, we +study some basic properties of the tensor products of groups including some relations between the generators of +⊗3G. Section 3 deals with the proof of our main theorem. +2 +Computing The Triple Tensor Products +This section contains some results to be used throughout the rest of the paper. We start with some familiar rules +of nonabelian tensor product of groups. +Proposition 2.1. ([3, Proposition 2]) Let G and H be two groups. Then for all g ∈ G and h ∈ H +(i) There are homomorphism of groups λ : G ⊗ H −→ G and λ′ : G ⊗ H −→ H such that λ(g ⊗ h) = ghg−1 and +λ′(g ⊗ h) = ghh−1. +(ii) λ(t) ⊗ h = tht−1 and g ⊗ λ′(t) = gtt−1, and thus λ(t) ⊗ λ′(t1) = [t, t1] for all t, t1 ∈ G ⊗ H. Hence, G acts +trivially on Kerλ′ and H acts trivially on Kerλ. +Proposition 2.2. ([3, Proposition 3]) Let G and H be two groups. The following relations hold in G ⊗ H for all +g, g′ ∈ G and h, h′ ∈ H (note [g, h] may be interpreted as ghg−1 ∈ G or ghh−1 ∈ H): +(i) g(g−1 ⊗ h) = (g ⊗ h)−1 = +h(g ⊗ h−1), +(ii) g′ ⊗ (ghh−1) = g′(g ⊗ h)(g ⊗ h)−1, +(iii) (ghg−1) ⊗ h′ = (g ⊗ h)h′(g ⊗ h)−1, +(iv) (g ⊗ h)(g′ ⊗ h′)(g ⊗ h)−1 = [g,h](g′ ⊗ h′), + +Finite split metacyclic... +5 +(v) [g ⊗ h, g′ ⊗ h′] = (ghg−1) ⊗ (g′h′h′−1). +Corollary 2.3. For all a, b, c, d ∈ G the following relations hold in G ⊗ G and ⊗3G (to simplify the notation we +shall identify b ⊗ c ⊗ d with (b ⊗ c) ⊗ d in ⊗3G): +(i) a(b ⊗ c) = (a ⊗ [b, c])(b ⊗ c), +(ii) a(b ⊗ c ⊗ d) = (b ⊗ c ⊗ d)(d ⊗ [b, c] ⊗ a). +In next two subsections, given the metacyclic group G with presentation (1), we list some useful relations +for the elements of ⊗3G proceeding in a series of steps. We start with the case that n is odd which is relatively +straightforward. Note by Proposition 2.2 that [a⊗b⊗c, a′⊗b′⊗c′] = [[a, b]⊗c, [a′, b′, c′]], for all a, b, c, a′, b′, c′ ∈ G. +Now by setting G = G ⊗ G and H = G in Proposition 2.1, it tells us immediately that Ker(λ′ : ⊗3G → G) is +central. Since Imλ′ = γ3(G) is cyclic, it follows that ⊗3G is abelian. +2.1 +The n odd case +Throughout this section we let G be an arbitrary finite split metacyclic group as in (1) and n is odd. +y(x ⊗ y ⊗ y) = x(x ⊗ y ⊗ y) = x ⊗ y ⊗ y. +(2) +Proof. Taking G = G ⊗ G and H = G in Proposition 2.1 we get the homomorphism λ : ⊗3G → G ⊗ G so that +λ(a ⊗ b ⊗ c) = (a ⊗ b)c(a ⊗ b)−1, for all a, b, c ∈ G. In addition, it follows from the Proposition 2.2(iii) and [3, +(5.1)] that x ⊗ y ⊗ y ∈ Kerλ. Thus Proposition 2.1 concludes the result. +(x ⊗ y ⊗ y)l−1 = 1. +(3) +Proof. It follows from [3, (5.3)] that (x ⊗ y)x(x ⊗ y)−1 = (x ⊗ y)1−l. According to the defining actions in ⊗3G, as +x⊗yxx−1 = [x, y, x] = [yl−1, x] = y−(l−1)2, then +(x ⊗ y ⊗ y)(l−1)3 = (x ⊗ y)1−l ⊗ y−(l−1)2 += (x ⊗ y) x(x ⊗ y)−1 ⊗ (x⊗y)xx−1 += [x ⊗ y ⊗ x, x ⊗ y ⊗ x] = 1. +On the other hand from (2) and [3, (5.3)] we have +(x ⊗ y ⊗ y) = x(x ⊗ y ⊗ y) = x(x ⊗ y) ⊗ xy = (x ⊗ y)l ⊗ yl = (x ⊗ y ⊗ y)l2, +which implies that (x ⊗ y ⊗ y)l2−1 = 1. Hence +1 = (x ⊗ y ⊗ y)(l−1)3−(l2−1) = (x ⊗ y ⊗ y)l3−4l2+3l = (x ⊗ y ⊗ y)4(l−1). + +Finite split metacyclic... +6 +Now since n is odd, it finishes the proof. +y(x ⊗ y ⊗ x) = x ⊗ y ⊗ x. +(4) +Proof. From [3, (5.3)] and (3) we have that +y(x ⊗ y ⊗ x) = ((x ⊗ y) ⊗ yx) += ((x ⊗ y) ⊗ y1−lx) += (x ⊗ y ⊗ y1−l)y1−l(x ⊗ y ⊗ x) += (x ⊗ y ⊗ y)1−l y1−l(x ⊗ y ⊗ x) += y1−l(x ⊗ y ⊗ x), +which implies x ⊗ y ⊗ x is fixed by yl. In addition, as l and n are relatively prime, there exist integers s, t such +that sl + tn = 1. Hence y(x ⊗ y ⊗ x) = ysl+tn(x ⊗ y ⊗ x) = ysl(x ⊗ y ⊗ x) = x ⊗ y ⊗ x. +x(x ⊗ y ⊗ x) = (x ⊗ y ⊗ x)l. +(5) +Proof. By applying [3, (5.3)] we see that x(x ⊗ y ⊗ x) = (x ⊗ y)l ⊗ x. Now the result follows by induction on l +together with using (4). +Setting i.j = i(1 + l + l2 + ... + lj−1), then +(x ⊗ y)i ⊗ xj = (x ⊗ y ⊗ x)i.j and (x ⊗ y)i ⊗ yk = (x ⊗ y ⊗ y)ik, for any integers i, j, k. +(6) +Proof. It follows by applying [3, (5.2)], (2), (4), and (5). +2.2 +The n even case +In this subsection we are concerned with the case that G is a finite split metacyclic group as in (1) and n is even. +As we’ll see, this case is slightly more complicated with respect to the odd case. For instance, the order of y ⊗y ⊗y +would be discussed here while in odd case it is easily determined as a direct factor of the abelian group ∇(G)⊗Gab +(see Section 3 for more details). +(y ⊗ y ⊗ y)l−1 = 1. +(7) +Proof. It is readily seen that x(y ⊗ y ⊗ y) = y ⊗ y ⊗ yl = (y ⊗ y ⊗ y)l. In addition, since y ⊗ y ⊗ y belongs to Kerλ +given in the proof of (2), it follows that x(y ⊗ y ⊗ y) = y ⊗ y ⊗ y. The result now follows easily. + +Finite split metacyclic... +7 +y(x ⊗ y ⊗ y) = x ⊗ y ⊗ y. +(8) +Proof. Using [11, (9)] and (7), +y(x ⊗ y ⊗ y) = y(x ⊗ y) ⊗ y += (x ⊗ y)(y ⊗ y)l−1 ⊗ y += x⊗y((y ⊗ y)l−1 ⊗ y)(x ⊗ y ⊗ y) += (x ⊗ y ⊗ y)(y ⊗ y ⊗ y)l−1 += x ⊗ y ⊗ y. +(x ⊗ y)i ⊗ yj = (x ⊗ y ⊗ y)ij, for any integers i, j. +(9) +Proof. It can be proved by induction on i together with using (8). +x(x ⊗ y ⊗ y) = (x ⊗ y ⊗ y)l2. +(10) +Proof. Invoking [11, (10)], (8) and (9), +x(x ⊗ y ⊗ y) = x(x ⊗ y) ⊗ xy += (x ⊗ y)l(y ⊗ y)−l(l−1)2/2 ⊗ yl += (x⊗y)l((y ⊗ y)−l(l−1)2/2 ⊗ yl)((x ⊗ y)l ⊗ yl) += (y ⊗ y ⊗ y)−l2(l−1)2/2(x ⊗ y ⊗ y)l2. +The result follows by (7). +(x ⊗ y ⊗ y)1+l+...+lm−1 = 1. +(11) +Proof. Clearly the assertion holds when (x⊗ y)1+l+...+lm−1 = 1. If (x⊗ y)1+l+...+lm−1 ̸= 1, then by Proposition 1.2 +as A2 = 1 we see (x⊗ y)1+l+...+lm−1 = (y ⊗ y)l−1. Hence (x⊗ y)1+l+...+lm−1 ⊗ y = (y ⊗ y)l−1 ⊗ y, and consequently +(7) concludes the result. +(x ⊗ y ⊗ y)4(l−1) = 1. +(12) + +Finite split metacyclic... +8 +Proof. As (y⊗y)2(l−1) = 1 by [11, (15)], then it follows that y fixes (x⊗y)2. So the element (x⊗y⊗y)2 = (x⊗y)2⊗y +belongs to Kerλ, where λ is the homomorphism given in the proof of (2). Now Proposition 2.1(ii) implies the group +G acts on (x ⊗ y ⊗ y)2 trivially. Consequently it follows by (10) that (x ⊗ y ⊗ y)2 = x(x ⊗ y ⊗ y)2 = (x ⊗ y ⊗ y)2l2. +Therefore (x⊗ y ⊗ y)2(l2−1) = 1. On the other hand, from [11, (10)] and Proposition 2.2 we have (x⊗ y)x(x⊗ y)−1 +=(x ⊗ y)1−l(y ⊗ y)(l−1)2/2. Thus (x ⊗ y)1−l = (x ⊗ y)x(x ⊗ y)−1(y ⊗ y)−(l−1)2/2. Hence by applying (7) we have +(x ⊗ y ⊗ y)(l−1)3 = (x ⊗ y)1−l ⊗ y−(l−1)2 += ((x ⊗ y)x(x ⊗ y)−1(y ⊗ y)−(l−1)2/2) ⊗ y−(l−1)2 += ((y ⊗ y)−(l−1)2/2 ⊗ y−(l−1)2)((x ⊗ y)x(x ⊗ y)−1 ⊗ y−(l−1)2) += (y ⊗ y ⊗ y) +(l−1)2 +2 +−(l−1)2((x ⊗ y)x(x ⊗ y)−1 ⊗ x⊗yxx−1) += [x ⊗ y ⊗ x, x ⊗ y ⊗ x] = 1. +Combining two last equations leads to the assertion. +y(x ⊗ y ⊗ x) = (x ⊗ y ⊗ x)(x ⊗ y ⊗ y)l−1. +(13) +Proof. Using Corollary 2.3, [11, (11)] and (7), +y(x ⊗ y ⊗ x) = ((x ⊗ [x, y] ⊗ y)(x ⊗ y ⊗ x) += [(x ⊗ y)l−1(y ⊗ y)−(l−1)2(l−2)/2 ⊗ y](x ⊗ y ⊗ x) += (x ⊗ y ⊗ x)(x ⊗ y ⊗ y)l−1. +(x ⊗ y)i ⊗ x = (x ⊗ y ⊗ x)i(x ⊗ y ⊗ y)( +i +2)(l−1)2, for any integer i. +(14) +Proof. It follows by induction on i, [3, (22)], (9), and (13). +(x ⊗ y ⊗ y)l2−1 = (y ⊗ y ⊗ x)l−1. +(15) +Proof. Use (10) and Corollary 2.3, +(x ⊗ y ⊗ y)l2 = x(x ⊗ y ⊗ y) = (y ⊗ [x, y] ⊗ x)(x ⊗ y ⊗ y) = (y ⊗ y ⊗ x)l−1(x ⊗ y ⊗ y). +x(x ⊗ y ⊗ x) = (x ⊗ y ⊗ x)l(x ⊗ y ⊗ y)(l−1)2. +(16) + +Finite split metacyclic... +9 +Proof. First note that G acts trivially on y ⊗ y ⊗ x. It follows by Corollary 2.3, [11, (11)], (14), and (15) that +x(x ⊗ y ⊗ x) = (x ⊗ [x, y] ⊗ x)(x ⊗ y ⊗ x) += [(x ⊗ y)l−1(y ⊗ y)−(l−1)2(l−2)/2 ⊗ x](x ⊗ y ⊗ x) += (y ⊗ y ⊗ x)−(l−1)2(l−2)/2((x ⊗ y)l−1 ⊗ x)(x ⊗ y ⊗ x) += (y ⊗ y ⊗ x)−(l−1)2(l−2)/2(x ⊗ y ⊗ x)l−1(x ⊗ y ⊗ y)(l−2)(l−1)3/2(x ⊗ y ⊗ x) += (x ⊗ y ⊗ x)l(x ⊗ y ⊗ y)−(l−2)(l−1)2. +So the result is obviously true by applying the last equation used in the proof of (12). +For any integers i and j we have: +(x ⊗ y)i ⊗ xj = +� +� +� +� +� +(x ⊗ y ⊗ x)i.j (x ⊗ y ⊗ y)(i +2).j(l−1)2 +if +j ≡ 0 or 1 (mod 4) +(x ⊗ y ⊗ x)i.j (x ⊗ y ⊗ y)((i +2).j+i)(l−1)2 +if +j ≡ 2 or 3 (mod 4) +(17) +where i.j = i(1 + l + l2 + ... + lj−1). +Proof. We proceed by induction on both i and j. Assuming i = 1, it follows by induction on j together with (16) +that +x ⊗ y ⊗ xj = +� +� +� +� +� +(x ⊗ y ⊗ x)1.j +if +j ≡ 0 or 1 (mod 4) +(x ⊗ y ⊗ x)1.j (x ⊗ y ⊗ y)(l−1)2 +if +j ≡ 2 or 3 (mod 4) +It suffices to prove the second part of the assertion and the first is similar. Using (13), l − 1 times, we have +(x ⊗ y)i+1 ⊗ xj = x⊗y((x ⊗ y)i ⊗ xj)(x ⊗ y ⊗ xj) += x⊗y((x ⊗ y ⊗ x)i.j(x ⊗ y ⊗ y)((i +2).j+i)(l−1)2)(x ⊗ y ⊗ x)1.j(x ⊗ y ⊗ y)(l−1)2 += (x ⊗ y ⊗ x)i.j(x ⊗ y ⊗ y)i.j(l−1)2(x ⊗ y ⊗ y)((i +2).j+i)(l−1)2(x ⊗ y ⊗ x)1.j(x ⊗ y ⊗ y)(l−1)2 += (x ⊗ y ⊗ x)(i+1).j(x ⊗ y ⊗ y)(( +i+1 +2 ).j+i+1)(l−1)2. +3 +Proof of The Main Theorem +In this section we prove the main theorem of the paper. The proof relies on the previous section together with +a computation method based on the crossed pairings. There is a significant difference between the even and odd +case, and w’ll treat them separately beginning with the odd case. + +Finite split metacyclic... +10 +3.1 +The n odd case +Let G be a finite split metacyclic group as in (1) and n is odd. We observe from Proposition 1.1 that in this case +G ⊗ G splits as +G ⊗ G ∼= ∇(G) × (G ∧ G) ∼= Cm × C(n,l−1) × C(n,l−1,1+l+...+lm−1) × C(n,1+l+...+lm−1), +where G ∧ G is isomorphic to the subgroup of G ⊗ G generated by ⟨x ⊗ y⟩. This is not the case in general when n +is even. The groups G ∧ G and G act on each other in such a way that G ∧ G ∼= ⟨x ⊗ y⟩ is fixed under the action +of G. So by applying [3, Proposition 10] one could describe ⊗3G as follows: +⊗3G ∼= (∇(G) × (G ∧ G)) ⊗ G ∼= (∇(G) ⊗ G) × (G ∧ G ⊗ G). +Now since the groups ∇(G) and G act on each other trivially, Proposition 2.4 in [4] allows us to express ∇(G) ⊗ G +as the tensor product ∇(G) ⊗ Gab of abelian groups. Therefore by Proposition 1.1 we conclude that +∇(G) ⊗ Gab ∼= Cm × C(n,l−1) × C2 +(m,n,l−1) × C(n,l−1,1+l+...+lm−1) × C(m,n,l−1,1+l+...+lm−1). +The main task then is to determine the cyclic invariants of G ∧ G ⊗ G. For this purpose first we expand the +arbitrary element (x ⊗ y)i⊗ yjxk of G ∧ G ⊗ G by invoking (4) and (6): +(x ⊗ y)i ⊗ yjxk = ((x ⊗ y)i ⊗ yj) yj((x ⊗ y)i ⊗ xk) += (x ⊗ y ⊗ y)ij yj(x ⊗ y ⊗ x)i.k += (x ⊗ y ⊗ y)ij(x ⊗ y ⊗ x)i.k. +This establishes what the generators of G ∧ G ⊗ G are. To find the orders of these generators, and any possible +relations among them, by considering our analysis of ⊗3G in Subsection 2.1, we use crossed pairing into suitable +cyclic groups. Recall that for groups G, H and L where G and H acting upon each other compatibly and acting +upon themselves by conjugation, a function Φ : G × H −→ L is called a crossed pairing if +Φ(gg′, h) = Φ(gg′,g h)Φ(g, h) +and +Φ(g, hh′) = Φ(g, h)Φ(hg,h h′), +for all g, g′ ∈ G, h, h′ ∈ H. Clearly any crossed pairing Φ : G × H −→ L determines a unique homomorphism +Φ∗ : G ⊗ H −→ L such that Φ∗(g ⊗ h) = Φ(g, h). +Now define Φ : (G ∧ G) × G → C(n,1+l+...+lm−1) × C(n,l−1,1+l+...+lm−1) by Φ((x ⊗ y)i, yjxk) = ai.k bij and let +((x⊗y)i, yjxk) = ((x⊗y)i′, yj′xk′). Then with setting r = (n, 1+l+...+lm−1), it follows that i′ = i+ur, j′ = j+vn, +and k′ = k + wm, for some integers u, v, w. So ai′k′bi′j′ = ai.k bij, which implies that Φ is well-defined. We prove + +Finite split metacyclic... +11 +that Φ is a crossed pairing. The following verifies the first rule for a crossed pairing: +Φ((x⊗y)i(x ⊗ y)i′, (x⊗y)iyjxk)Φ((x ⊗ y)i, yjxk) = Φ((x ⊗ y)i′, yj+i(l−1)(1−lk)xk)Φ((x ⊗ y)i, yjxk) += ai′.kbi′j+i′i(l−1)(1−lk)ai.k bij += a(i+i′).k b(i+i′)j += Φ((x ⊗ y)i(x ⊗ y)i′ ⊗ yjxk). +The other rule follows by [3, (22), (5.2), (5.3)]: +Φ((x ⊗ y)i, yjxk)Φ(yjxk(x ⊗ y)i,yjxk yj′xk′) = Φ((x ⊗ y)i, yjxk)Φ((x ⊗ y)ilk, yj′lk+j(1−lk′ )xk′) += ai.kbijailk.k′ bilk(j′lk+j(1−lk′ )) += ai(1+l+...+lk−1)+i(lk+lk+1+...+lk+k′−1) bi(j+j′lk) += ai.(k+k′) bi(j+j′lk) += Φ((x ⊗ y)i, yj+j′lkxk+k′) += Φ((x ⊗ y)i, yjxkyj′xk′). +Thus Φ induces a homomorphism Φ∗ : G ∧ G ⊗ G → C(n,1+l+...+lm−1) ×C(n,l−1,1+l+...+lm−1), which shows that +the generators x ⊗ y ⊗ x and x ⊗ y ⊗ y are independent and have the orders divided by (n, 1 + l + ... + lm−1) and +(n, l−1, 1+l+...+lm−1), respectively. On the other hand we simply note by Proposition 1.1 and (6) that (x⊗y ⊗ +x)(n,1+l+...+lm−1) = 1. Likewise it follows from Proposition 1.1, (2) and (3) that (x ⊗ y ⊗ y)(n,l−1,1+l+...+lm−1) = 1. +Hence |x⊗ y ⊗ x| = (n, 1 + l + ...+ lm−1) and |x⊗ y ⊗ y| = (n, l − 1, 1 + l+ ...+ lm−1); from which we conclude that +G ∧ G ⊗ G ∼= C(n,1+l+...+lm−1) × C(n,l−1,1+l+...+lm−1), +as desired. Consequently as x⊗y ⊗[x, y] = x⊗y ⊗yl−1 = (x⊗y ⊗y)l−1 = 1 by (2) and (3), it follows by definition +that ∧3G ∼= G ∧ G ⊗ G. +Now we are ready to complete the proof of the main theorem in odd case by computing M(2)(G). Our method +relies on a tensor product approach due to Burns and Ellis [5]. Let γ♯ +3(G) be the quotient group ∧3G/τ(G) where +τ(G) is the normal subgroup of ∧3G generated by the elements ⟨a, b, c⟩ = ((a∧b)∧bc)((b∧c)∧ca)((c∧a)∧ab), for all +a, b, c ∈ G. By the Hall–Witt commutator identity, the homomorphism [ , , ] : ∧3G −→ G induces a homomorphism +[ , , ] : γ♯ +3(G) −→ G. It is well-known [5, Theorem 2.9] that the surjection ker([ , , ] : ∧3G −→ G) ։ M(2)(G) +given in [5, Theorem 2.6] gives rise to the natural isomorphism +M(2)(G) ∼= ker([ , , ] : γ♯ +3(G) −→ G). +(18) + +Finite split metacyclic... +12 +So in order to describe M(2)(G), first, it is required to evaluate the subgroup τ(G) of ∧3G. By invoking (2) and +(4) together with the initial relations of G ∧ G we have +⟨x, y, y⟩ = (x ∧ y ∧ yy)(y ∧ y ∧ yx)(y ∧ x ∧ xy) += (x ∧ y ∧ y)(y ∧ x ∧ yl) += (x ∧ y ∧ y)((x ∧ y)−1 ∧ y)l += (x ∧ y ∧ y)(x ∧ y ∧ y)−l += (x ∧ y ∧ y)1−l = 1. +Analogously (2), (3) and (4) imply that +⟨x, y, x⟩ = (x ∧ y ∧ yx)(y ∧ x ∧ xx)(x ∧ x ∧ xy) += (x ∧ y ∧ y1−lx)((x ∧ y)−1 ∧ x) += (x ∧ y ∧ y1−l) y1−l(x ∧ y ∧ x)(x ∧ y ∧ x)−1 += (x ∧ y ∧ y)1−l = 1. +Hence τ(G) is trivial and consequently γ♯ +3(G) ∼= ∧3G. As γ3(G) = ⟨y(l−1)2⟩, then it readily follows by (18) that +M(2)(G) ∼= C(n,l−1,1+l+...+lm−1) × C(n,1+l+...+lm−1)(n,(l−1)2)/n. +3.2 +The n even case +As can be expected, the n even case presents more difficulties in computation of ⊗3G; here Proposition 1.2 does +not give us a splitting of G ⊗ G into ∇(G) × (G ∧ G), the orders of generators are not completely determined by +the analysis in Subsection 2.2, and the identities found there are a bit more complex (e.g., compare the n odd case +with the n even case in (6) and (17)). The reason is that for the even case the subgroup ⟨x ⊗ y⟩ is not fixed under +the action of G (see [11, (9) and (10)]). +So in order to satisfy the hypotheses of [3, Proposition 10] we let G ⊗ G ∼= A × B, where A = ⟨X, Z⟩ and +B = ⟨Y, T ⟩ are the subgroups of G ⊗ G constructed by using Proposition 1.2. Thus like the discussion at the +beginning of Subsection 3.1 it follows that +⊗3G ∼= (A ⊗ G) × (B ⊗ G) ∼= Cm × C(m,n,l−1) × C(n,l−1,1+l+...+lm−1) × C(m,n,l−1,1+l+...+lm−1) × (B ⊗ G), +whence it is enough to work on the cyclic invariants of the subgroup B ⊗ G. +Assume Y iT j ∈ B and ysxt ∈ G for some integers i, j, s, t, and put Yx = Y ⊗ x, Yy = Y ⊗ y, Tx = T ⊗ x, Ty = +T ⊗y. From (8), (9), (13), (17), and the argument in the proof of (2) which shows that Yx is fixed under the action + +Finite split metacyclic... +13 +of G, we have that +Y iT j ⊗ ysxt = (Y iT j ⊗ ys) ys(Y iT j ⊗ xt) += (T j ⊗ ys)(Y i ⊗ ys) ys(T j ⊗ xt) ys(Y i ⊗ xt) += T js +y +Y is +y +ys(T j.t +x +T k +y ) ysY it +x += T js +y +Y is +y +(T j.t +x +T k+s(l−1)j.t +y +) Y it +x += Y it +x Y is +y +T j.t +x +T k+s(l−1)j.t+js +y +, +where k is the correspondence power of Ty which already occurred in (17). This tells us that B ⊗ G is generated +by the four elements Yx, Yy, Tx, and Ty. +It remains to determine the relations among these generators. As noted above, Yx is fixed under the action of G. +From Proposition 1.2, as Y n = Y 2(l−1) = 1, it follows that Y n +x = Y 2(l−1) +x += 1. Also we have 1 = Y ⊗xm = (Y ⊗x)m. +So Y (m,n,2(l−1)) +x += 1. Similarly and by (7) we obtain Y (n,l−1) +y += 1. Since T n = 1, it is readily seen by (8) that +T n +y = 1. This together with (11) and (12) conclude that T (n,4(l−1),1+l+...+lm−1) +y += 1. Furthermore, (15) gives us +Y l−1 +x += T l2−1 +y +. Now, as the relation T 1+l+...+lm−1 = A(l−1)(m−1)m/4 in Proposition 1.2 affects our discussion, we +proceed with the following two cases: +Case 1) If T 1+l+...+lm−1 = 1. +Put (i, j) = (1, m) in (17). It gives either T 1+l+...+lm−1 +x += 1 or T 1+l+...+lm−1 +x += T −(l−1)2 +y +. On the other hand, +taking (i, j) = (n, 1) in (17) it implies either T n +x = T +−( +n +2)(l−1)2 +y +or T n +x = T +−(( +n +2)+n)(l−1)2 +y +. By the fact T n +y = 1 it is +now trivial that T n +x = 1. So we obtain the last relation equivalent to +� +� +� +� +� +� +� +T (n,1+l+...+lm−1) +x += 1, +if +m ≡ 0 or 1 (mod 4) +T n +x = 1, T 1+l+...+lm−1 +x += T −(l−1)2 +y +, +if +m ≡ 2 or 3 (mod 4) +Case 2) If T 1+l+...+lm−1 ̸= 1. +If l − 1 is divisible by 4, then clearly (l − 1)(m − 1)m/4 is even. When l − 1 isn’t divisible by 4 and m ≡ +0 or 1 (mod 4), then again (l − 1)(m − 1)m/4 is even. Hence T 1+l+...+lm−1 = A(l−1)(m−1)m/4 = 1 and we are +reduced to the first case. So it only remains the case when l − 1 isn’t divisible by 4 and m ≡ 2 or 3 (mod 4). It is +obvious that (l − 1)(m − 1)m/4 is odd, whence T 1+l+...+lm−1 = Y l−1. By applying (14) we see +Y l−1 +x += Y l−1 ⊗ x = T 1+l+...+lm−1 ⊗ x = T 1+l+...+lm−1 +x +T (1+l+...+lm−1 +2 +)(l−1)2 +y +. +As (m, n, 2(l − 1)) divides l − 1, then Y l−1 +x += 1. Therefore it follows by (11) that T 1+l+...+lm−1 +x += 1. Consequently +from the argument in the first case we deduce that T (n,1+l+...+lm−1) +x += 1. + +Finite split metacyclic... +14 +Now by imposing the relation x ⊗ y ⊗ [x, y] = x ⊗ y ⊗ yl−1 = (x ⊗ y ⊗ y)l−1 = 1 to G ∧ G ⊗ G = ⟨Tx, Ty⟩, it is +readily obtained that +∧3G ∼= C(n,1+l+..+lm−1) × C(n,l−1,1+l+...+lm−1). +Finally, we evaluate the subgroup τ(G) of ∧3G. Likewise the odd case at the end of Subsection 3.1, we first observe +that ⟨x, y, y⟩ = 1. Also by invoking (13) we get +⟨x, y, x⟩ = (x ∧ y ∧ yx)(y ∧ x ∧ xx)(x ∧ x ∧ xy) += (x ∧ y ∧ y1−lx)((x ∧ y)−1 ∧ x) += (x ∧ y ∧ y1−l) y1−l(x ∧ y ∧ x)(x ∧ y ∧ x)−1 += (x ∧ y ∧ y)l(1−l) = 1. +As a consequence τG) ∼= ⟨1⟩ and then γ♯ +3(G) ∼= ∧3G. As before, the isomorphism (18) now implies that +M(2)(G) ∼= C(n,l−1,1+l+...+lm−1) × C(n,1+l+...+lm−1)(n,(l−1)2)/n. +Statements and Declarations +Competing interests: On behalf of all authors, the corresponding author states that there is no conflict of +interest. +References +[1] Baer, R.: Representations of groups as quotient groups, I, II, and III. Trans. Amer. Math. Soc. 58, 295–419 +(1945) +[2] Beyl, F.R.: The Schur multiplicator of metacyclic groups. Proc. Amer. Math. Soc. 40, 413–418 (1973) +[3] Brown, R., Johnson, D.L., Robertson, E.F.: Some computations of nonabelian tensor products of groups. J. +Algebra 111, 177–202 (1987) +[4] Brown, R., Loday, J.-L.: Van Kampen theorems for diagrams of spaces. Topology 26, 311–335 (1987) +[5] Burns, J., Ellis, G.: On the nilpotent multipliers of a group. Math. Z. 226, 405–428 (1997) +[6] Ellis, G.: The nonabelian tensor product of finite groups is finite. J. Algebra 111, 203–205 (1987) + +Finite split metacyclic... +15 +[7] Fasihi, F., Jafari, S.H.: A tensor product approach to compute 2-nilpotent multipliers of groups. Asian-Eur. +J. Math., 15(5), 2250090-1–2250090-10 (2022) +[8] Hall, P.: The classification of prime-power groups. J. Reine Angew. Math. 182, 130–141 (1940) +[9] Hall, P.: Verbal and marginal subgroups. J. Reine Angew. Math. 182, 156–157 (1940) +[10] Jafari, S.H., Davarpanah, S.M., Fasihi, F.: On the triple tensor products of groups of order p4. Comm. +Algebra, 50(6), 2672–2685 (2022) +[11] Johnson, D.L.: The nonabelian thensor square of a finite split metacyclic group. Proc. Edinburgh Math. Soc. +30, 91–96 (1987) +[12] Miller, C.: The second homology group of a group. Proc. Amer. Math. Soc. 3, 588–595 (1952) +[13] The GAP Group: +GAP—Groups, algorithms and programming. Version 4.11 (2020), Available at: +http://www.gap-system.org. +E-mail address: saeedaofi2017@gmail.com, s.hadi jafari@yahoo.com + diff --git a/N9FPT4oBgHgl3EQfmTW7/content/tmp_files/load_file.txt b/N9FPT4oBgHgl3EQfmTW7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2544d72592f79300105237da4b2913507b6fa21 --- /dev/null +++ b/N9FPT4oBgHgl3EQfmTW7/content/tmp_files/load_file.txt @@ -0,0 +1,581 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf,len=580 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='13125v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='GR] 30 Jan 2023 Finite split metacyclic groups and their 2-nilpotent multipliers S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Aofi Al-Akbi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hadi Jafari Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran January 31, 2023 Abstract There has been a great importance in understanding the nilpotent multipliers of finite groups in recent past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Let a group G be presented as the quotient of a free group F by a normal subgroup R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Given a positive integer c, the c-nilpotent multiplier of the group G is the abelian group M(c)(G) = (R ∩ γc+1(F))/γc+1(R, F), where γ1(R, F) = R, γc+1(R, F) = [γc(R, F), F], and γc+1(F) = γc+1(F, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In particular, M(1)(G) is the Schur multiplier of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The crucial aspect of the research in to the c-nilpotent multipliers of groups includes either establishing their structures, or estimating their sizes and exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' One reason for studying the c-nilpotent multiplier is its relevance to the isologism theory of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The study of Schur multiplier of finite metacyclic groups goes back to the paper by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Beyl in 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In this article, we study the 2-nilpotent multiplier of finite split metacyclic groups with the help of their nonabelian tensor squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In particular, we give a complete description of the triple tensor product, the triple exterior product, and the 2-nilpotent multiplier of such groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Keywords Metacyclic group .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Group action .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Nonabelian tensor product .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Nilpotent multiplier Mathematics Subject Classification 20F05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 20C25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 20J99 1 Definitions And Motivation A group G is said to be metacyclic if it possesses a cyclic normal subgroup N such that G/N is also cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It is clear that a metacyclic group G can be written G = HN with H ⩽ G, N ✂ G and both H and N cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' If H ∩ N = 1, then G is called the split metacyclic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It was already known to H¨older that finite metacyclic groups can be presented on two generators and three defining relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' When G = HN is split, by taking generators x of H and y of N, it is well-known that G has a presentation of the form ⟨x, y | xm = yn = 1, xyx−1 = yl⟩, (1) 1 Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 2 where the integers m, n, and l satisfying lm ≡ 1 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In the rest of the paper we will fix m, n, and l as the integers of the above presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The purpose of this paper is to show that the 2-nilpotent multiplier M(2)(G) is the direct product of two cyclic groups of orders (n, l − 1, 1 + l + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' + lm−1) and 1 n(n, 1 + l + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' + lm−1)(n, (l − 1)2), where by (r, s) we mean the greatest common divisor of the integers r and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Given a positive integer c, recall that the c-nilpotent multiplier of a group G which is presented as the quotient of a free group F by a normal subgroup R, is the abelian group M(c)(G) = (R ∩ γc+1(F))/γc+1(R, F), where γ1(R, F) = R, γc+1(R, F) = [γc(R, F), F] and γc+1(F) = γc+1(F, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It was shown by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Baer [1] that these invariants are, up to group isomorphism, independent of the choice of free presentation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The group M(G) = M(1)(G) is more known as the Schur multiplier of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' One motivation to study the c-nilpotent multipliers is its relevance to the isologism theory of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hall [8, 9] in which groups can be classified into isologism classes (see [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 406] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Another important reason is that, determining the structures of c-nilpotent multipliers is essential in studying generalized capability and covering groups, and would be generally useful in developing such a framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Furthermore, although the above formula of M(c)(G) looks simple enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' it is mostly a hard but interesting problem to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In particular, according to the method proposed in [5], which is based on the notion of triple exterior product, we hope our different approach of studying 2-nilpotent multipliers can be extended to more general cases of c-nilpotent multipliers and provides some new insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Let’s first recall the definition of the nonabelian tensor product of two groups G and H, by supposing that they act on each other satisfying certain compatibility conditions [3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 178-179], and act on themselves by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The nonabelian tensor product G ⊗ H, was introduced by Brown and Loday in [4], is defined as a group generated by the symbols g ⊗ h, where g ∈ G and h ∈ H, with defining relations gg′ ⊗ h = (gg′ ⊗ gh)(g ⊗ h) and g ⊗ hh′ = (g ⊗ h)(hg ⊗ h′h), for all g, g′ ∈ G and h, h′ ∈ H, where gg′ = gg′g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It was shown in [6] with homological methods, involving the original approach in [3, 4], that G ⊗ H is a finite group, when G and H are finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Since the compatible actions are satisfied when a group acts on itself by conjugation, it leads to define the nonabelian tensor square G⊗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The nonabelian exterior square G∧G is the quotient of G⊗G modulo the central subgroup ∇(G) generated by the elements g ⊗g for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The image of the generator g ⊗g′ in G∧G is written g ∧ g′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The commutator map induces a homomorphism [ , ] : G ∧ G −→ G mapping g ∧ g′ to [g, g′] = gg′g−1g′−1, in which its kernel is isomorphic to M(G) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The first attempt at determining the tensor square of a finite split metacyclic group (1) was made by Brown, Johnson, and Robertson [3] where they achieved the favorable special case when n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 3 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' ([3, Proposition 15]) Let G be a metacyclic group with presentation (1) and n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Then G ⊗ G is the direct product of four cyclic groups with generators x ⊗ x, y ⊗ y, (x ⊗ y)(y ⊗ x), x ⊗ y, of orders m, (n, l − 1), (n, l − 1, 1 + l + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' + lm−1), and (n, 1 + l + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' + lm−1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Furthermore, ∇(G)=⟨x ⊗ x, y ⊗ y, (x ⊗ y)(y ⊗ x)⟩ and M(G) is cyclic of order 1 n(n, l − 1)(n, 1 + l + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' + lm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Next Johnson [11] completed such a computations be evaluating G ⊗ G for even n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' ([11, Proposition 16]) Let G be a metacyclic group with presentation (1) and n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Then G⊗G is the abelian group with generators X = x⊗x, Y = y⊗y, Z = (x⊗y)(y⊗x), T = x⊗y, and A = (y⊗y)l−1, and relations Xm = Y n = Zl−1 = Zn = Z1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = T n = A2 = Am = 1, Y l−1 = A, T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = A(l−1)(m−1)m/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Furthermore, M(G) is cyclic of order 1 n(n, l − 1)(n, 1 + l + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' + lm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' With the above assumptions, one could easily observe that ∇(G) ∼= Cm × C(n,2(l−1)) × C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Here we need to give a brief recall of the notion of triple tensor (exterior) product of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The notation and terminology are the same as those in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Since conjugation yields compatible actions, there is a diagonal action of G on G ⊗ G such that g(g1 ⊗ g2) = gg1 ⊗ gg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Also the tensor square G ⊗ G acts on G by conjugation in G via the homomorphism [ , ] : G ⊗ G −→ [G, G] induced by the commutator map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Evidently these actions are compatible and we can thus construct the triple tensor product ⊗3G = (G ⊗ G) ⊗ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Also by [4, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='11] the triple exterior product ∧3G = (G ∧ G) ∧ G is obtained from the tensor product (G ∧ G) ⊗ G by imposing the additional relations (g ∧ g′) ⊗ [g, g′] = 1 for all g, g′ ∈ G (see [10] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Recently, the second author in joint work with some others [7, 10] established the explicit structures of the triple tensor (exterior) products of certain groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Like the above results, we’ll see below that although the structures of ∧3G and M(2)(G) remain invariant for any integer n, but the evaluation of ⊗3G for even n is somewhat different from the odd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Because for even case the computations depend on the relation T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = A(l−1)(m−1)m/4 which appeared above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The main result of this paper is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Let G be a finite split metacyclic group with presentation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (i) If n is odd, then ⊗3G ∼= Cm × C(n,l−1) × C2 (m,n,l−1) × C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C2 (n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(m,n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1), (ii) If n is even, then under the assumptions of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2, ⊗3G ∼= (A ⊗ Gab) × (B ⊗ G) in which A = ⟨X, Z⟩ and B = ⟨Y, T ⟩ are subgroups of G ⊗ G such that A ⊗ Gab ∼= Cm × C(m,n,l−1) × C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(m,n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1), and B ⊗ G is the abelian group Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 4 generated by Yx = Y ⊗ x, Yy = Y ⊗ y, Tx = T ⊗ x, and Ty = T ⊗ y, subject to the relations Y (m,n,2(l−1)) x = Y (n,l−1) y = T (n,4(l−1),1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) y = 1, Y l−1 x = T l2−1 y , and either � � � � � � � T (n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) x = 1, if m ≡ 0 or 1 (mod 4) T n x = 1, T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 x = T −(l−1)2 y , if m ≡ 2 or 3 (mod 4) when T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = 1, or T (n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) x = 1 when T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 ̸= 1, (iii) ∧3G ∼= C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1), (iv) M(2)(G) ∼= C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1)(n,(l−1)2)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It should be mentioned that the HAP package (Ellis, 2008) of GAP [13] has methods for checking the above results in the case when G is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' We applied them for several of metacyclic groups by performing the relevant computations (for more details see [10, Section 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In Section 2, we study some basic properties of the tensor products of groups including some relations between the generators of ⊗3G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Section 3 deals with the proof of our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 2 Computing The Triple Tensor Products This section contains some results to be used throughout the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' We start with some familiar rules of nonabelian tensor product of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' ([3, Proposition 2]) Let G and H be two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Then for all g ∈ G and h ∈ H (i) There are homomorphism of groups λ : G ⊗ H −→ G and λ′ : G ⊗ H −→ H such that λ(g ⊗ h) = ghg−1 and λ′(g ⊗ h) = ghh−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (ii) λ(t) ⊗ h = tht−1 and g ⊗ λ′(t) = gtt−1, and thus λ(t) ⊗ λ′(t1) = [t, t1] for all t, t1 ∈ G ⊗ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hence, G acts trivially on Kerλ′ and H acts trivially on Kerλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' ([3, Proposition 3]) Let G and H be two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The following relations hold in G ⊗ H for all g, g′ ∈ G and h, h′ ∈ H (note [g, h] may be interpreted as ghg−1 ∈ G or ghh−1 ∈ H): (i) g(g−1 ⊗ h) = (g ⊗ h)−1 = h(g ⊗ h−1), (ii) g′ ⊗ (ghh−1) = g′(g ⊗ h)(g ⊗ h)−1, (iii) (ghg−1) ⊗ h′ = (g ⊗ h)h′(g ⊗ h)−1, (iv) (g ⊗ h)(g′ ⊗ h′)(g ⊗ h)−1 = [g,h](g′ ⊗ h′), Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 5 (v) [g ⊗ h, g′ ⊗ h′] = (ghg−1) ⊗ (g′h′h′−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' For all a, b, c, d ∈ G the following relations hold in G ⊗ G and ⊗3G (to simplify the notation we shall identify b ⊗ c ⊗ d with (b ⊗ c) ⊗ d in ⊗3G): (i) a(b ⊗ c) = (a ⊗ [b, c])(b ⊗ c), (ii) a(b ⊗ c ⊗ d) = (b ⊗ c ⊗ d)(d ⊗ [b, c] ⊗ a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In next two subsections, given the metacyclic group G with presentation (1), we list some useful relations for the elements of ⊗3G proceeding in a series of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' We start with the case that n is odd which is relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Note by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2 that [a⊗b⊗c, a′⊗b′⊗c′] = [[a, b]⊗c, [a′, b′, c′]], for all a, b, c, a′, b′, c′ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Now by setting G = G ⊗ G and H = G in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1, it tells us immediately that Ker(λ′ : ⊗3G → G) is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Since Imλ′ = γ3(G) is cyclic, it follows that ⊗3G is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1 The n odd case Throughout this section we let G be an arbitrary finite split metacyclic group as in (1) and n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' y(x ⊗ y ⊗ y) = x(x ⊗ y ⊗ y) = x ⊗ y ⊗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Taking G = G ⊗ G and H = G in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1 we get the homomorphism λ : ⊗3G → G ⊗ G so that λ(a ⊗ b ⊗ c) = (a ⊗ b)c(a ⊗ b)−1, for all a, b, c ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In addition, it follows from the Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2(iii) and [3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1)] that x ⊗ y ⊗ y ∈ Kerλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Thus Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1 concludes the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (x ⊗ y ⊗ y)l−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It follows from [3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='3)] that (x ⊗ y)x(x ⊗ y)−1 = (x ⊗ y)1−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' According to the defining actions in ⊗3G, as x⊗yxx−1 = [x, y, x] = [yl−1, x] = y−(l−1)2, then (x ⊗ y ⊗ y)(l−1)3 = (x ⊗ y)1−l ⊗ y−(l−1)2 = (x ⊗ y) x(x ⊗ y)−1 ⊗ (x⊗y)xx−1 = [x ⊗ y ⊗ x, x ⊗ y ⊗ x] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' On the other hand from (2) and [3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='3)] we have (x ⊗ y ⊗ y) = x(x ⊗ y ⊗ y) = x(x ⊗ y) ⊗ xy = (x ⊗ y)l ⊗ yl = (x ⊗ y ⊗ y)l2, which implies that (x ⊗ y ⊗ y)l2−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hence 1 = (x ⊗ y ⊗ y)(l−1)3−(l2−1) = (x ⊗ y ⊗ y)l3−4l2+3l = (x ⊗ y ⊗ y)4(l−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 6 Now since n is odd, it finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' y(x ⊗ y ⊗ x) = x ⊗ y ⊗ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' From [3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='3)] and (3) we have that y(x ⊗ y ⊗ x) = ((x ⊗ y) ⊗ yx) = ((x ⊗ y) ⊗ y1−lx) = (x ⊗ y ⊗ y1−l)y1−l(x ⊗ y ⊗ x) = (x ⊗ y ⊗ y)1−l y1−l(x ⊗ y ⊗ x) = y1−l(x ⊗ y ⊗ x), which implies x ⊗ y ⊗ x is fixed by yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In addition, as l and n are relatively prime, there exist integers s, t such that sl + tn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hence y(x ⊗ y ⊗ x) = ysl+tn(x ⊗ y ⊗ x) = ysl(x ⊗ y ⊗ x) = x ⊗ y ⊗ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' x(x ⊗ y ⊗ x) = (x ⊗ y ⊗ x)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' By applying [3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='3)] we see that x(x ⊗ y ⊗ x) = (x ⊗ y)l ⊗ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Now the result follows by induction on l together with using (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Setting i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j = i(1 + l + l2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' + lj−1), then (x ⊗ y)i ⊗ xj = (x ⊗ y ⊗ x)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j and (x ⊗ y)i ⊗ yk = (x ⊗ y ⊗ y)ik, for any integers i, j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It follows by applying [3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2)], (2), (4), and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2 The n even case In this subsection we are concerned with the case that G is a finite split metacyclic group as in (1) and n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' As we’ll see, this case is slightly more complicated with respect to the odd case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' For instance, the order of y ⊗y ⊗y would be discussed here while in odd case it is easily determined as a direct factor of the abelian group ∇(G)⊗Gab (see Section 3 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (y ⊗ y ⊗ y)l−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It is readily seen that x(y ⊗ y ⊗ y) = y ⊗ y ⊗ yl = (y ⊗ y ⊗ y)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' In addition, since y ⊗ y ⊗ y belongs to Kerλ given in the proof of (2), it follows that x(y ⊗ y ⊗ y) = y ⊗ y ⊗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The result now follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 7 y(x ⊗ y ⊗ y) = x ⊗ y ⊗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Using [11, (9)] and (7), y(x ⊗ y ⊗ y) = y(x ⊗ y) ⊗ y = (x ⊗ y)(y ⊗ y)l−1 ⊗ y = x⊗y((y ⊗ y)l−1 ⊗ y)(x ⊗ y ⊗ y) = (x ⊗ y ⊗ y)(y ⊗ y ⊗ y)l−1 = x ⊗ y ⊗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (x ⊗ y)i ⊗ yj = (x ⊗ y ⊗ y)ij, for any integers i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It can be proved by induction on i together with using (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' x(x ⊗ y ⊗ y) = (x ⊗ y ⊗ y)l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Invoking [11, (10)], (8) and (9), x(x ⊗ y ⊗ y) = x(x ⊗ y) ⊗ xy = (x ⊗ y)l(y ⊗ y)−l(l−1)2/2 ⊗ yl = (x⊗y)l((y ⊗ y)−l(l−1)2/2 ⊗ yl)((x ⊗ y)l ⊗ yl) = (y ⊗ y ⊗ y)−l2(l−1)2/2(x ⊗ y ⊗ y)l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The result follows by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (x ⊗ y ⊗ y)1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Clearly the assertion holds when (x⊗ y)1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' If (x⊗ y)1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 ̸= 1, then by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2 as A2 = 1 we see (x⊗ y)1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = (y ⊗ y)l−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hence (x⊗ y)1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 ⊗ y = (y ⊗ y)l−1 ⊗ y, and consequently (7) concludes the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (x ⊗ y ⊗ y)4(l−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (12) Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' As (y⊗y)2(l−1) = 1 by [11, (15)], then it follows that y fixes (x⊗y)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' So the element (x⊗y⊗y)2 = (x⊗y)2⊗y belongs to Kerλ, where λ is the homomorphism given in the proof of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Now Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1(ii) implies the group G acts on (x ⊗ y ⊗ y)2 trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Consequently it follows by (10) that (x ⊗ y ⊗ y)2 = x(x ⊗ y ⊗ y)2 = (x ⊗ y ⊗ y)2l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Therefore (x⊗ y ⊗ y)2(l2−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' On the other hand, from [11, (10)] and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2 we have (x⊗ y)x(x⊗ y)−1 =(x ⊗ y)1−l(y ⊗ y)(l−1)2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Thus (x ⊗ y)1−l = (x ⊗ y)x(x ⊗ y)−1(y ⊗ y)−(l−1)2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hence by applying (7) we have (x ⊗ y ⊗ y)(l−1)3 = (x ⊗ y)1−l ⊗ y−(l−1)2 = ((x ⊗ y)x(x ⊗ y)−1(y ⊗ y)−(l−1)2/2) ⊗ y−(l−1)2 = ((y ⊗ y)−(l−1)2/2 ⊗ y−(l−1)2)((x ⊗ y)x(x ⊗ y)−1 ⊗ y−(l−1)2) = (y ⊗ y ⊗ y) (l−1)2 2 −(l−1)2((x ⊗ y)x(x ⊗ y)−1 ⊗ x⊗yxx−1) = [x ⊗ y ⊗ x, x ⊗ y ⊗ x] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Combining two last equations leads to the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' y(x ⊗ y ⊗ x) = (x ⊗ y ⊗ x)(x ⊗ y ⊗ y)l−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='3, [11, (11)] and (7), y(x ⊗ y ⊗ x) = ((x ⊗ [x, y] ⊗ y)(x ⊗ y ⊗ x) = [(x ⊗ y)l−1(y ⊗ y)−(l−1)2(l−2)/2 ⊗ y](x ⊗ y ⊗ x) = (x ⊗ y ⊗ x)(x ⊗ y ⊗ y)l−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (x ⊗ y)i ⊗ x = (x ⊗ y ⊗ x)i(x ⊗ y ⊗ y)( i 2)(l−1)2, for any integer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It follows by induction on i, [3, (22)], (9), and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (x ⊗ y ⊗ y)l2−1 = (y ⊗ y ⊗ x)l−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Use (10) and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='3, (x ⊗ y ⊗ y)l2 = x(x ⊗ y ⊗ y) = (y ⊗ [x, y] ⊗ x)(x ⊗ y ⊗ y) = (y ⊗ y ⊗ x)l−1(x ⊗ y ⊗ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' x(x ⊗ y ⊗ x) = (x ⊗ y ⊗ x)l(x ⊗ y ⊗ y)(l−1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (16) Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' First note that G acts trivially on y ⊗ y ⊗ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It follows by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='3, [11, (11)], (14), and (15) that x(x ⊗ y ⊗ x) = (x ⊗ [x, y] ⊗ x)(x ⊗ y ⊗ x) = [(x ⊗ y)l−1(y ⊗ y)−(l−1)2(l−2)/2 ⊗ x](x ⊗ y ⊗ x) = (y ⊗ y ⊗ x)−(l−1)2(l−2)/2((x ⊗ y)l−1 ⊗ x)(x ⊗ y ⊗ x) = (y ⊗ y ⊗ x)−(l−1)2(l−2)/2(x ⊗ y ⊗ x)l−1(x ⊗ y ⊗ y)(l−2)(l−1)3/2(x ⊗ y ⊗ x) = (x ⊗ y ⊗ x)l(x ⊗ y ⊗ y)−(l−2)(l−1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' So the result is obviously true by applying the last equation used in the proof of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' For any integers i and j we have: (x ⊗ y)i ⊗ xj = � � � � � (x ⊗ y ⊗ x)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j (x ⊗ y ⊗ y)(i 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j(l−1)2 if j ≡ 0 or 1 (mod 4) (x ⊗ y ⊗ x)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j (x ⊗ y ⊗ y)((i 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j+i)(l−1)2 if j ≡ 2 or 3 (mod 4) (17) where i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j = i(1 + l + l2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' + lj−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' We proceed by induction on both i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Assuming i = 1, it follows by induction on j together with (16) that x ⊗ y ⊗ xj = � � � � � (x ⊗ y ⊗ x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j if j ≡ 0 or 1 (mod 4) (x ⊗ y ⊗ x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j (x ⊗ y ⊗ y)(l−1)2 if j ≡ 2 or 3 (mod 4) It suffices to prove the second part of the assertion and the first is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Using (13), l − 1 times, we have (x ⊗ y)i+1 ⊗ xj = x⊗y((x ⊗ y)i ⊗ xj)(x ⊗ y ⊗ xj) = x⊗y((x ⊗ y ⊗ x)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j(x ⊗ y ⊗ y)((i 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j+i)(l−1)2)(x ⊗ y ⊗ x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j(x ⊗ y ⊗ y)(l−1)2 = (x ⊗ y ⊗ x)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j(x ⊗ y ⊗ y)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j(l−1)2(x ⊗ y ⊗ y)((i 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j+i)(l−1)2(x ⊗ y ⊗ x)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j(x ⊗ y ⊗ y)(l−1)2 = (x ⊗ y ⊗ x)(i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j(x ⊗ y ⊗ y)(( i+1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='j+i+1)(l−1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 3 Proof of The Main Theorem In this section we prove the main theorem of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The proof relies on the previous section together with a computation method based on the crossed pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' There is a significant difference between the even and odd case, and w’ll treat them separately beginning with the odd case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1 The n odd case Let G be a finite split metacyclic group as in (1) and n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' We observe from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1 that in this case G ⊗ G splits as G ⊗ G ∼= ∇(G) × (G ∧ G) ∼= Cm × C(n,l−1) × C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1), where G ∧ G is isomorphic to the subgroup of G ⊗ G generated by ⟨x ⊗ y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' This is not the case in general when n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The groups G ∧ G and G act on each other in such a way that G ∧ G ∼= ⟨x ⊗ y⟩ is fixed under the action of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' So by applying [3, Proposition 10] one could describe ⊗3G as follows: ⊗3G ∼= (∇(G) × (G ∧ G)) ⊗ G ∼= (∇(G) ⊗ G) × (G ∧ G ⊗ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Now since the groups ∇(G) and G act on each other trivially, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='4 in [4] allows us to express ∇(G) ⊗ G as the tensor product ∇(G) ⊗ Gab of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Therefore by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1 we conclude that ∇(G) ⊗ Gab ∼= Cm × C(n,l−1) × C2 (m,n,l−1) × C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(m,n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The main task then is to determine the cyclic invariants of G ∧ G ⊗ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' For this purpose first we expand the arbitrary element (x ⊗ y)i⊗ yjxk of G ∧ G ⊗ G by invoking (4) and (6): (x ⊗ y)i ⊗ yjxk = ((x ⊗ y)i ⊗ yj) yj((x ⊗ y)i ⊗ xk) = (x ⊗ y ⊗ y)ij yj(x ⊗ y ⊗ x)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='k = (x ⊗ y ⊗ y)ij(x ⊗ y ⊗ x)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' This establishes what the generators of G ∧ G ⊗ G are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' To find the orders of these generators, and any possible relations among them, by considering our analysis of ⊗3G in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1, we use crossed pairing into suitable cyclic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Recall that for groups G, H and L where G and H acting upon each other compatibly and acting upon themselves by conjugation, a function Φ : G × H −→ L is called a crossed pairing if Φ(gg′, h) = Φ(gg′,g h)Φ(g, h) and Φ(g, hh′) = Φ(g, h)Φ(hg,h h′), for all g, g′ ∈ G, h, h′ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Clearly any crossed pairing Φ : G × H −→ L determines a unique homomorphism Φ∗ : G ⊗ H −→ L such that Φ∗(g ⊗ h) = Φ(g, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Now define Φ : (G ∧ G) × G → C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) by Φ((x ⊗ y)i, yjxk) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='k bij and let ((x⊗y)i, yjxk) = ((x⊗y)i′, yj′xk′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Then with setting r = (n, 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1), it follows that i′ = i+ur, j′ = j+vn, and k′ = k + wm, for some integers u, v, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' So ai′k′bi′j′ = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='k bij, which implies that Φ is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' We prove Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 11 that Φ is a crossed pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The following verifies the first rule for a crossed pairing: Φ((x⊗y)i(x ⊗ y)i′, (x⊗y)iyjxk)Φ((x ⊗ y)i, yjxk) = Φ((x ⊗ y)i′, yj+i(l−1)(1−lk)xk)Φ((x ⊗ y)i, yjxk) = ai′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='kbi′j+i′i(l−1)(1−lk)ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='k bij = a(i+i′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='k b(i+i′)j = Φ((x ⊗ y)i(x ⊗ y)i′ ⊗ yjxk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The other rule follows by [3, (22), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='3)]: Φ((x ⊗ y)i, yjxk)Φ(yjxk(x ⊗ y)i,yjxk yj′xk′) = Φ((x ⊗ y)i, yjxk)Φ((x ⊗ y)ilk, yj′lk+j(1−lk′ )xk′) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='kbijailk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='k′ bilk(j′lk+j(1−lk′ )) = ai(1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lk−1)+i(lk+lk+1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lk+k′−1) bi(j+j′lk) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (k+k′) bi(j+j′lk) = Φ((x ⊗ y)i, yj+j′lkxk+k′) = Φ((x ⊗ y)i, yjxkyj′xk′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Thus Φ induces a homomorphism Φ∗ : G ∧ G ⊗ G → C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) ×C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1), which shows that the generators x ⊗ y ⊗ x and x ⊗ y ⊗ y are independent and have the orders divided by (n, 1 + l + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' + lm−1) and (n, l−1, 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' On the other hand we simply note by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1 and (6) that (x⊗y ⊗ x)(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Likewise it follows from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1, (2) and (3) that (x ⊗ y ⊗ y)(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hence |x⊗ y ⊗ x| = (n, 1 + l + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+ lm−1) and |x⊗ y ⊗ y| = (n, l − 1, 1 + l+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+ lm−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' from which we conclude that G ∧ G ⊗ G ∼= C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Consequently as x⊗y ⊗[x, y] = x⊗y ⊗yl−1 = (x⊗y ⊗y)l−1 = 1 by (2) and (3), it follows by definition that ∧3G ∼= G ∧ G ⊗ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Now we are ready to complete the proof of the main theorem in odd case by computing M(2)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Our method relies on a tensor product approach due to Burns and Ellis [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Let γ♯ 3(G) be the quotient group ∧3G/τ(G) where τ(G) is the normal subgroup of ∧3G generated by the elements ⟨a, b, c⟩ = ((a∧b)∧bc)((b∧c)∧ca)((c∧a)∧ab), for all a, b, c ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' By the Hall–Witt commutator identity, the homomorphism [ , , ] : ∧3G −→ G induces a homomorphism [ , , ] : γ♯ 3(G) −→ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It is well-known [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='9] that the surjection ker([ , , ] : ∧3G −→ G) ։ M(2)(G) given in [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='6] gives rise to the natural isomorphism M(2)(G) ∼= ker([ , , ] : γ♯ 3(G) −→ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' (18) Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 12 So in order to describe M(2)(G), first, it is required to evaluate the subgroup τ(G) of ∧3G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' By invoking (2) and (4) together with the initial relations of G ∧ G we have ⟨x, y, y⟩ = (x ∧ y ∧ yy)(y ∧ y ∧ yx)(y ∧ x ∧ xy) = (x ∧ y ∧ y)(y ∧ x ∧ yl) = (x ∧ y ∧ y)((x ∧ y)−1 ∧ y)l = (x ∧ y ∧ y)(x ∧ y ∧ y)−l = (x ∧ y ∧ y)1−l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Analogously (2), (3) and (4) imply that ⟨x, y, x⟩ = (x ∧ y ∧ yx)(y ∧ x ∧ xx)(x ∧ x ∧ xy) = (x ∧ y ∧ y1−lx)((x ∧ y)−1 ∧ x) = (x ∧ y ∧ y1−l) y1−l(x ∧ y ∧ x)(x ∧ y ∧ x)−1 = (x ∧ y ∧ y)1−l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hence τ(G) is trivial and consequently γ♯ 3(G) ∼= ∧3G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' As γ3(G) = ⟨y(l−1)2⟩, then it readily follows by (18) that M(2)(G) ∼= C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1)(n,(l−1)2)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2 The n even case As can be expected, the n even case presents more difficulties in computation of ⊗3G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' here Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2 does not give us a splitting of G ⊗ G into ∇(G) × (G ∧ G), the orders of generators are not completely determined by the analysis in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2, and the identities found there are a bit more complex (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=', compare the n odd case with the n even case in (6) and (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' The reason is that for the even case the subgroup ⟨x ⊗ y⟩ is not fixed under the action of G (see [11, (9) and (10)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' So in order to satisfy the hypotheses of [3, Proposition 10] we let G ⊗ G ∼= A × B, where A = ⟨X, Z⟩ and B = ⟨Y, T ⟩ are the subgroups of G ⊗ G constructed by using Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Thus like the discussion at the beginning of Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1 it follows that ⊗3G ∼= (A ⊗ G) × (B ⊗ G) ∼= Cm × C(m,n,l−1) × C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(m,n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × (B ⊗ G), whence it is enough to work on the cyclic invariants of the subgroup B ⊗ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Assume Y iT j ∈ B and ysxt ∈ G for some integers i, j, s, t, and put Yx = Y ⊗ x, Yy = Y ⊗ y, Tx = T ⊗ x, Ty = T ⊗y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' From (8), (9), (13), (17), and the argument in the proof of (2) which shows that Yx is fixed under the action Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 13 of G, we have that Y iT j ⊗ ysxt = (Y iT j ⊗ ys) ys(Y iT j ⊗ xt) = (T j ⊗ ys)(Y i ⊗ ys) ys(T j ⊗ xt) ys(Y i ⊗ xt) = T js y Y is y ys(T j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='t x T k y ) ysY it x = T js y Y is y (T j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='t x T k+s(l−1)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='t y ) Y it x = Y it x Y is y T j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='t x T k+s(l−1)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='t+js y , where k is the correspondence power of Ty which already occurred in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' This tells us that B ⊗ G is generated by the four elements Yx, Yy, Tx, and Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It remains to determine the relations among these generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' As noted above, Yx is fixed under the action of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' From Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2, as Y n = Y 2(l−1) = 1, it follows that Y n x = Y 2(l−1) x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Also we have 1 = Y ⊗xm = (Y ⊗x)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' So Y (m,n,2(l−1)) x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Similarly and by (7) we obtain Y (n,l−1) y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Since T n = 1, it is readily seen by (8) that T n y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' This together with (11) and (12) conclude that T (n,4(l−1),1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Furthermore, (15) gives us Y l−1 x = T l2−1 y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Now, as the relation T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = A(l−1)(m−1)m/4 in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='2 affects our discussion, we proceed with the following two cases: Case 1) If T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Put (i, j) = (1, m) in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It gives either T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 x = 1 or T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 x = T −(l−1)2 y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' On the other hand, taking (i, j) = (n, 1) in (17) it implies either T n x = T −( n 2)(l−1)2 y or T n x = T −(( n 2)+n)(l−1)2 y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' By the fact T n y = 1 it is now trivial that T n x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' So we obtain the last relation equivalent to � � � � � � � T (n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) x = 1, if m ≡ 0 or 1 (mod 4) T n x = 1, T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 x = T −(l−1)2 y , if m ≡ 2 or 3 (mod 4) Case 2) If T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' If l − 1 is divisible by 4, then clearly (l − 1)(m − 1)m/4 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' When l − 1 isn’t divisible by 4 and m ≡ 0 or 1 (mod 4), then again (l − 1)(m − 1)m/4 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Hence T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = A(l−1)(m−1)m/4 = 1 and we are reduced to the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' So it only remains the case when l − 1 isn’t divisible by 4 and m ≡ 2 or 3 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' It is obvious that (l − 1)(m − 1)m/4 is odd, whence T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 = Y l−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' By applying (14) we see Y l−1 x = Y l−1 ⊗ x = T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 ⊗ x = T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 x T (1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 2 )(l−1)2 y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' As (m, n, 2(l − 1)) divides l − 1, then Y l−1 x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Therefore it follows by (11) that T 1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1 x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Consequently from the argument in the first case we deduce that T (n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Finite split metacyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' 14 Now by imposing the relation x ⊗ y ⊗ [x, y] = x ⊗ y ⊗ yl−1 = (x ⊗ y ⊗ y)l−1 = 1 to G ∧ G ⊗ G = ⟨Tx, Ty⟩, it is readily obtained that ∧3G ∼= C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='.+lm−1) × C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Finally, we evaluate the subgroup τ(G) of ∧3G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Likewise the odd case at the end of Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='1, we first observe that ⟨x, y, y⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Also by invoking (13) we get ⟨x, y, x⟩ = (x ∧ y ∧ yx)(y ∧ x ∧ xx)(x ∧ x ∧ xy) = (x ∧ y ∧ y1−lx)((x ∧ y)−1 ∧ x) = (x ∧ y ∧ y1−l) y1−l(x ∧ y ∧ x)(x ∧ y ∧ x)−1 = (x ∧ y ∧ y)l(1−l) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' As a consequence τG) ∼= ⟨1⟩ and then γ♯ 3(G) ∼= ∧3G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' As before, the isomorphism (18) now implies that M(2)(G) ∼= C(n,l−1,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1) × C(n,1+l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='+lm−1)(n,(l−1)2)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' Statements and Declarations Competing interests: On behalf of all authors, the corresponding author states that there is no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=' References [1] Baer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content=': Representations of groups as quotient groups, I, II, and III.' metadata={'source': 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s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='hadi jafari@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FPT4oBgHgl3EQfmTW7/content/2301.13125v1.pdf'} diff --git a/NNAyT4oBgHgl3EQfUPdn/content/tmp_files/2301.00121v1.pdf.txt b/NNAyT4oBgHgl3EQfUPdn/content/tmp_files/2301.00121v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6fd4eadd852ab669911d9348a4491cc660d67209 --- /dev/null +++ b/NNAyT4oBgHgl3EQfUPdn/content/tmp_files/2301.00121v1.pdf.txt @@ -0,0 +1,1942 @@ +arXiv:2301.00121v1 [math.QA] 31 Dec 2022 +Circular bidiagonal pairs +Paul Terwilliger and Arjana ˇZitnik +Abstract +A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry +is on the diagonal, or the subdiagonal, or in the top-right corner; (ii) each subdiagonal +entry is nonzero, and the entry in the top-right corner is nonzero. Let F denote a field, +and let V denote a nonzero finite-dimensional vector space over F. We consider an +ordered pair of F-linear maps A : V → V and A∗ : V → V that satisfy the following +two conditions: +• there exists a basis for V with respect to which the matrix representing A is +circular bidiagonal and the matrix representing A∗ is diagonal; +• there exists a basis for V with respect to which the matrix representing A∗ is +circular bidiagonal and the matrix representing A is diagonal. +We call such a pair a circular bidiagonal pair on V . We classify the circular bidiagonal +pairs up to affine equivalence. There are two infinite families of solutions, which we +describe in detail. +Keywords. Bidiagonal pair; Hessenberg pair; Leonard pair; tridiagonal pair. +2020 Mathematics Subject Classification. Primary: 17B37; Secondary: 15A21. +1 +Introduction +In a celebrated paper [2], Richard Askey and James Wilson introduced the q-Racah family +of orthogonal polynomials. In [26], Doug Leonard showed that the q-Racah polynomials +are the most general orthogonal polynomials that have orthogonal polynomial duals. In [3, +Theorem 5.1], Eiichi Bannai and Tatsuro Ito gave a comprehensive version of Leonard’s +theorem. In an effort to clarify and simplify the Leonard theorem, in [29] the first author +introduced the concept of a Leonard pair. Roughly speaking, a Leonard pair consists of two +diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on +an eigenbasis of the other one in an irreducible tridiagonal fashion. In [29, Definition 1.4] +there appears an “oriented” version of a Leonard pair, called a Leonard system. In [29, +Theorem 1.9] the Leonard systems are classified up to isomorphism. The article [36] contains +a modern treatment of this classification, along with a detailed account of the history. By +[29, Theorem 1.12] a Leonard pair satisfies two polynomial relations called the tridiagonal +relations. Some notable papers about Leonard pairs are [30–35]. +In [19] Tatsuro Ito, Kenichiro Tanabe, and the first author introduced the concept of a +tridiagonal pair as a generalization of a Leonard pair. The concept of a tridiagonal system +1 + +was also introduced. In [18, Corollary 18.1] the tridiagonal systems over an algebraically +closed field are classified up to isomorphism. In [24, Section 1.4] it is shown how a tridiagonal +system induces a tensor product factorization of the underlying vector space. In [22,23] the +tridiagonal pairs are related to some finite-dimensional irreducible modules for Uq(� +sl2). It +is shown in [19, Theorem 10.1] that every tridiagonal pair satisfies the tridiagonal relations. +Some notable papers about tridiagonal pairs are [5–9,20,21,27]. +Over the past 20 years, there appeared in the literature some variations on the Leonard pair +and tridiagonal pair concepts. In the next few paragraphs, we summarize these variations. +In [14] Ali Godjali introduced the concept of a Hessenberg pair as a generalization of a +tridiagonal pair. +He showed in [14, Corollary 1.9] that every Hessenberg pair induces a +split decomposition of the underlying vector space. In [15] Godjali considers a special case +of Hessenberg pair, called a TH pair. He defines a TH system, and classifies these up to +isomorphism [15, Theorem 6.3]. In [16, Section 18] the TH systems are characterized in +terms of West/South Vandermonde matrices. +In [10] Darren Funk-Neubauer introduced the concept of a bidiagonal pair as a variation on a +tridiagonal pair. In [10, Theorem 5.1] the bidiagonal pairs are classified up to isomorphism. +In [10, Theorems 5.10, 5.11] this classification is interpreted using the equitable presentations +of sl2 and Uq(sl2). In [11] Funk-Neubauer introduces the concept of a bidiagonal triple. +In [11, Theorem 4.1] he shows how every bidiagonal pair extends to a bidiagonal triple. +In [11, Theorem 4.3] the bidiagonal triples are classified up to isomorphism. See [12] for +related work. +In [4] Pascal Baseilhac, Azat Gainutdinov, and Thao Vu introduced the concept of a cyclic +tridiagonal pair, as a generalization of a tridiagonal pair. They used cyclic tridiagonal pairs +to study a higher-order generalization of the Onsager algebra. In [4, Appendix A] some +examples of cyclic tridiagonal pairs are given. It remains an open problem to classify the +cyclic tridiagonal pairs up to isomorphism. +In [25] Jae-ho Lee introduced the concept of a circular Hessenberg pair. This is a special +case of a TH pair, and also a special case of a cyclic tridiagonal pair. In [25, Theorem 5.6] +the circular Hessenberg pairs are classified under the assumption that the pair satisfies the +tridiagonal relations. The classification yields four infinite families of solutions [25, Exam- +ples 5.1–5.4]. +In the present paper, we introduce the concept of a circular bidiagonal pair. +This is a +variation on a bidiagonal pair, and a special case of a circular Hessenberg pair. The reason +we focus on this special case, is that it affords a classification without assuming in advance +that the tridiagonal relations are satisfied. We will display two infinite families of circular +bidiagonal pairs. We will introduce the notion of affine equivalence. Our main result is that +every circular bidiagonal pair is affine equivalent to a member of one of the two families. In +the next section, we formally define a circular bidiagonal pair and give a detailed statement +of our results. +2 + +2 +Definitions and statement of results +In this section, we introduce the concept of a circular bidiagonal pair. To define the concept, +we first explain what it means for a square matrix to be circular bidiagonal. The following +matrices are circular bidiagonal: + + + + +3 +0 +0 +1 +1 +4 +0 +0 +0 +−1 +1 +0 +0 +0 +−1 +2 + + + + , + + + + +2 +0 +0 +−1 +−1 +3 +0 +0 +0 +1 +0 +0 +0 +0 +−1 +−1 + + + + , + + + + +0 +0 +0 +1 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 + + + + . +Circular bidiagonal means (i) each nonzero entry is on the diagonal, or the subdiagonal, or +in the top-right corner; (ii) each subdiagonal entry is nonzero, and the entry in the top-right +corner is nonzero. +Next, we define a circular bidiagonal pair. For the rest of this paper, F denotes a field. +Definition 2.1. Let V denote a nonzero vector space over F with finite dimension. By a +circular bidiagonal pair on V , we mean an ordered pair of F-linear maps A : V → V and +A∗ : V → V that satisfy the following two conditions: +(i) there exists a basis for V with respect to which the matrix representing A is circular +bidiagonal and the matrix representing A∗ is diagonal; +(ii) there exists a basis for V with respect to which the matrix representing A∗ is circular +bidiagonal and the matrix representing A is diagonal. +Definition 2.2. The circular bidiagonal pair in Definition 2.1 is said to be over F. +Definition 2.3. Referring to Definition 2.1, assume that A, A∗ is a circular bidiagonal pair +on V . Then the pair A∗, A is a circular bidiagonal pair on V , called the dual of A, A∗. +Next, we give some examples of circular bidiagonal pairs. Our first example is elementary. +Let V denote a vector space over F that has dimension one. Then any ordered pair of F-linear +maps A : V → V and A∗ : V → V is a circular bidiagonal pair on V . +Our next example is more substantial. Consider the vector space V = F5 (column vectors). +Assume that q ∈ F is a primitive 5th root of unity. Consider the matrices +A = + + + + + + +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 + + + + + + +, +A∗ = diag(1, q, q2, q3, q4). +These matrices satisfy +A5 = I, +(A∗)5 = I, +A∗A = qAA∗, +3 + +where I denotes the identity matrix. We claim that the pair A, A∗ acts on V as a circular +bidiagonal pair. To see this, we check that A, A∗ satisfy the conditions in Definition 2.1. The +matrix A is circular bidiagonal and the matrix A∗ is diagonal. Therefore, condition (i) in +Definition 2.1 is satisfied by the basis for V consisting of the columns of I. Define a matrix +P = + + + + + + +1 +1 +1 +1 +1 +1 +q +q2 +q3 +q4 +1 +q2 +q4 +q6 +q8 +1 +q3 +q6 +q9 +q12 +1 +q4 +q8 +q12 +q16 + + + + + + +. +The matrix P is Vandermonde, and hence invertible. One checks that A∗P = PA. In this +equation, take the transpose of each side to obtain PA∗ = A−1P. Rearranging this equation, +we obtain AP = P(A∗)−1. These results show that condition (ii) of Definition 2.1 is satisfied +by the basis for V consisting of the columns of P. We have shown that the pair A, A∗ acts +on V as a circular bidiagonal pair. +The previous circular bidiagonal pair is a member of an infinite family of circular bidiagonal +pairs. Before describing this family, we bring in some notation. For the rest of this paper, +every vector space and algebra mentioned is understood to be over F. Pick an integer d ≥ 1. +Let Matd+1(F) denote the algebra consisting of the d+1 by d+1 matrices that have all entries +in F. We index the rows and columns by 0, 1, 2, . . . , d. Let Fd+1 denote the vector space +consisting of the column vectors that have d + 1 coordinates and all entries in F. We index +the coordinates by 0, 1, 2, . . ., d. Note that Matd+1(F) acts on Fd+1 by left multiplication. +Let I ∈ Matd+1(F) denote the identity matrix. +Lemma 2.4. Pick an integer d ≥ 1, and consider the vector space V = Fd+1. Assume that +q ∈ F is a primitive nth root of unity, where n = d + 1. Define matrices A, A∗ ∈ Matd+1(F) +as follows. We have A0,d = 1, and Ai,i−1 = 1 for 1 ≤ i ≤ d. All other entries of A are zero. +The matrix A∗ is diagonal, with A∗ +i,i = qi for 0 ≤ i ≤ d. Then the pair A, A∗ acts on V as a +circular bidiagonal pair. Moreover +An = I, +(A∗)n = I, +A∗A = qAA∗. +(1) +Proof. The relations (1) are readily checked. Define a matrix P ∈ Matd+1(F) that has (i, j)- +entry qij for 0 ≤ i, j ≤ d. The matrix P is Vandermonde, and hence invertible. One checks +that A∗P = PA and AP = P(A∗)−1. Consequently, the pair A, A∗ acts on V as a circular +bidiagonal pair. +Note 2.5. The relation on the right in (1) is a defining relation for the quantum torus +algebra; see for example [17]. +For the next example, we return to the vector space V = F5. Assume that q ∈ F is a primitive +5th root of unity. Pick ε ∈ F that is not among 1, q, q2, q3, q4. Consider the matrices +A = + + + + + + +ε +0 +0 +0 +1 − ε +1 − q−1ε +q−1ε +0 +0 +0 +0 +1 − q−2ε +q−2ε +0 +0 +0 +0 +1 − q−3ε +q−3ε +0 +0 +0 +0 +1 − q−4ε +q−4ε + + + + + + +, +A∗ = diag(1, q, q2, q3, q4). +4 + +One checks that +A5 = I, +(A∗)5 = I, +qAA∗ − A∗A +q − 1 += εI. +We will show that the pair A, A∗ acts on V as a circular bidiagonal pair. This is a special +case of the following result. +Lemma 2.6. Pick an integer d ≥ 1, and consider the vector space V = Fd+1. Assume that +q ∈ F is a primitive nth root of unity, where n = d + 1. Pick ε ∈ F that is not among +1, q, q2, . . . , qd. Define a matrix A = A(q, ε) in Matd+1(F) as follows. We have Ai,i = q−iε for +0 ≤ i ≤ d. We have A0,d = 1 − ε, and Ai,i−1 = 1 − q−iε for 1 ≤ i ≤ d. All other entries of A +are zero. We define a diagonal matrix A∗ = A∗(q) in Matd+1(F) with A∗ +i,i = qi for 0 ≤ i ≤ d. +Then the pair A, A∗ acts on V as a circular bidiagonal pair. Moreover +An = I, +(A∗)n = I, +qAA∗ − A∗A +q − 1 += εI. +(2) +Proof. Define a matrix P = P(q, ε) in Matd+1(F) with (i, j)-entry +Pi,j = qij (εq−i; q)j +(εq; q)j +(0 ≤ i, j ≤ d). +(3) +The above notation is explained in Section 3. The following two relations are verified by +matrix multiplication: +A(q, ε)P(q, ε) = P(q, ε)A∗(q−1), +(4) +A∗(q)P(q, ε) = P(q, ε)A(q−1, ε). +(5) +We claim that P(q, ε) is invertible. To prove the claim, we show that +P(q, ε)P(q−1, ε) = (q; q)d +(εq; q)d +I. +(6) +Abbreviate Y = P(q, ε)P(q−1, ε). Observe that (4), (5) remain valid if we replace q by q−1. +By this observation, Y commutes with A(q, ε) and A∗(q). The matrix Y commutes with +A∗(q) = diag(1, q, . . . , qd), so Y is diagonal. Write Y = diag(y0, y1, . . . , yd). For 1 ≤ i ≤ d +we compare the (i, i − 1)-entry on each side of A(q, ε)Y = Y A(q, ε); this yields yi−1 = yi. +Consequently y0 = y1 = · · · = yd, so Y = y0I. We have +P(q, ε)P(q−1, ε) = y0I. +(7) +For the product on the left in (7), we compute the (0, 0)-entry using matrix multiplication, +and express the result in terms of basic hypergeometric series [13]; this yields +y0 = +d +� +j=0 +(ε; q)j +(εq; q)j += 2φ1 +� +q−d, ε +εq +����q, 1 +� += (q; q)d +(εq; q)d +. +5 + +In the above line, the last equality is the q-Vandermonde summation formula [13, Appendix +II]: +2φ1 +� +q−d, b +c +����q, cqd +b +� += (b−1c; q)d +(c; q)d +with b = ε and c = εq. We have verified (6), and the claim is proven. By the claim and (4), +(5) the pair A, A∗ acts on V as a circular bidiagonal pair. Concerning the relations in (2), +the last two are verified by matrix multiplication, and the first is obtained from the second +using (4). +Note 2.7. For the circular bidiagonal pair in Lemma 2.6, if we set ε = 0 then we get the +circular bidiagonal pair in Lemma 2.4. +Remark 2.8. Referring to Lemma 2.6, the number of primitive nth roots of unity depends +on F and n; this number might be zero. For example, if Char(F) divides n then F does not +contain a primitive nth root of unity. +Definition 2.9. The circular bidiagonal pair A, A∗ in Lemma 2.6 will be called CBP(F; d, q, ε). +For the next example, we return to the vector space V = F5. Assume that Char(F) = 5. +Pick γ ∈ F that is not among 0, 1, 2, 3, 4. Consider the matrices +A = + + + + + + +γ +0 +0 +0 +−γ +−1 − γ +1 + γ +0 +0 +0 +0 +−2 − γ +2 + γ +0 +0 +0 +0 +−3 − γ +3 + γ +0 +0 +0 +0 +−4 − γ +4 + γ + + + + + + +, +A∗ = diag(0, 1, 2, 3, 4). +One checks that +A5 = A, +(A∗)5 = A∗, +AA∗ − A∗A + A − A∗ = γI. +We will show that the pair A, A∗ acts on V as a circular bidiagonal pair. This is a special +case of the following result. +Lemma 2.10. Pick an integer d ≥ 1, and consider the vector space V = Fd+1. Assume that +n = d + 1 is prime, and that Char(F) = n. Pick γ ∈ F that is not among 0, 1, 2, . . ., d. We +define a matrix A = A(γ) in Matd+1(F) as follows. We have Ai,i = i + γ for 0 ≤ i ≤ d. We +have A0,d = −γ, and Ai,i−1 = −i − γ for 1 ≤ i ≤ d. All other entries of A are zero. We +define a diagonal matrix A∗ ∈ Matd+1(F) with A∗ +i,i = i for 0 ≤ i ≤ d. Then the pair A, A∗ +acts on V as a circular bidiagonal pair. Moreover +An = A, +(A∗)n = A∗, +AA∗ − A∗A + A − A∗ = γI. +(8) +Proof. Define a matrix P = P(γ) in Matd+1(F) with (i, j)-entry +Pi,j = (−i − γ)j +(1 − γ)j +(0 ≤ i, j ≤ d). +(9) +6 + +The above notation is explained in Section 3. The following two relations are verified by +matrix multiplication: +A(γ)P(γ) = P(γ)A∗, +(10) +A∗P(γ) = P(γ)A(−γ). +(11) +We claim that P(γ) is invertible. To prove the claim, we show that +P(γ)P(−γ) = +d! +(1 − γ)d +I. +(12) +Abbreviate Y = P(γ)P(−γ). +Observe that (10), (11) remain valid if we replace γ by +−γ. By this observation, Y commutes with A(γ) and A∗. The matrix Y commutes with +A∗ = diag(0, 1, 2, . . . , d), so Y is diagonal. Write Y = diag(y0, y1, . . . , yd). For 1 ≤ i ≤ d +we compare the (i, i − 1)-entry on each side of A(γ)Y = Y A(γ); this yields yi−1 = yi. +Consequently y0 = y1 = · · · = yd, so Y = y0I. We have +P(γ)P(−γ) = y0I. +(13) +For the product on the left in (13), we compute the (0, 0)-entry using matrix multiplication, +and express the result in terms of hypergeometric series [1]; this yields +y0 = +d +� +j=0 +(−γ)j +(1 − γ)j += 2F1 +�−d, −γ +1 − γ +����1 +� += +d! +(1 − γ)d +. +In the above line, the last equality is the Vandermonde summation formula [1, Chapter 2]: +2F1 +� +−d, b +c +���� 1 +� += (c − b)d +(c)d +with b = −γ and c = 1−γ. We have verified (12), and the claim is proven. By the claim and +(10), (11) the pair A, A∗ acts on V as a circular bidiagonal pair. Concerning the relations in +(8), the last two are verified by matrix multiplication, and the first is obtain from the second +using (10). +Definition 2.11. The circular bidiagonal pair A, A∗ in Lemma 2.10 will be called CBP(F; d, γ). +Next, we define the notion of isomorphism for circular bidiagonal pairs. +Definition 2.12. Let A, A∗ denote a circular bidiagonal pair on a vector space V , and let +B, B∗ denote a circular bidiagonal pair on a vector space V. By an isomorphism of circular +bidiagonal pairs from A, A∗ to B, B∗ we mean an isomorphism of vector spaces σ : V → V +such that σA = Bσ and σA∗ = B∗σ. We say that the circular bidiagonal pairs A, A∗ and +B, B∗ are isomorphic whenever there exists an isomorphism of circular bidiagonal pairs from +A, A∗ to B, B∗. +In Section 7, we use the concepts of isomorphism and duality to intrepret the proof of +Lemmas 2.6, 2.10. +Next, we show that the circular bidiagonal pairs in Lemmas 2.6, 2.10 are mutually noniso- +morphic. +7 + +Lemma 2.13. The following (i), (ii) hold for d ≥ 1. +(i) Assume that Char(F) ̸= d + 1. +Then circular bidiagonal pairs CBP(F; d, q, ε) and +CBP(F; d, q′, ε′) are isomorphic if and only if both +q = q′, +ε = ε′. +(ii) Assume that Char(F) = d + 1. +Then circular bidiagonal pairs CBP(F; d, γ) and +CBP(F; d, γ′) are isomorphic if and only if γ = γ′. +The proof of Lemma 2.13 will be completed in Section 5. +Next, we describe how to adjust a circular bidiagonal pair to obtain another circular bidiag- +onal pair. +Lemma 2.14. Let A, A∗ denote a circular bidiagonal pair on a vector space V . Pick scalars +s, s∗, t, t∗ in F with s, s∗ nonzero. Then the pair sA + tI, s∗A∗ + t∗I is a circular bidigonal +pair on V . +Proof. Routine. +Definition 2.15. Referring to Lemma 2.14, the pair sA + tI, s∗A∗ + t∗I is called an affine +transformation of A, A∗. +Next, we define the notion of affine equivalence for circular bidiagonal pairs. +Definition 2.16. Let A, A∗ and B, B∗ denote circular bidiagonal pairs over F. We say that +A, A∗ and B, B∗ are affine equivalent whenever there exists an affine transformation of A, A∗ +that is isomorphic to B, B∗. +Next, we apply the concept of affine equivalence to the circular bidiagonal pairs in Lemmas +2.6, 2.10. +Lemma 2.17. The following (i), (ii) hold for d ≥ 1. +(i) Assume that Char(F) ̸= d + 1. +Then circular bidiagonal pairs CBP(F; d, q, ε) and +CBP(F; d, q′, ε′) are affine equivalent if and only if both +q = q′, +ε′ ∈ {ε, qε, q2ε, . . . , qdε}. +(ii) Assume that Char(F) = d + 1. +Then circular bidiagonal pairs CBP(F; d, γ) and +CBP(F; d, γ′) are affine equivalent if and only if +γ′ − γ ∈ {0, 1, 2, . . ., d}. +The proof of Lemma 2.17 will be completed in Section 6. +The following is our main result. +8 + +Theorem 2.18. Pick an integer d ≥ 1. Let A, A∗ denote a circular bidiagonal pair on a +vector space of dimension d + 1. First assume that Char(F) ̸= d + 1. Then A, A∗ is affine +equivalent to CBP(F; d, q, ε) for at least one ordered pair q, ε. Next assume that Char(F) = +d + 1. Then A, A∗ is affine equivalent to CBP(F; d, γ) for at least one γ. +The proof of Theorem 2.18 will be completed in Section 4. +We have a comment. +Lemma 2.19. The following (i), (ii) hold for d ≥ 1. +(i) Assume that Char(F) ̸= d + 1, and write A, A∗ for CBP(F; d, q, ε). Then A, A∗ is +isomorphic to qA, q−1A∗. +(ii) Assume that Char(F) = d + 1, and write A, A∗ for CBP(F; d, γ). Then A, A∗ is iso- +morphic to A − I, A∗ − I. +The proof of Lemma 2.19 will be completed in Section 8. +In Section 9, we discuss how circular bidiagonal pairs are related to the circular Hessenberg +pairs introduced by Jae-ho Lee [25]. +3 +Preliminaries +In this section, we review some basic concepts and notation that will be used throughout +the paper. Recall the natural numbers N = {0, 1, 2, . . .} and integers Z = {0, ±1, ±2, . . .}. +Recall the field F from Section 2. For a, q ∈ F and r ∈ N define +(a; q)r = (1 − a)(1 − aq)(1 − aq2) · · ·(1 − aqr−1). +We interpret (a; q)0 = 1. For a ∈ F and r ∈ N define +(a)r = a(a + 1)(a + 2) · · ·(a + r − 1). +We interpret (a)0 = 1. Let λ denote an indeterminate. The algebra F[λ] consists of the +polynomials in λ that have all coefficients in F. Fix an integer d ≥ 1, and let V denote +a vector space with dimension d + 1. Let End(V ) denote the algebra consisting of the F- +linear maps from V to V . Next we recall how each basis {vi}d +i=0 of V yields an algebra +isomorphism End(V ) → Matd+1(F). For A ∈ End(V ) and X ∈ Matd+1(F), we say that +X represents A with respect to {vi}d +i=0 whenever Avj = �d +i=0 Xi,jvi for 0 ≤ j ≤ d. The +isomorphism sends A to the unique matrix in Matd+1(F) that represents A with respect to +{vi}d +i=0. For A ∈ End(V ), we say that A is diagonalizable whenever V is spanned by the +eigenspaces of A. We say that A is multiplicity-free whenever A is diagonalizable, and each +eigenspace of A has dimension one. Assume that A is multiplicity-free, and let {Vi}d +i=0 denote +an ordering of the eigenspaces of A. The sum V = �d +i=0 Vi is direct. For 0 ≤ i ≤ d let +θi ∈ F denote the eigenvalue of A for Vi. By construction, the scalars {θi}d +i=0 are mutually +distinct. For 0 ≤ i ≤ d define Ei ∈ End(V ) such that (Ei − I)Vi = 0 and EiVj = 0 if i ̸= j +(0 ≤ j ≤ d). Thus Ei is the projection V → Vi. We call Ei the primitive idempotent of A +9 + +associated with Vi (or θi). We have (i) EiEj = δi,jEi (0 ≤ i, j ≤ d); (ii) I = �d +i=0 Ei; (iii) +Vi = EiV (0 ≤ i ≤ d); (iv) tr(Ei) = 1 (0 ≤ i ≤ d); (v) A = �d +i=0 θiEi; (vi) AEi = θiEi = EiA +(0 ≤ i ≤ d). Moreover +Ei = +� +0≤j≤d +j̸=i +A − θjI +θi − θj +(0 ≤ i ≤ d). +(14) +Let M denote the subalgebra of End(V ) generated by A. The vector space M has a basis +{Ai}d +i=0, and also 0 = �d +i=0(A − θiI). Moreover, the elements {Ei}d +i=0 form a basis for the +vector space M. Pick scalars s, t ∈ F with s ̸= 0. The map sA + tI is multiplicity-free, with +eigenvalues {sθi + t}d +i=0. For 0 ≤ i ≤ d, the map Ei is the primitive idempotent of sA + tI +associated with sθi + t. Abbreviate n = d + 1. A scalar q ∈ F is called a primitive nth root +of unity whenever qn = 1 and qi ̸= 1 for 1 ≤ i ≤ d. If q ∈ F is a primitive nth root of unity, +then in the algebra F[λ], +λn − 1 = (λ − 1)(λ − q) · · ·(λ − qd). +If Char(F) = n, then in the algebra F[λ], +λn − λ = λ(λ − 1)(λ − 2) · · ·(λ − d). +This fact is a version of Fermat’s little theorem [28, Theorem 1.50]. For i, j ∈ Z we say that +i ≡ j (mod n) whenever n divides i − j. +4 +The proof of Theorem 2.18 +In this section, our goal is to prove Theorem 2.18. Throughout this section, we fix an integer +d ≥ 1, a vector space V with dimension d + 1, and a circular bidiagonal pair A, A∗ on V . +The following result is a special case of [15, Lemma 2.1]; we will give a short proof for the +sake of completeness. +Lemma 4.1. (See [15, Lemma 2.1].) Each of A, A∗ is multiplicity-free. +Proof. We first consider A. +The map A is diagonalizable by Definition 2.1(ii); we show +that each eigenspace of A has dimension one. To do this, it suffices to show that A has +d + 1 eigenspaces. Let {vi}d +i=0 denote a basis for V that satisfies Definition 2.1(i). Let the +matrix B ∈ Matd+1(F) represent A with respect to {vi}d +i=0. By construction, B is circular +bidiagonal. In particular, for B each entry on the subdiagonal is nonzero and each entry +below the subdiagonal is zero. For 0 ≤ r ≤ d we examine the entries of Br. For 0 ≤ i, j ≤ d +the (i, j)-entry of Br is nonzero if i − j = r, and zero if i − j > r. Therefore, the matrices +{Br}d +r=0 are linearly independent. By this and linear algebra, the maps {Ar}d +r=0 are linearly +independent. Consequently, the minimal polynomial of A has degree d + 1. This minimal +polynomial has no repeated roots, since A is diagonalizable. Therefore, A has d + 1 distinct +eigenvalues and hence d + 1 eigenspaces. We have shown that A is multiplicity-free. One +similarly shows that A∗ is multiplicity-free. +10 + +Definition 4.2. Let M (resp. M∗) denote the subalgebra of End(V ) generated by A (resp. +A∗). +Note that {Ai}d +i=0 is a basis for M, and {(A∗)i}d +i=0 is a basis for M∗. +Definition 4.3. Let {Ei}d +i=0 (resp. {E∗ +i }d +i=0) denote an ordering of the primitive idempotents +of A (resp. A∗). For 0 ≤ i ≤ d let 0 ̸= vi ∈ EiV and 0 ̸= v∗ +i ∈ E∗ +i V . Note that {vi}d +i=0 (resp. +{v∗ +i }d +i=0) is a basis for V . The ordering {Ei}d +i=0 (resp. {E∗ +i }d +i=0) is called standard whenever +the basis {vi}d +i=0 (resp. {v∗ +i }d +i=0) satisfies Definition 2.1(ii) (resp. Definition 2.1(i)). +Next we explain how the standard orderings in Definition 4.3 are not unique. In this ex- +planation, we discuss the primitive idempotents of A; a similar discussion applies to the +primitive idempotents of A∗. +Definition 4.4. Let E and F denote primitive idempotents of A. Let us write E → F +whenever there exists α ∈ F such that (A∗ − αI)EV = FV . +Lemma 4.5. For every primitive idempotent E of A, there exists a unique primitive idem- +potent F of A such that E → F. Moreover E ̸= F. +Proof. Since A∗ acts on the eigenspaces of A in a circular bidiagonal fashion. +Lemma 4.6. Let {Ei}d +i=0 denote an ordering of the primitive idempotents of A. This order- +ing is standard if and only if E0 → E1 → · · · → Ed → E0. +Proof. By Definitions 2.1 and 4.3. +Lemma 4.7. There are exactly d + 1 standard orderings of the primitive idempotents of A. +Proof. By Lemmas 4.5 and 4.6, for every primitive idempotent E of A, there exists a unique +standard ordering {Ei}d +i=0 of the primitive idempotents of A such that E = E0. The result +follows. +Definition 4.8. For the rest of this section, we fix a standard ordering {Ei}d +i=0 of the +primitive idempotents of A, and a standard ordering {E∗ +i }d +i=0 of the primitive idempotents +of A∗. For 0 ≤ i ≤ d let θi (resp. θ∗ +i ) denote the eigenvalue of A (resp. A∗) for Ei (resp. E∗ +i ). +Note that {Ei}d +i=0 is a basis for M, and {E∗ +i }d +i=0 is a basis for M∗. +We have a comment about the subscript i in Ei, E∗ +i , θi, θ∗ +i . Due to the circular nature +of a circular bidiagonal pair, our calculations involving these subscripts will be carried out +modulo n, where n = d + 1. The details are explained in the following definition. +Definition 4.9. For X ∈ {E, E∗, θ, θ∗} and i ∈ Z, we define Xi = Xr where 0 ≤ r ≤ d and +i ≡ r (mod n). +Definition 4.10. For 0 ≤ i ≤ d define +ai = tr(AE∗ +i ), +a∗ +i = tr(A∗Ei). +(15) +Lemma 4.11. The following (i), (ii) hold for 0 ≤ i ≤ d: +11 + +(i) tr(AE∗ +i ) = tr(E∗ +i AE∗ +i ) = tr(E∗ +i A); +(ii) tr(A∗Ei) = tr(EiA∗Ei) = tr(EiA∗). +Proof. (i) By linear algebra, tr(XY ) = tr(Y X) for all X, Y ∈ End(V ). The result follows +from this and (E∗ +i )2 = E∗ +i . +(ii) Similar to the proof of (i). +Lemma 4.12. For 0 ≤ i ≤ d we have +E∗ +i AE∗ +i = aiE∗ +i , +EiA∗Ei = a∗ +i Ei. +Proof. We verify the equation on the left. Abbreviate A = End(V ). The primitive idem- +potent E∗ +i has rank one, so E∗ +i is a basis for E∗ +i AE∗ +i . Therefore, there exists αi ∈ F such +that E∗ +i AE∗ +i = αiE∗ +i . In this equation, take the trace of each side and use (15) along with +Lemma 4.11(i) and tr(E∗ +i ) = 1 to obtain ai = αi. We have verified the equation on the left. +The equation on the right is similarly verified. +Lemma 4.13. For 0 ≤ i ≤ d we have +(A − aiI)E∗ +i V = E∗ +i+1V, +(A∗ − a∗ +i I)EiV = Ei+1V. +Proof. We verifiy the equation on the left. By Definition 4.4 and Lemma 4.6, there exists +αi ∈ F such that (A − αiI)E∗ +i V = E∗ +i+1V . In this equation, apply E∗ +i to each side and +evaluate the result using Lemma 4.12; this yields +0 = E∗ +i (A − αiI)E∗ +i V = (ai − αi)E∗ +i V. +Of course E∗ +i V ̸= 0, so αi = ai. We have verified the equation on the left. The equation on +the right is similarly verified. +Lemma 4.14. The following (i), (ii) hold for 0 ≤ i, j ≤ d. +(i) E∗ +i AE∗ +j = +� +0, +if i − j ̸∈ {0, 1} (mod n); +̸= 0, +if i − j ≡ 1 (mod n). +(ii) EiA∗Ej = +� +0, +if i − j ̸∈ {0, 1} (mod n); +̸= 0, +if i − j ≡ 1 (mod n). +Proof. By Lemma 4.13. +The following generalization of Lemma 4.14 will be useful. +Lemma 4.15. The following (i), (ii) hold for 0 ≤ i, j, r ≤ d. +(i) E∗ +i ArE∗ +j = +� +0, +if i − j ̸∈ {0, 1, . . . , r} (mod n); +̸= 0, +if i − j ≡ r (mod n). +12 + +(ii) Ei(A∗)rEj = +� +0, +if i − j ̸∈ {0, 1, . . . , r} (mod n); +̸= 0, +if i − j ≡ r (mod n). +Proof. This is a routine consequence of Lemma 4.14. +Lemma 4.16. The following holds for 0 ≤ i, j ≤ d: +(i) EiE∗ +j ̸= 0; +(ii) E∗ +i Ej ̸= 0. +Proof. (i) Using (14) and Lemma 4.15(i), +E∗ +j+dEiE∗ +j = E∗ +j+d +� � +0≤ℓ≤d +ℓ̸=i +A − θℓI +θi − θℓ +� +E∗ +j = E∗ +j+dAdE∗ +j +� +0≤ℓ≤d +ℓ̸=i +1 +θi − θℓ +̸= 0. +Therefore EiE∗ +j ̸= 0. +(ii) Similar to the proof of (i). +Lemma 4.17. In each of (i)–(iv) below, we give a basis for the vector space End(V ): +(i) EiE∗ +j (0 ≤ i, j ≤ d); +(ii) Ai(A∗)j (0 ≤ i, j ≤ d); +(iii) E∗ +i Ej (0 ≤ i, j ≤ d); +(iv) (A∗)iAj (0 ≤ i, j ≤ d). +Proof. (i) The dimension of End(V ) is (d + 1)2, and this is the number of vectors listed. +Therefore, it suffices to show that the listed vectors are linearly independent. Assume that +0 = +d +� +i=0 +d +� +j=0 +αi,jEiE∗ +j +(αi,j ∈ F). +We show that αr,s = 0 for 0 ≤ r, s ≤ d. Let r, s be given. We have +0 = Er +� +d +� +i=0 +d +� +j=0 +αi,jEiE∗ +j +� +E∗ +s = αr,sErE∗ +s. +We have ErE∗ +s ̸= 0 by Lemma 4.16(i), so αr,s = 0. +(ii) By (i) and the notes below Definitions 4.2, 4.8. +(iii), (iv) Similar to the proof of (i), (ii). +The next three lemmas contain results about A and {E∗ +i }d +i=0; similar results hold for A∗ and +{Ei}d +i=0. +Lemma 4.18. Let θ denote an eigenvalue of A, and let 0 ̸= ξ ∈ V denote a corresponding +eigenvector. Then the following (i)–(iii) hold: +13 + +(i) the vector E∗ +i ξ is a basis for E∗ +i V (0 ≤ i ≤ d); +(ii) the vectors {E∗ +i ξ}d +i=0 form a basis for V ; +(iii) the basis {E∗ +i ξ}d +i=0 satisfies Definition 2.1(i). +Proof. (i) The dimension of E∗ +i V is one, so it suffices to show that E∗ +i ξ ̸= 0. There exists an +integer j (0 ≤ j ≤ d) such that θ = θj. The subspace EjV has dimension one and contains +ξ, so ξ spans EjV . Therefore, E∗ +i ξ spans E∗ +i EjV . We have E∗ +i Ej ̸= 0 by Lemma 4.16(ii), so +E∗ +i EjV ̸= 0. By these comments E∗ +i ξ ̸= 0. +(ii) By (i) and since the sum V = �d +i=0 E∗ +i V is direct. +(iii) By Definition 4.8. +Lemma 4.19. We refer to the basis {E∗ +i ξ}d +i=0 in Lemma 4.18. Let B (resp. B∗) denote +the matrix in Matd+1(F) that represents A (resp. A∗) with respect to {E∗ +i ξ}d +i=0. Then the +following (i)–(iv) hold: +(i) B is circular bidiagonal with constant row sum θ; +(ii) Bi,i = ai for 0 ≤ i ≤ d; +(iii) B0,d = θ − a0, and Bi,i−1 = θ − ai for 1 ≤ i ≤ d; +(iv) B∗ is diagonal, with B∗ +i,i = θ∗ +i for 0 ≤ i ≤ d. +Proof. (i) The matrix B is circular bidiagonal by Definition 2.1(i) and Lemma 4.18(iii). +Define a vector 1 ∈ Fd+1 that has all entries 1. We have B1 = θ1, because ξ = �d +i=0 E∗ +i ξ is +an eigenvector for A with eigenvalue θ. By B1 = θ1, the matrix B has constant row sum θ. +(ii) By Lemma 4.12. +(iii) By (i), (ii) above. +(iv) The matrix B∗ is diagonal by Definition 2.1(i) and Lemma 4.18(iii). By Definition 4.8 +we obtain B∗ +i,i = θ∗ +i for 0 ≤ i ≤ d. +Lemma 4.20. Let θ denote an eigenvalue of A. Then θ ̸= ai for 0 ≤ i ≤ d. +Proof. Let 0 ̸= ξ ∈ V denote an eigenvector for A with eigenvalue θ, and let B ∈ Matd+1(F) +represent A with respect to the basis {E∗ +i ξ}d +i=0. The matrix B is circular bidiagonal by +Lemma 4.19(i). Therefore, B0,d ̸= 0 and Bi,i−1 ̸= 0 for 1 ≤ i ≤ d. The result follows in view +of Lemma 4.19(iii). +Our next general goal is to obtain a relation involving A and A∗. +The next two lemmas contain results about A and {E∗ +i }d +i=0; similar results hold for A∗ and +{Ei}d +i=0. +Lemma 4.21. The following (i), (ii) hold for 0 ≤ i ≤ d: +(i) E∗ +i A = E∗ +i AE∗ +i + E∗ +i AE∗ +i−1; +(ii) AE∗ +i = E∗ +i AE∗ +i + E∗ +i+1AE∗ +i . +14 + +Proof. (i) Using Lemma 4.14(i) we have +E∗ +i A = E∗ +i AI = +d +� +j=0 +E∗ +i AE∗ +j = E∗ +i AE∗ +i + E∗ +i AE∗ +i−1. +(ii) Using Lemma 4.14(i) we have +AE∗ +i = IAE∗ +i = +d +� +j=0 +E∗ +j AE∗ +i = E∗ +i AE∗ +i + E∗ +i+1AE∗ +i . +Lemma 4.22. For 0 ≤ i ≤ d, +AE∗ +i − aiE∗ +i = E∗ +i+1AE∗ +i = E∗ +i+1A − ai+1E∗ +i+1. +Proof. The equation on the left follows from Lemma 4.12 and Lemma 4.21(ii). The equation +on the right follows from Lemma 4.12 and Lemma 4.21(i). +We bring in some notation. Define +M∗AM∗ = Span{XAY |X, Y ∈ M∗}. +Proposition 4.23. The following (i), (ii) hold. +(i) M∗AM∗ + M∗ = AM∗ + M∗; +(ii) M∗AM∗ + M∗ = M∗A + M∗. +Proof. (i) To obtain the inclusion ⊆, we use Lemmas 4.12, 4.14(i), 4.22 to obtain +M∗AM∗ = Span{E∗ +i AE∗ +j |0 ≤ i, j ≤ d} += Span{E∗ +i AE∗ +j |0 ≤ i, j ≤ d, i − j ∈ {0, 1} (mod n)} += Span{E∗ +i AE∗ +i |0 ≤ i ≤ d} + Span{E∗ +i+1AE∗ +i |0 ≤ i ≤ d} +⊆ AM∗ + M∗. +The inclusion ⊇ holds since I ∈ M∗. +(ii) Similar to the proof of (i). +Proposition 4.24. There exists a unique sequence q, α, β, γ of scalars in F such that +qAA∗ − A∗A + αA − βA∗ = γI. +(16) +Proof. First, we show that the sequence q, α, β, γ exists. Using Proposition 4.23, we obtain +A∗A ∈ M∗A ⊆ M∗A + M∗ = AM∗ + M∗. +15 + +Therefore, there exist X, Y ∈ M∗ such that A∗A = AX + Y . The elements {(A∗)r}d +r=0 form +a basis for M∗. Write +X = +d +� +r=0 +αr(A∗)r, +Y = +d +� +r=0 +βr(A∗)r, +αr, βr ∈ F. +We claim that αr = 0 and βr = 0 for 2 ≤ r ≤ d. To prove the claim, we assume that it is +false, and get a contradiction. There exists an integer r (2 ≤ r ≤ d) such that αr ̸= 0 or +βr ̸= 0. Define +m = max{r|2 ≤ r ≤ d, αr ̸= 0 or βr ̸= 0}. +By construction 2 ≤ m ≤ d. Also, αr = 0 and βr = 0 for m + 1 ≤ r ≤ d. Therefore +X = +m +� +r=0 +αr(A∗)r, +Y = +m +� +r=0 +βr(A∗)r. +Let 0 ≤ i ≤ d. By Lemma 4.15(ii) and the construction, we have +Em+iXEi = αmEm+i(A∗)mEi, +Em+iY Ei = βmEm+i(A∗)mEi. +Also, by Lemma 4.14(ii) and m ≥ 2, we have Em+iA∗Ei = 0. We may now argue +0 = Em+iA∗Eiθi += Em+iA∗AEi += Em+i(AX + Y )Ei += θm+iEm+iXEi + Em+iY Ei += (θm+iαm + βm)Em+i(A∗)mEi. +We have Em+i(A∗)mEi ̸= 0 by Lemma 4.15(ii). By these comments 0 = θm+iαm + βm for +0 ≤ i ≤ d. In particular, +0 = θ0αm + βm, +0 = θ1αm + βm. +We have θ0 ̸= θ1, so αm = 0 and βm = 0. This contradicts the definition of m, so the claim +is proved. By the claim, X = α0I + α1A∗ and Y = β0I + β1A∗. Using this to evaluate +A∗A = AX + Y , we obtain (16) with +q = α1, +α = α0, +β = −β1, +γ = −β0. +We have shown that the sequence q, α, β, γ exists. This sequence is unique, because the +following maps are linearly independent by Lemma 4.17(ii): +AA∗, +A, +A∗, +I. +Definition 4.25. The sequence q, α, β, γ from Proposition 4.24 is called the profile of A, A∗. +16 + +Lemma 4.26. The following (i), (ii) hold for 0 ≤ i ≤ d: +(i) qθi+1 = θi + β; +(ii) θ∗ +i+1 = qθ∗ +i + α. +Proof. (i) In the equation (16), multiply each term on the left by Ei+1 and on the right by +Ei. Simplify the result using Ei+1A∗Ei ̸= 0. +(ii) In the equation (16), multiply each term on the left by E∗ +i+1 and on the right by E∗ +i . +Simplify the result using E∗ +i+1AE∗ +i ̸= 0. +Lemma 4.27. The following (i), (ii) hold for 0 ≤ i ≤ d: +(i) ai +� +θ∗ +i (q − 1) + α +� += βθ∗ +i + γ; +(ii) a∗ +i +� +θi(1 − q) + β +� += αθi − γ. +Proof. (i) In the equation (16), multiply each term on the left and right by E∗ +i . Simplify the +result using E∗ +i AE∗ +i = aiE∗ +i . +(ii) In the equation (16), multiply each term on the left and right by Ei. Simplify the result +using EiA∗Ei = a∗ +i Ei. +Lemma 4.28. The scalars α, β satisfy the following inequalities. +(i) Assume that q ̸= 1. Then +α ̸= (1 − q)θ∗ +0, +β ̸= (q − 1)θ0. +(ii) Assume that q = 1. Then +α ̸= 0, +β ̸= 0. +Proof. The inequality about α is from Lemma 4.26(ii) with i = 0 and θ∗ +1 ̸= θ∗ +0. The inequality +about β is from Lemma 4.26(i) with i = 0 and θ1 ̸= θ0. +Lemma 4.29. For 0 ≤ i ≤ d we have +θi+1 − θ0 +θ1 − θ0 += +i +� +ℓ=0 +q−ℓ, +θ∗ +i+1 − θ∗ +0 +θ∗ +1 − θ∗ +0 += +i +� +ℓ=0 +qℓ. +(17) +Proof. We verify the equation on the left. Using Lemma 4.26(i), +θi+1 − q−i−1θ0 += θi+1 − q−1θi + q−1(θi − q−1θi−1) + q−2(θi−1 − q−1θi−2) + · · · + q−i(θ1 − q−1θ0) += (1 + q−1 + q−2 + · · · + q−i)q−1β. +Observe that +θi+1 − θ0 = θi+1 − q−i−1θ0 + (q−i−1 − 1)θ0 += +� +1 + q−1 + q−2 + · · · + q−i�� +q−1β + (q−1 − 1)θ0 +� += +� +1 + q−1 + q−2 + · · · + q−i� +(θ1 − θ0). +This verifies the equation on the left in (17). The equation on the right in (17) is similarly +verified. +17 + +Lemma 4.30. We have �d +ℓ=0 qℓ = 0, and �i +ℓ=0 qℓ ̸= 0 for 0 ≤ i ≤ d − 1. +Proof. We have θ∗ +d+1 = θ∗ +0, and θ∗ +i+1 ̸= θ∗ +0 for 0 ≤ i ≤ d − 1. The result follows in view of +Lemma 4.29. +Recall n = d + 1. +Lemma 4.31. The scalar q is related to Char(F) in the following way. +(i) Assume that Char(F) ̸= n. Then q is a primitive nth root of unity. +(ii) Assume that Char(F) = n. Then q = 1. +Proof. First suppose that q = 1. Then by Lemma 4.30, n = 0 in F and 1, 2, . . . , d are nonzero +in F. Therefore Char(F) = n. Next suppose that q ̸= 1. Then by Lemma 4.30, qn = 1 and +qj ̸= 1 for 1 ≤ j ≤ d. Therefore q is a primitive nth root of unity. In this case Char(F) ̸= n; +otherwise 0 = qn − 1 = (q − 1)n, forcing q = 1 for a contradiction. The result follows from +these comments. +In Definitions 4.8, 4.10 and Proposition 4.24, we introduced various parameters that describe +the circular bidiagonal pair A, A∗. Next, we consider how these parameters are affected by +an affine transformation of A, A∗. +Pick scalars s, s∗, t, t∗ in F with s, s∗ nonzero. Consider the circular bidiagonal pair +A∨ = sA + tI, +(A∗)∨ = s∗A∗ + t∗I. +(18) +Note that {Ei}d +i=0 and {E∗ +i }d +i=0 are orderings of the primitive idempotents of A∨ and (A∗)∨, +respectively. These orderings are standard. For 0 ≤ i ≤ d let θ∨ +i (resp. (θ∗ +i )∨) denote the +eigenvalue of A∨ (resp. (A∗)∨) for Ei (resp. E∗ +i ). Also, define +a∨ +i = tr(A∨E∗ +i ), +(a∗ +i )∨ = tr +� +(A∗)∨Ei +� +. +Let q∨, α∨, β∨, γ∨ denote the profile of A∨, (A∗)∨. +Lemma 4.32. We refer to the circular bidiagonal pair A∨, (A∗)∨ from (18). In the tables +below, we describe various parameters for A∨, (A∗)∨ in terms of the corresponding parameters +for A, A∗. +parameter +parameter description +θ∨ +i +sθi + t +(θ∗ +i )∨ +s∗θ∗ +i + t∗ +a∨ +i +sai + t +(a∗ +i )∨ +s∗a∗ +i + t∗ +parameter +parameter description +q∨ +q +α∨ +s∗α + t∗(1 − q) +β∨ +sβ + t(q − 1) +γ∨ +ss∗γ + ts∗α − st∗β + tt∗(1 − q) +18 + +Proof. The first table is verified using Definitions 4.8, 4.10 and the discussion below (18). +The second table is verified using Proposition 4.24 and the comment above Lemma 4.32. +Next, we define what it means for the circular bidiagonal pair A, A∗ to be normalized. +Definition 4.33. Let E (resp. E∗) denote a primitive idempotent of A (resp. A∗). Let θ +(resp. θ∗) denote the corresponding eigenvalue. The circular bidiagonal pair A, A∗ is said to +be normalized with respect to E and E∗ whenever α, β, θ, θ∗ satisfy the requirements in the +table below: +case +α +β +θ +θ∗ +Char(F) ̸= n +0 +0 +1 +1 +Char(F) = n +1 +1 +0 +0 +Next, we put A, A∗ in normalized form by applying an affine transformation. +Lemma 4.34. We normalize the circular bidiagonal pair A, A∗ as follows. +(i) Assume that Char(F) ̸= n. Then the circular bidiagonal pair +(q − 1)A − βI +(q − 1)θ0 − β , +(1 − q)A∗ − αI +(1 − q)θ∗ +0 − α +is normalized with respect to E0 and E∗ +0. +(ii) Assume that Char(F) = n. Then the circular bidiagonal pair +A − θ0I +β +, +A∗ − θ∗ +0I +α +is normalized with respect to E0 and E∗ +0. +Proof. This is readily checked using Lemmas 4.28, 4.31, 4.32 and Definition 4.33. +Lemma 4.35. Assume that the circular bidiagonal pair A, A∗ is normalized with respect to +E0 and E∗ +0. +(i) Assume that Char(F) ̸= n. Then +qAA∗ − A∗A +q − 1 += εI, +where ε = γ/(q − 1). Moreover +θi = q−i, +θ∗ +i = qi, +ai = q−iε, +a∗ +i = qiε, +(0 ≤ i ≤ d). +(ii) Assume that Char(F) = n. Then +AA∗ − A∗A + A − A∗ = γI. +Moreover +θi = i, +θ∗ +i = i, +ai = i + γ, +a∗ +i = i − γ, +(0 ≤ i ≤ d). +19 + +Proof. Evaluate Proposition 4.24 and Lemmas 4.26, 4.27 using Definition 4.33. +Lemma 4.36. Assume that the circular bidiagonal pair A, A∗ is normalized with respect to +E0 and E∗ +0. +(i) Assume that Char(F) ̸= n. +Then the scalar ε from Lemma 4.35(i) is not among +1, q, q2, . . . , qd. +(ii) Assume that Char(F) = n. +Then the scalar γ from Lemma 4.35(ii) is not among +0, 1, 2, . . ., d. +Proof. Use Lemma 4.20 and the data in Lemma 4.35. +Proposition 4.37. Assume that the circular bidiagonal pair A, A∗ is normalized with respect +to E0 and E∗ +0. +(i) Assume that Char(F) ̸= n. Then the circular bidiagonal pair A, A∗ is isomorphic to +CBP(F; d, q, ε), where q, ε are from Lemma 4.35(i). +(ii) Assume that Char(F) = n. Then the circular bidiagonal pair A, A∗ is isomorphic to +CBP(F; d, γ), where γ is from Lemma 4.35(ii). +Proof. (i) By Lemma 4.31(i), q is a primitive nth root of unity. By Lemma 4.36(i), ε is not +among 1, q, q2, . . . , qd. The circular bidiagonal pair CBP(F; d, q, ε) is described in Lemma +2.6. We have θ0 = 1 by Lemma 4.35(i). Therefore 1 is an eigenvalue of A; let 0 ̸= ξ ∈ V +denote a corresponding eigenvector. Consider the basis {E∗ +i ξ}d +i=0 of V from Lemma 4.18. +Let B (resp. B∗) denote the matrix in Matd+1(F) that represents A (resp. A∗) with respect +to {E∗ +i ξ}d +i=0. Using Lemma 4.19 (with θ = 1) and the data in Lemma 4.35(i), we find that +CBP(F; d, q, ε) is equal to B, B∗. The result follows. +(ii) By Lemma 4.31(ii), Char(F) = n. By Lemma 4.36(ii), γ is not among 0, 1, 2, . . . , d. The +circular bidiagonal pair CBP(F; d, γ) is described in Lemma 2.10. We have θ0 = 0 by Lemma +4.35(ii). Therefore 0 is an eigenvalue of A; let 0 ̸= ξ ∈ V denote a corresponding eigenvector. +Consider the basis {E∗ +i ξ}d +i=0 of V from Lemma 4.18. Let B (resp. B∗) denote the matrix in +Matd+1(F) that represents A (resp. A∗) with respect to {E∗ +i ξ}d +i=0. Using Lemma 4.19 (with +θ = 0) and the data in Lemma 4.35(ii), we find that CBP(F; d, γ) is equal to B, B∗. The +result follows. +Theorem 2.18 is immediate from Lemma 4.34 and Proposition 4.37. +5 +The proof of Lemma 2.13 +In this section, we prove Lemma 2.13. +Proof of Lemma 2.13 (i) Assume that CBP(F; d, q, ε) and CBP(F; d, q′, ε′) are isomorphic. +We will show that q = q′ and ε = ε′. +Write A, A∗ for CBP(F; d, q, ε) and B, B∗ for +CBP(F; d, q′, ε′). By Definition 2.12, there exists an invertible P ∈ Matd+1(F) such that +PA = BP and PA∗ = B∗P. The matrix A∗ is diagonal, and its diagonal entries are mu- +tually distinct. The matrix B∗ is diagonal, and its diagonal entries are mutually distinct. +20 + +Examining the entries of PA∗ = B∗P, we find that there exists a permutation p of the set +{0, 1, 2, . . ., d} such that for 0 ≤ i, j ≤ d, +j = p(i) +⇔ +Pi,j ̸= 0 +⇔ +A∗ +j,j = B∗ +i,i. +We have A∗ +0,0 = 1 = B∗ +0,0. Therefore p(0) = 0. Next we show that P is diagonal. The +matrices A and B are circular bidiagonal. For 1 ≤ i ≤ d we examine the +� +i, p(i − 1) +� +-entry +in PA = BP; this gives +Pi,p(i)Ap(i),p(i−1) = Bi,i−1Pi−1,p(i−1). +By construction Bi,i−1 ̸= 0 and Pi−1,p(i−1) ̸= 0. Therefore Ap(i),p(i−1) ̸= 0. Of course p(i) ̸= +p(i − 1), so p(i) = p(i − 1) + 1. Using induction and p(0) = 0, we obtain p(i) = i for +0 ≤ i ≤ d. We have shown that P is diagonal. Consequently P commutes with A∗, so +A∗ = B∗. Therefore q = q′. Also, for 0 ≤ i ≤ d we have Ai,i = Bi,i, which implies that +ε = ε′. We are done in one logical direction. We now consider the opposite logical direction. +Assume that q = q′ and ε = ε′. Then CBP(F; d, q, ε) and CBP(F; d, q′, ε′) are the same, and +hence isomorphic. +(ii) Similar to the proof of (i). +✷ +6 +The proof of Lemma 2.17 +In this section, we prove Lemma 2.17. +We begin with some comments about the matrices A = A(q, ε) and A∗ = A∗(q) from Lemma +2.6. +Lemma 6.1. With the above notation, +(A∗ − εI)A(q, qε) = qA(q, ε)(A∗ − εI). +Proof. This is routinely verified by matrix multiplication, using the data in Lemma 2.6. +Lemma 6.2. The map A∗ − εI from Lemma 6.1 is an isomorphism of circular bidiagonal +pairs, from A(q, qε), A∗ to qA(q, ε), A∗. +Proof. Define the matrix P = A∗ − εI. The matrix P is invertible, since ε is not among +1, q, q2, . . . , qd. By Lemma 6.1 and the construction, +PA(q, qε) = qA(q, ε)P, +PA∗ = A∗P. +The result follows in view of Definition 2.12. +Corollary 6.3. The circular bidiagonal pairs CBP(F; d, q, ε) and CBP(F; d, q, qε) are affine +equivalent. +Proof. By Definitions 2.9, 2.16 and Lemma 6.2. +21 + +Corollary 6.4. The following circular bidiagonal pairs are mutually affine equivalent: +CBP(F; d, q, qiε) +i ∈ {0, 1, 2, . . ., d}. +Proof. By Corollary 6.3, and since affine equivalence is an equivalence relation. +Next, we have some comments about the matrices A = A(γ) and A∗ from Lemma 2.10. +Lemma 6.5. With the above notation, +(A∗ + γI)A(γ − 1) = +� +A(γ) − I +� +(A∗ + γI). +Proof. This is routinely verified by matrix multiplication, using the data in Lemma 2.10. +Lemma 6.6. The map A∗ + γI from Lemma 6.5 is an isomorphism of circular bidiagonal +pairs, from A(γ − 1), A∗ to A(γ) − I, A∗. +Proof. Define the matrix P = A∗ + γI. The matrix P is invertible, since γ is not among +0, 1, 2, . . . , d. By Lemma 6.5 and the construction, +PA(γ − 1) = +� +A(γ) − I +� +P, +PA∗ = A∗P. +The result follows in view of Definition 2.12. +Corollary 6.7. The circular bidiagonal pairs CBP(F; d, γ) and CBP(F; d, γ − 1) are affine +equivalent. +Proof. By Definitions 2.11, 2.16 and Lemma 6.6. +Corollary 6.8. The following circular bidiagonal pairs are mutually affine equivalent: +CBP(F; d, γ + i) +i ∈ {0, 1, 2, . . ., d}. +Proof. By Corollary 6.7, and since affine equivalence is an equivalence relation. +Proof of Lemma 2.17 (i) Assume that CBP(F; d, q, ε) and CBP(F; d, q′, ε′) are affine equiva- +lent. We will show that q = q′ and ε′ ∈ {ε, qε, q2ε, . . . , qdε}. Write A, A∗ for CBP(F; d, q, ε) +and B, B∗ for CBP(F; d, q′, ε′). The profile of A, A∗ is q, α, β, γ where +α = 0, +β = 0, +γ = ε(q − 1). +(19) +The profile of B, B∗ is q′, α′, β′, γ′ where +α′ = 0, +β′ = 0, +γ′ = ε′(q′ − 1). +(20) +By assumption, there exist scalars s, s∗, t, t∗ in F with s, s∗ nonzero such that sA+tI, s∗A∗+ +t∗I is isomorphic to B, B∗. Define A∨ = sA + tI and (A∗)∨ = s∗A∗ + t∗I. The profile q∨, +α∨, β∨, γ∨ of A∨, (A∗)∨ is described in Lemma 4.32. The circular bidiagonal pairs A∨, (A∗)∨ +and B, B∗ are isomorphic, so they have the same profile: +q∨ = q′, +α∨ = α′, +β∨ = β′, +γ∨ = γ′. +22 + +Evaluate the above equations using (19), (20) and the second table in Lemma 4.32. This +yields +q = q′, +t = 0, +t∗ = 0, +ss∗ε = ε′. +By construction, the circular bidiagonal pairs sA, s∗A∗ and B, B∗ are isomorphic. Therefore +sA has the same eigenvalues as B, and s∗A∗ has the same eigenvalues as B∗. The scalar +1 is an eigenvalue of A, so the scalar s is an eigenvalue of sA. The eigenvalues of B are +1, q, q2, . . . , qd. By these comments s ∈ {1, q, q2, . . . , qd}. The scalar 1 is an eigenvalue of A∗, +so the scalar s∗ is an eigenvalue of s∗A∗. The eigenvalues of B∗ are 1, q, q2, . . . , qd. By these +comments s∗ ∈ {1, q, q2, . . . , qd}. We may now argue +ε′ = ss∗ε ∈ {ε, qε, q2ε, . . . , qdε}. +Next, we reverse the logical direction. Assume that q′ = q and ε′ ∈ {ε, qε, q2ε, . . . , qdε}. +Then CBP(F; d, q, ε) and CBP(F; d, q′, ε′) are affine equivalent by Corollary 6.4. +(ii) Assume that CBP(F; d, γ) and CBP(F; d, γ′) are affine equivalent. We will show that +γ′ − γ ∈ {0, 1, 2, . . ., d}. Write A, A∗ for CBP(F; d, γ) and B, B∗ for CBP(F; d, γ′). The +profile of A, A∗ is q, α, β, γ where +q = 1, +α = 1, +β = 1. +(21) +The profile of B, B∗ is q′, α′, β′, γ′ where +q′ = 1, +α′ = 1, +β′ = 1. +(22) +By assumption, there exist scalars s, s∗, t, t∗ in F with s, s∗ nonzero such that sA+tI, s∗A∗+ +t∗I is isomorphic to B, B∗. Define A∨ = sA + tI and (A∗)∨ = s∗A∗ + t∗I. The profile q∨, +α∨, β∨, γ∨ of A∨, (A∗)∨ is described in Lemma 4.32. The circular bidiagonal pairs A∨, (A∗)∨ +and B, B∗ are isomorphic, so they have the same profile: +q∨ = q′, +α∨ = α′, +β∨ = β′, +γ∨ = γ′. +Evaluate the above equations using (21), (22) and the second table in Lemma 4.32. This +yields +s = 1, +s∗ = 1, +γ′ − γ = t − t∗. +By construction, the circular bidiagonal pairs A + tI, A∗ + t∗I and B, B∗ are isomorphic. +Therefore A+tI has the same eigenvalues as B, and A∗+t∗I has the same eigenvalues as B∗. +The scalar 0 is an eigenvalue of A, so the scalar t is an eigenvalue of A + tI. The eigenvalues +of B are 0, 1, 2, . . ., d. By these comments t ∈ {0, 1, 2, . . ., d}. The scalar 0 is an eigenvalue +of A∗, so the scalar t∗ is an eigenvalue of A∗ + t∗I. The eigenvalues of B∗ are 0, 1, 2, . . . , d. +By these comments t∗ ∈ {0, 1, 2, . . ., d}. We may now argue +γ′ − γ = t − t∗ ∈ {0, 1, 2, . . ., d}. +Next, we reverse the logical direction. Assume that γ′−γ ∈ {0, 1, 2, . . ., d}. Then CBP(F; d, γ) +and CBP(F; d, γ′) are affine equivalent by Corollary 6.8. +✷ +23 + +7 +Isomorphism and duality +For circular bidiagonal pairs, the concepts of duality and isomorphism were explained in +Definitions 2.3 and 2.12, respectively. In this section, we use these concepts to interpret the +proof of Lemmas 2.6, 2.10. +Proposition 7.1. We refer to the circular bidiagonal pair CBP(F; d, q, ε) in Lemma 2.6. The +matrix P(q, ε) from (3) is an isomorphism of circular bidiagonal pairs from A(q−1, ε), A∗(q−1) +to A∗(q), A(q, ε). +Proof. We showed in the proof of Lemma 2.6 that P(q, ε) is invertible. The result follows +from this along with (4), (5) and Definition 2.12. +Corollary 7.2. The following are dual, up to isomorphism of circular bidiagonal pairs: +CBP(F; d, q, ε); +CBP(F; d, q−1; ε). +Proof. By Definition 2.3 and Proposition 7.1. +Proposition 7.3. We refer to the circular bidiagonal pair CBP(F; d, γ) in Lemma 2.10. +The matrix P(γ) from (9) is an isomorphism of circular bidiagonal pairs from A(−γ), A∗ to +A∗, A(γ). +Proof. We showed in the proof of Lemma 2.10 that P(γ) is invertible. The result follows +from this along with (10), (11) and Definition 2.12. +Corollary 7.4. The following are dual, up to isomorphism of circular bidiagonal pairs: +CBP(F; d, γ), +CBP(F; d, −γ). +Proof. By Definition 2.3 and Proposition 7.3. +8 +The proof of Lemma 2.19 +Our goal in this section is to prove Lemma 2.19. +Our proof strategy is to display the +isomorphism involved. We will give a detailed description of this isomorphism. +Recall the circular bidiagonal pair CBP(F; d, q, ε) from Lemma 2.6. +Definition 8.1. Referring to CBP(F; d, q, ε), define a matrix R = R(q, ε) in Matd+1(F) with +entries R0,d = 1 − ε and Ri,i−1 = qi − ε for 1 ≤ i ≤ d. All other entries of R are zero. We +call R the raising matrix for CBP(F; d, q, ε). +Example 8.2. Referring to Definition 8.1, assume that d = 4. Then +R = + + + + + + +0 +0 +0 +0 +1 − ε +q − ε +0 +0 +0 +0 +0 +q2 − ε +0 +0 +0 +0 +0 +q3 − ε +0 +0 +0 +0 +0 +q4 − ε +0 + + + + + + +. +24 + +Lemma 8.3. With reference to Definition 8.1, the following (i), (ii) hold: +(i) Rd+1 = (−1)d(ε; q)d+1I; +(ii) R is invertible. +Proof. (i) By matrix multiplication. +(ii) By (i) and since (ε; q)d+1 ̸= 0. +Next, we explain how R is related to the matrices A = A(q, ε) and A∗ = A∗(q) from Lemma +2.6. +Lemma 8.4. With the above notation, we have +(i) A∗A − εI = R = q(AA∗ − εI); +(ii) qAR = RA and q−1A∗R = RA∗. +Proof. (i) By matrix multiplication, using the data in Lemma 2.6 and Definition 8.1. +(ii) Observe that +qAR − RA = qA(A∗A − εI) − q(AA∗ − εI)A = 0; +q−1A∗R − RA∗ = q−1A∗q(AA∗ − εI) − (A∗A − εI)A∗ = 0. +Next, we explain how R is related to the primitive idempotents of A and A∗. For 0 ≤ i ≤ d, +let Ei (resp. E∗ +i ) denote the primitive idempotent of A (resp. A∗) for the eigenvalue q−i +(resp. qi). +Lemma 8.5. With the above notation, we have +(i) REi = Ei+1R and RE∗ +i = E∗ +i+1R for 0 ≤ i ≤ d; +(ii) REiV = Ei+1V and RE∗ +i V = E∗ +i+1V for 0 ≤ i ≤ d. +Proof. (i) To obtain REi = Ei+1R, multiply each side of (14) on the left by R and on the +right by R−1. Evaluate the result using θr = q−r (0 ≤ r ≤ d) along with qA = RAR−1. The +equation RE∗ +i = E∗ +i+1R is similar obtained. +(ii) By (i) above and Lemma 8.3(ii). +Proposition 8.6. With the above notation, R is an isomorphism of circular bidiagonal pairs +from A, A∗ to qA, q−1A∗. +Proof. By Definition 2.12 and Lemma 8.4(ii). +We turn our attention to the circular bidiagonal pair CBP(F; d, γ) from Lemma 2.10. +Definition 8.7. Referring to CBP(F; d, γ), define a matrix R = R(γ) in Matd+1(F) with +entries R0,d = γ and Ri,i−1 = i + γ for 1 ≤ i ≤ d. All other entries of R are zero. We call R +the raising matrix for CBP(F; d, γ). +25 + +Example 8.8. Referring to Definition 8.7, assume that d = 4. Then +R = + + + + + + +0 +0 +0 +0 +γ +1 + γ +0 +0 +0 +0 +0 +2 + γ +0 +0 +0 +0 +0 +3 + γ +0 +0 +0 +0 +0 +4 + γ +0 + + + + + + +. +Lemma 8.9. With reference to Definition 8.7, the following (i), (ii) hold: +(i) Rd+1 = (γ)d+1I; +(ii) R is invertible. +Proof. (i) By matrix multiplication. +(ii) By (i) and since (γ)d+1 ̸= 0. +Next, we describe how R is related to the matrices A = A(γ) and A∗ from Lemma 2.10. +Lemma 8.10. With the above notation, +(i) A∗ − A + γI = R = AA∗ − A∗A; +(ii) (A − I)R = RA and (A∗ − I)R = RA∗. +Proof. (i) By matrix multiplication, using the data in Lemma 2.10 and Definition 8.7. +(ii) We have +[A, R] = [A, A∗ − A + γI] = [A, A∗] = R, +[A∗, R] = [A∗, A∗ − A + γI] = −[A∗, A] = [A, A∗] = R. +Next, we describe how R is related to the primitive idempotents of A and A∗. For 0 ≤ i ≤ d, +let Ei (resp. E∗ +i ) denote the primitive idempotent of A (resp. A∗) for the eigenvalue i. +Lemma 8.11. With the above notation, +(i) REi = Ei+1R and RE∗ +i = E∗ +i+1R for 0 ≤ i ≤ d; +(ii) REiV = Ei+1V and RE∗ +i V = E∗ +i+1V for 0 ≤ i ≤ d. +Proof. (i) To obtain REi = Ei+1R, multiply each side of (14) on the left by R and on the +right by R−1. Evaluate the result using θr = r (0 ≤ r ≤ d) along with A − I = RAR−1. The +equation RE∗ +i = E∗ +i+1R is similar obtained. +(ii) By (i) and Lemma 8.9(ii). +Proposition 8.12. With the above notation, R is an isomorphism of circular bidiagonal +pairs from A, A∗ to A − I, A∗ − I. +Proof. By Definition 2.12 and Lemma 8.10(ii). +Lemma 2.19 is immediate from Propositions 8.6 and 8.12. +26 + +9 +Circular Hessenberg pairs +In [25] Jae-ho Lee introduced the concept of a circular Hessenberg pair. A circular bidiagonal +pair is a special case of a circular Hessenberg pair. In Lemmas 2.6 and 2.10, we gave some +examples of a circular bidiagonal pair. In the present section, we describe these examples +using the notation of [25]. +In [25] it is assumed that d ≥ 3; we make the same assumption throughout this section. +Example 9.1. Recall CBP(F; d, q, ε) from Lemma 2.6. This corresponds to [25, Example 5.1] +with parameters +a = 0, +b = 0, +c = 1, +a∗ = 0, +b∗ = 1, +c∗ = 0, +y = 1 − ε, +z = 0. +Here are some related parameters. Referring to [25, Example 5.1], +θi = q−i, +θ∗ +i = qi +(0 ≤ i ≤ d); +φi = (qi − 1)(qi − ε), +ϑi = (qi − 1)(1 − ε) +(1 ≤ i ≤ d). +Referring to [25, Proposition 6.12], +bi = 0 +(0 ≤ i ≤ d − 1); +ai = q−iε +(0 ≤ i ≤ d); +ci = 1 − q−iε +(1 ≤ i ≤ d); +ξ = 1 − ε. +Referring to [25, Proposition 6.11], +b∗ +i = 0 +(0 ≤ i ≤ d − 1); +a∗ +i = qiε +(0 ≤ i ≤ d); +c∗ +i = 1 − qiε +(1 ≤ i ≤ d); +ξ∗ = 1 − ε. +Example 9.2. Recall CBP(F; d, γ) from Lemma 2.10. This corresponds to [25, Example 5.2] +with parameters +a = 0, +b = 1, +c = 0, +a∗ = 0, +b∗ = 1, +c∗ = 0, +y = −γ, +z = 0. +Here are some related parameters. Referring to [25, Example 5.2], +θi = i, +θ∗ +i = i +(0 ≤ i ≤ d); +φi = −i(i + γ), +ϑi = −iγ +(1 ≤ i ≤ d). +Referring to [25, Proposition 6.12], +bi = 0 +(0 ≤ i ≤ d − 1); +ai = i + γ +(0 ≤ i ≤ d); +ci = −i − γ +(1 ≤ i ≤ d); +ξ = −γ. +Referring to [25, Proposition 6.11], +b∗ +i = 0 +(0 ≤ i ≤ d − 1); +a∗ +i = i − γ +(0 ≤ i ≤ d); +c∗ +i = γ − i +(1 ≤ i ≤ d); +ξ∗ = γ. +27 + +10 +Acknowledgement +The first author thanks Jae-ho Lee for many conversations about circular bidiagonal pairs +and circular Hessenberg pairs. The authors thank ˇStefko Miklaviˇc for giving this paper a +close reading and offering valuable comments. +References +[1] G. Andrews, R. Askey, R. Roy. Special functions. Encyclopedia of Mathematics and its +Applications, 71. Cambridge University Press, Cambridge, 1999. +[2] R. Askey and J. Wilson. A set of orthogonal polynomials that generalize the Racah +coefficients or 6–j symbols. SIAM J. Math. Anal. 10 (1979) 1008–1016. +[3] E. Bannai and T. 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Graphs Combin. 37 (2021) +1687–1748; arXiv:2003.09668. +Paul Terwilliger +Department of Mathematics +University of Wisconsin +480 Lincoln Drive +Madison, WI 53706-1388 USA +email: terwilli@math.wisc.edu +Arjana ˇZitnik +Faculty of Mathematics and Physics +University of Ljubljana, and IMFM +Jadranska 19, 1000 Ljubljana, Slovenia +email: Arjana.Zitnik@fmf.uni-lj.si +30 + diff --git a/NNAyT4oBgHgl3EQfUPdn/content/tmp_files/load_file.txt b/NNAyT4oBgHgl3EQfUPdn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a163940533e2d686039d0bd74434c7c3e205d048 --- /dev/null +++ b/NNAyT4oBgHgl3EQfUPdn/content/tmp_files/load_file.txt @@ -0,0 +1,1707 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf,len=1706 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='00121v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='QA] 31 Dec 2022 Circular bidiagonal pairs Paul Terwilliger and Arjana ˇZitnik Abstract A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) each subdiagonal entry is nonzero, and the entry in the top-right corner is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let F denote a field, and let V denote a nonzero finite-dimensional vector space over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We consider an ordered pair of F-linear maps A : V → V and A∗ : V → V that satisfy the following two conditions: there exists a basis for V with respect to which the matrix representing A is circular bidiagonal and the matrix representing A∗ is diagonal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' there exists a basis for V with respect to which the matrix representing A∗ is circular bidiagonal and the matrix representing A is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We call such a pair a circular bidiagonal pair on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We classify the circular bidiagonal pairs up to affine equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' There are two infinite families of solutions, which we describe in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Bidiagonal pair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Hessenberg pair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Leonard pair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' tridiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Primary: 17B37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Secondary: 15A21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 1 Introduction In a celebrated paper [2], Richard Askey and James Wilson introduced the q-Racah family of orthogonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [26], Doug Leonard showed that the q-Racah polynomials are the most general orthogonal polynomials that have orthogonal polynomial duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [3, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1], Eiichi Bannai and Tatsuro Ito gave a comprehensive version of Leonard’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In an effort to clarify and simplify the Leonard theorem, in [29] the first author introduced the concept of a Leonard pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Roughly speaking, a Leonard pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in an irreducible tridiagonal fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [29, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4] there appears an “oriented” version of a Leonard pair, called a Leonard system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [29, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='9] the Leonard systems are classified up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The article [36] contains a modern treatment of this classification, along with a detailed account of the history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By [29, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12] a Leonard pair satisfies two polynomial relations called the tridiagonal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Some notable papers about Leonard pairs are [30–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [19] Tatsuro Ito, Kenichiro Tanabe, and the first author introduced the concept of a tridiagonal pair as a generalization of a Leonard pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The concept of a tridiagonal system 1 was also introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [18, Corollary 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1] the tridiagonal systems over an algebraically closed field are classified up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [24, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4] it is shown how a tridiagonal system induces a tensor product factorization of the underlying vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [22,23] the tridiagonal pairs are related to some finite-dimensional irreducible modules for Uq(� sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' It is shown in [19, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1] that every tridiagonal pair satisfies the tridiagonal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Some notable papers about tridiagonal pairs are [5–9,20,21,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Over the past 20 years, there appeared in the literature some variations on the Leonard pair and tridiagonal pair concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In the next few paragraphs, we summarize these variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [14] Ali Godjali introduced the concept of a Hessenberg pair as a generalization of a tridiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' He showed in [14, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='9] that every Hessenberg pair induces a split decomposition of the underlying vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [15] Godjali considers a special case of Hessenberg pair, called a TH pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' He defines a TH system, and classifies these up to isomorphism [15, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [16, Section 18] the TH systems are characterized in terms of West/South Vandermonde matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [10] Darren Funk-Neubauer introduced the concept of a bidiagonal pair as a variation on a tridiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [10, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1] the bidiagonal pairs are classified up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [10, Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='11] this classification is interpreted using the equitable presentations of sl2 and Uq(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [11] Funk-Neubauer introduces the concept of a bidiagonal triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [11, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1] he shows how every bidiagonal pair extends to a bidiagonal triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [11, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3] the bidiagonal triples are classified up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' See [12] for related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [4] Pascal Baseilhac, Azat Gainutdinov, and Thao Vu introduced the concept of a cyclic tridiagonal pair, as a generalization of a tridiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' They used cyclic tridiagonal pairs to study a higher-order generalization of the Onsager algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [4, Appendix A] some examples of cyclic tridiagonal pairs are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' It remains an open problem to classify the cyclic tridiagonal pairs up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [25] Jae-ho Lee introduced the concept of a circular Hessenberg pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This is a special case of a TH pair, and also a special case of a cyclic tridiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [25, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6] the circular Hessenberg pairs are classified under the assumption that the pair satisfies the tridiagonal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The classification yields four infinite families of solutions [25, Exam- ples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In the present paper, we introduce the concept of a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This is a variation on a bidiagonal pair, and a special case of a circular Hessenberg pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The reason we focus on this special case, is that it affords a classification without assuming in advance that the tridiagonal relations are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We will display two infinite families of circular bidiagonal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We will introduce the notion of affine equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Our main result is that every circular bidiagonal pair is affine equivalent to a member of one of the two families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In the next section, we formally define a circular bidiagonal pair and give a detailed statement of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 2 2 Definitions and statement of results In this section, we introduce the concept of a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' To define the concept, we first explain what it means for a square matrix to be circular bidiagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following matrices are circular bidiagonal: \uf8eb \uf8ec \uf8ec \uf8ed 3 0 0 1 1 4 0 0 0 −1 1 0 0 0 −1 2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , \uf8eb \uf8ec \uf8ec \uf8ed 2 0 0 −1 −1 3 0 0 0 1 0 0 0 0 −1 −1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , \uf8eb \uf8ec \uf8ec \uf8ed 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Circular bidiagonal means (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) each subdiagonal entry is nonzero, and the entry in the top-right corner is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we define a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For the rest of this paper, F denotes a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let V denote a nonzero vector space over F with finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By a circular bidiagonal pair on V , we mean an ordered pair of F-linear maps A : V → V and A∗ : V → V that satisfy the following two conditions: (i) there exists a basis for V with respect to which the matrix representing A is circular bidiagonal and the matrix representing A∗ is diagonal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) there exists a basis for V with respect to which the matrix representing A∗ is circular bidiagonal and the matrix representing A is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pair in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1 is said to be over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1, assume that A, A∗ is a circular bidiagonal pair on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the pair A∗, A is a circular bidiagonal pair on V , called the dual of A, A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we give some examples of circular bidiagonal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Our first example is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let V denote a vector space over F that has dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then any ordered pair of F-linear maps A : V → V and A∗ : V → V is a circular bidiagonal pair on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Our next example is more substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consider the vector space V = F5 (column vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that q ∈ F is a primitive 5th root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consider the matrices A = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , A∗ = diag(1, q, q2, q3, q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' These matrices satisfy A5 = I, (A∗)5 = I, A∗A = qAA∗, 3 where I denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We claim that the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' To see this, we check that A, A∗ satisfy the conditions in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix A is circular bidiagonal and the matrix A∗ is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore, condition (i) in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1 is satisfied by the basis for V consisting of the columns of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define a matrix P = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 1 1 1 1 1 q q2 q3 q4 1 q2 q4 q6 q8 1 q3 q6 q9 q12 1 q4 q8 q12 q16 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix P is Vandermonde, and hence invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' One checks that A∗P = PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In this equation, take the transpose of each side to obtain PA∗ = A−1P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Rearranging this equation, we obtain AP = P(A∗)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' These results show that condition (ii) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1 is satisfied by the basis for V consisting of the columns of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have shown that the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The previous circular bidiagonal pair is a member of an infinite family of circular bidiagonal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Before describing this family, we bring in some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For the rest of this paper, every vector space and algebra mentioned is understood to be over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick an integer d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let Matd+1(F) denote the algebra consisting of the d+1 by d+1 matrices that have all entries in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We index the rows and columns by 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let Fd+1 denote the vector space consisting of the column vectors that have d + 1 coordinates and all entries in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We index the coordinates by 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Note that Matd+1(F) acts on Fd+1 by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let I ∈ Matd+1(F) denote the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick an integer d ≥ 1, and consider the vector space V = Fd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that q ∈ F is a primitive nth root of unity, where n = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define matrices A, A∗ ∈ Matd+1(F) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have A0,d = 1, and Ai,i−1 = 1 for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' All other entries of A are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix A∗ is diagonal, with A∗ i,i = qi for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Moreover An = I, (A∗)n = I, A∗A = qAA∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The relations (1) are readily checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define a matrix P ∈ Matd+1(F) that has (i, j)- entry qij for 0 ≤ i, j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix P is Vandermonde, and hence invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' One checks that A∗P = PA and AP = P(A∗)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consequently, the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Note 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The relation on the right in (1) is a defining relation for the quantum torus algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' see for example [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For the next example, we return to the vector space V = F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that q ∈ F is a primitive 5th root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick ε ∈ F that is not among 1, q, q2, q3, q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consider the matrices A = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ε 0 0 0 1 − ε 1 − q−1ε q−1ε 0 0 0 0 1 − q−2ε q−2ε 0 0 0 0 1 − q−3ε q−3ε 0 0 0 0 1 − q−4ε q−4ε \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , A∗ = diag(1, q, q2, q3, q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 4 One checks that A5 = I, (A∗)5 = I, qAA∗ − A∗A q − 1 = εI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We will show that the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This is a special case of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick an integer d ≥ 1, and consider the vector space V = Fd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that q ∈ F is a primitive nth root of unity, where n = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick ε ∈ F that is not among 1, q, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define a matrix A = A(q, ε) in Matd+1(F) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have Ai,i = q−iε for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have A0,d = 1 − ε, and Ai,i−1 = 1 − q−iε for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' All other entries of A are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We define a diagonal matrix A∗ = A∗(q) in Matd+1(F) with A∗ i,i = qi for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Moreover An = I, (A∗)n = I, qAA∗ − A∗A q − 1 = εI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define a matrix P = P(q, ε) in Matd+1(F) with (i, j)-entry Pi,j = qij (εq−i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)j (εq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)j (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (3) The above notation is explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following two relations are verified by matrix multiplication: A(q, ε)P(q, ε) = P(q, ε)A∗(q−1), (4) A∗(q)P(q, ε) = P(q, ε)A(q−1, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (5) We claim that P(q, ε) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' To prove the claim, we show that P(q, ε)P(q−1, ε) = (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)d (εq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)d I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (6) Abbreviate Y = P(q, ε)P(q−1, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Observe that (4), (5) remain valid if we replace q by q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By this observation, Y commutes with A(q, ε) and A∗(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix Y commutes with A∗(q) = diag(1, q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qd), so Y is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Write Y = diag(y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , yd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 1 ≤ i ≤ d we compare the (i, i − 1)-entry on each side of A(q, ε)Y = Y A(q, ε);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' this yields yi−1 = yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consequently y0 = y1 = · · · = yd, so Y = y0I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have P(q, ε)P(q−1, ε) = y0I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (7) For the product on the left in (7), we compute the (0, 0)-entry using matrix multiplication, and express the result in terms of basic hypergeometric series [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' this yields y0 = d � j=0 (ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)j (εq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)j = 2φ1 � q−d, ε εq ����q, 1 � = (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)d (εq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 5 In the above line, the last equality is the q-Vandermonde summation formula [13, Appendix II]: 2φ1 � q−d, b c ����q, cqd b � = (b−1c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)d (c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)d with b = ε and c = εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have verified (6), and the claim is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By the claim and (4), (5) the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Concerning the relations in (2), the last two are verified by matrix multiplication, and the first is obtained from the second using (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Note 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For the circular bidiagonal pair in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6, if we set ε = 0 then we get the circular bidiagonal pair in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6, the number of primitive nth roots of unity depends on F and n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' this number might be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For example, if Char(F) divides n then F does not contain a primitive nth root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pair A, A∗ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6 will be called CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For the next example, we return to the vector space V = F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that Char(F) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick γ ∈ F that is not among 0, 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consider the matrices A = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed γ 0 0 0 −γ −1 − γ 1 + γ 0 0 0 0 −2 − γ 2 + γ 0 0 0 0 −3 − γ 3 + γ 0 0 0 0 −4 − γ 4 + γ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , A∗ = diag(0, 1, 2, 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' One checks that A5 = A, (A∗)5 = A∗, AA∗ − A∗A + A − A∗ = γI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We will show that the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This is a special case of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick an integer d ≥ 1, and consider the vector space V = Fd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that n = d + 1 is prime, and that Char(F) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick γ ∈ F that is not among 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We define a matrix A = A(γ) in Matd+1(F) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have Ai,i = i + γ for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have A0,d = −γ, and Ai,i−1 = −i − γ for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' All other entries of A are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We define a diagonal matrix A∗ ∈ Matd+1(F) with A∗ i,i = i for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Moreover An = A, (A∗)n = A∗, AA∗ − A∗A + A − A∗ = γI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define a matrix P = P(γ) in Matd+1(F) with (i, j)-entry Pi,j = (−i − γ)j (1 − γ)j (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (9) 6 The above notation is explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following two relations are verified by matrix multiplication: A(γ)P(γ) = P(γ)A∗, (10) A∗P(γ) = P(γ)A(−γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (11) We claim that P(γ) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' To prove the claim, we show that P(γ)P(−γ) = d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (1 − γ)d I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (12) Abbreviate Y = P(γ)P(−γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Observe that (10), (11) remain valid if we replace γ by −γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By this observation, Y commutes with A(γ) and A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix Y commutes with A∗ = diag(0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , d), so Y is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Write Y = diag(y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , yd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 1 ≤ i ≤ d we compare the (i, i − 1)-entry on each side of A(γ)Y = Y A(γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' this yields yi−1 = yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consequently y0 = y1 = · · · = yd, so Y = y0I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have P(γ)P(−γ) = y0I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (13) For the product on the left in (13), we compute the (0, 0)-entry using matrix multiplication, and express the result in terms of hypergeometric series [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' this yields y0 = d � j=0 (−γ)j (1 − γ)j = 2F1 �−d, −γ 1 − γ ����1 � = d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (1 − γ)d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In the above line, the last equality is the Vandermonde summation formula [1, Chapter 2]: 2F1 � −d, b c ���� 1 � = (c − b)d (c)d with b = −γ and c = 1−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have verified (12), and the claim is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By the claim and (10), (11) the pair A, A∗ acts on V as a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Concerning the relations in (8), the last two are verified by matrix multiplication, and the first is obtain from the second using (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pair A, A∗ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10 will be called CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we define the notion of isomorphism for circular bidiagonal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let A, A∗ denote a circular bidiagonal pair on a vector space V , and let B, B∗ denote a circular bidiagonal pair on a vector space V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By an isomorphism of circular bidiagonal pairs from A, A∗ to B, B∗ we mean an isomorphism of vector spaces σ : V → V such that σA = Bσ and σA∗ = B∗σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We say that the circular bidiagonal pairs A, A∗ and B, B∗ are isomorphic whenever there exists an isomorphism of circular bidiagonal pairs from A, A∗ to B, B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In Section 7, we use the concepts of isomorphism and duality to intrepret the proof of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we show that the circular bidiagonal pairs in Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10 are mutually noniso- morphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 7 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold for d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Assume that Char(F) ̸= d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then circular bidiagonal pairs CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q′, ε′) are isomorphic if and only if both q = q′, ε = ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that Char(F) = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then circular bidiagonal pairs CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ′) are isomorphic if and only if γ = γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='13 will be completed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we describe how to adjust a circular bidiagonal pair to obtain another circular bidiag- onal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let A, A∗ denote a circular bidiagonal pair on a vector space V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick scalars s, s∗, t, t∗ in F with s, s∗ nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the pair sA + tI, s∗A∗ + t∗I is a circular bidigonal pair on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='14, the pair sA + tI, s∗A∗ + t∗I is called an affine transformation of A, A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we define the notion of affine equivalence for circular bidiagonal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let A, A∗ and B, B∗ denote circular bidiagonal pairs over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We say that A, A∗ and B, B∗ are affine equivalent whenever there exists an affine transformation of A, A∗ that is isomorphic to B, B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we apply the concept of affine equivalence to the circular bidiagonal pairs in Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold for d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Assume that Char(F) ̸= d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then circular bidiagonal pairs CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q′, ε′) are affine equivalent if and only if both q = q′, ε′ ∈ {ε, qε, q2ε, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qdε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that Char(F) = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then circular bidiagonal pairs CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ′) are affine equivalent if and only if γ′ − γ ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='17 will be completed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following is our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 8 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick an integer d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let A, A∗ denote a circular bidiagonal pair on a vector space of dimension d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' First assume that Char(F) ̸= d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then A, A∗ is affine equivalent to CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) for at least one ordered pair q, ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next assume that Char(F) = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then A, A∗ is affine equivalent to CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) for at least one γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18 will be completed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have a comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold for d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Assume that Char(F) ̸= d + 1, and write A, A∗ for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then A, A∗ is isomorphic to qA, q−1A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that Char(F) = d + 1, and write A, A∗ for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then A, A∗ is iso- morphic to A − I, A∗ − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19 will be completed in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In Section 9, we discuss how circular bidiagonal pairs are related to the circular Hessenberg pairs introduced by Jae-ho Lee [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 3 Preliminaries In this section, we review some basic concepts and notation that will be used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Recall the natural numbers N = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='} and integers Z = {0, ±1, ±2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Recall the field F from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For a, q ∈ F and r ∈ N define (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)r = (1 − a)(1 − aq)(1 − aq2) · · ·(1 − aqr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We interpret (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For a ∈ F and r ∈ N define (a)r = a(a + 1)(a + 2) · · ·(a + r − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We interpret (a)0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let λ denote an indeterminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The algebra F[λ] consists of the polynomials in λ that have all coefficients in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Fix an integer d ≥ 1, and let V denote a vector space with dimension d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let End(V ) denote the algebra consisting of the F- linear maps from V to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next we recall how each basis {vi}d i=0 of V yields an algebra isomorphism End(V ) → Matd+1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For A ∈ End(V ) and X ∈ Matd+1(F), we say that X represents A with respect to {vi}d i=0 whenever Avj = �d i=0 Xi,jvi for 0 ≤ j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The isomorphism sends A to the unique matrix in Matd+1(F) that represents A with respect to {vi}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For A ∈ End(V ), we say that A is diagonalizable whenever V is spanned by the eigenspaces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We say that A is multiplicity-free whenever A is diagonalizable, and each eigenspace of A has dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that A is multiplicity-free, and let {Vi}d i=0 denote an ordering of the eigenspaces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The sum V = �d i=0 Vi is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d let θi ∈ F denote the eigenvalue of A for Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By construction, the scalars {θi}d i=0 are mutually distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d define Ei ∈ End(V ) such that (Ei − I)Vi = 0 and EiVj = 0 if i ̸= j (0 ≤ j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Thus Ei is the projection V → Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We call Ei the primitive idempotent of A 9 associated with Vi (or θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have (i) EiEj = δi,jEi (0 ≤ i, j ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) I = �d i=0 Ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iii) Vi = EiV (0 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iv) tr(Ei) = 1 (0 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (v) A = �d i=0 θiEi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (vi) AEi = θiEi = EiA (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Moreover Ei = � 0≤j≤d j̸=i A − θjI θi − θj (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (14) Let M denote the subalgebra of End(V ) generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The vector space M has a basis {Ai}d i=0, and also 0 = �d i=0(A − θiI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Moreover, the elements {Ei}d i=0 form a basis for the vector space M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick scalars s, t ∈ F with s ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The map sA + tI is multiplicity-free, with eigenvalues {sθi + t}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d, the map Ei is the primitive idempotent of sA + tI associated with sθi + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Abbreviate n = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A scalar q ∈ F is called a primitive nth root of unity whenever qn = 1 and qi ̸= 1 for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' If q ∈ F is a primitive nth root of unity, then in the algebra F[λ], λn − 1 = (λ − 1)(λ − q) · · ·(λ − qd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' If Char(F) = n, then in the algebra F[λ], λn − λ = λ(λ − 1)(λ − 2) · · ·(λ − d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This fact is a version of Fermat’s little theorem [28, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For i, j ∈ Z we say that i ≡ j (mod n) whenever n divides i − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 4 The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18 In this section, our goal is to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Throughout this section, we fix an integer d ≥ 1, a vector space V with dimension d + 1, and a circular bidiagonal pair A, A∗ on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following result is a special case of [15, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' we will give a short proof for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (See [15, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=') Each of A, A∗ is multiplicity-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We first consider A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The map A is diagonalizable by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1(ii);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' we show that each eigenspace of A has dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' To do this, it suffices to show that A has d + 1 eigenspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let {vi}d i=0 denote a basis for V that satisfies Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let the matrix B ∈ Matd+1(F) represent A with respect to {vi}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By construction, B is circular bidiagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In particular, for B each entry on the subdiagonal is nonzero and each entry below the subdiagonal is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ r ≤ d we examine the entries of Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i, j ≤ d the (i, j)-entry of Br is nonzero if i − j = r, and zero if i − j > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore, the matrices {Br}d r=0 are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By this and linear algebra, the maps {Ar}d r=0 are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consequently, the minimal polynomial of A has degree d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This minimal polynomial has no repeated roots, since A is diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore, A has d + 1 distinct eigenvalues and hence d + 1 eigenspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have shown that A is multiplicity-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' One similarly shows that A∗ is multiplicity-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 10 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' M∗) denote the subalgebra of End(V ) generated by A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Note that {Ai}d i=0 is a basis for M, and {(A∗)i}d i=0 is a basis for M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let {Ei}d i=0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' {E∗ i }d i=0) denote an ordering of the primitive idempotents of A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d let 0 ̸= vi ∈ EiV and 0 ̸= v∗ i ∈ E∗ i V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Note that {vi}d i=0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' {v∗ i }d i=0) is a basis for V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The ordering {Ei}d i=0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' {E∗ i }d i=0) is called standard whenever the basis {vi}d i=0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' {v∗ i }d i=0) satisfies Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1(ii) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next we explain how the standard orderings in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3 are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In this ex- planation, we discuss the primitive idempotents of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' a similar discussion applies to the primitive idempotents of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let E and F denote primitive idempotents of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let us write E → F whenever there exists α ∈ F such that (A∗ − αI)EV = FV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For every primitive idempotent E of A, there exists a unique primitive idem- potent F of A such that E → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Moreover E ̸= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Since A∗ acts on the eigenspaces of A in a circular bidiagonal fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let {Ei}d i=0 denote an ordering of the primitive idempotents of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This order- ing is standard if and only if E0 → E1 → · · · → Ed → E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' There are exactly d + 1 standard orderings of the primitive idempotents of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6, for every primitive idempotent E of A, there exists a unique standard ordering {Ei}d i=0 of the primitive idempotents of A such that E = E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For the rest of this section, we fix a standard ordering {Ei}d i=0 of the primitive idempotents of A, and a standard ordering {E∗ i }d i=0 of the primitive idempotents of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d let θi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' θ∗ i ) denote the eigenvalue of A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A∗) for Ei (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' E∗ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Note that {Ei}d i=0 is a basis for M, and {E∗ i }d i=0 is a basis for M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have a comment about the subscript i in Ei, E∗ i , θi, θ∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Due to the circular nature of a circular bidiagonal pair, our calculations involving these subscripts will be carried out modulo n, where n = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The details are explained in the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For X ∈ {E, E∗, θ, θ∗} and i ∈ Z, we define Xi = Xr where 0 ≤ r ≤ d and i ≡ r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d define ai = tr(AE∗ i ), a∗ i = tr(A∗Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (15) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold for 0 ≤ i ≤ d: 11 (i) tr(AE∗ i ) = tr(E∗ i AE∗ i ) = tr(E∗ i A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) tr(A∗Ei) = tr(EiA∗Ei) = tr(EiA∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) By linear algebra, tr(XY ) = tr(Y X) for all X, Y ∈ End(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows from this and (E∗ i )2 = E∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d we have E∗ i AE∗ i = aiE∗ i , EiA∗Ei = a∗ i Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We verify the equation on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Abbreviate A = End(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The primitive idem- potent E∗ i has rank one, so E∗ i is a basis for E∗ i AE∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore, there exists αi ∈ F such that E∗ i AE∗ i = αiE∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In this equation, take the trace of each side and use (15) along with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='11(i) and tr(E∗ i ) = 1 to obtain ai = αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have verified the equation on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The equation on the right is similarly verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d we have (A − aiI)E∗ i V = E∗ i+1V, (A∗ − a∗ i I)EiV = Ei+1V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We verifiy the equation on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6, there exists αi ∈ F such that (A − αiI)E∗ i V = E∗ i+1V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In this equation, apply E∗ i to each side and evaluate the result using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' this yields 0 = E∗ i (A − αiI)E∗ i V = (ai − αi)E∗ i V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Of course E∗ i V ̸= 0, so αi = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have verified the equation on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The equation on the right is similarly verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold for 0 ≤ i, j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) E∗ i AE∗ j = � 0, if i − j ̸∈ {0, 1} (mod n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ̸= 0, if i − j ≡ 1 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) EiA∗Ej = � 0, if i − j ̸∈ {0, 1} (mod n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ̸= 0, if i − j ≡ 1 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following generalization of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='14 will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold for 0 ≤ i, j, r ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) E∗ i ArE∗ j = � 0, if i − j ̸∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , r} (mod n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ̸= 0, if i − j ≡ r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 12 (ii) Ei(A∗)rEj = � 0, if i − j ̸∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , r} (mod n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ̸= 0, if i − j ≡ r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This is a routine consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following holds for 0 ≤ i, j ≤ d: (i) EiE∗ j ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) E∗ i Ej ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Using (14) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='15(i), E∗ j+dEiE∗ j = E∗ j+d � � 0≤ℓ≤d ℓ̸=i A − θℓI θi − θℓ � E∗ j = E∗ j+dAdE∗ j � 0≤ℓ≤d ℓ̸=i 1 θi − θℓ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore EiE∗ j ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In each of (i)–(iv) below, we give a basis for the vector space End(V ): (i) EiE∗ j (0 ≤ i, j ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Ai(A∗)j (0 ≤ i, j ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iii) E∗ i Ej (0 ≤ i, j ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iv) (A∗)iAj (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) The dimension of End(V ) is (d + 1)2, and this is the number of vectors listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore, it suffices to show that the listed vectors are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that 0 = d � i=0 d � j=0 αi,jEiE∗ j (αi,j ∈ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We show that αr,s = 0 for 0 ≤ r, s ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let r, s be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have 0 = Er � d � i=0 d � j=0 αi,jEiE∗ j � E∗ s = αr,sErE∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have ErE∗ s ̸= 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='16(i), so αr,s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) By (i) and the notes below Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iii), (iv) Similar to the proof of (i), (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The next three lemmas contain results about A and {E∗ i }d i=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' similar results hold for A∗ and {Ei}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let θ denote an eigenvalue of A, and let 0 ̸= ξ ∈ V denote a corresponding eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the following (i)–(iii) hold: 13 (i) the vector E∗ i ξ is a basis for E∗ i V (0 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) the vectors {E∗ i ξ}d i=0 form a basis for V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iii) the basis {E∗ i ξ}d i=0 satisfies Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) The dimension of E∗ i V is one, so it suffices to show that E∗ i ξ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' There exists an integer j (0 ≤ j ≤ d) such that θ = θj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The subspace EjV has dimension one and contains ξ, so ξ spans EjV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore, E∗ i ξ spans E∗ i EjV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have E∗ i Ej ̸= 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='16(ii), so E∗ i EjV ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By these comments E∗ i ξ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) By (i) and since the sum V = �d i=0 E∗ i V is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iii) By Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We refer to the basis {E∗ i ξ}d i=0 in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' B∗) denote the matrix in Matd+1(F) that represents A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A∗) with respect to {E∗ i ξ}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the following (i)–(iv) hold: (i) B is circular bidiagonal with constant row sum θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Bi,i = ai for 0 ≤ i ≤ d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iii) B0,d = θ − a0, and Bi,i−1 = θ − ai for 1 ≤ i ≤ d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iv) B∗ is diagonal, with B∗ i,i = θ∗ i for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) The matrix B is circular bidiagonal by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1(i) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define a vector 1 ∈ Fd+1 that has all entries 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have B1 = θ1, because ξ = �d i=0 E∗ i ξ is an eigenvector for A with eigenvalue θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By B1 = θ1, the matrix B has constant row sum θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iii) By (i), (ii) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (iv) The matrix B∗ is diagonal by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1(i) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8 we obtain B∗ i,i = θ∗ i for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let θ denote an eigenvalue of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then θ ̸= ai for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let 0 ̸= ξ ∈ V denote an eigenvector for A with eigenvalue θ, and let B ∈ Matd+1(F) represent A with respect to the basis {E∗ i ξ}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix B is circular bidiagonal by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore, B0,d ̸= 0 and Bi,i−1 ̸= 0 for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows in view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Our next general goal is to obtain a relation involving A and A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The next two lemmas contain results about A and {E∗ i }d i=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' similar results hold for A∗ and {Ei}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold for 0 ≤ i ≤ d: (i) E∗ i A = E∗ i AE∗ i + E∗ i AE∗ i−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) AE∗ i = E∗ i AE∗ i + E∗ i+1AE∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='14(i) we have E∗ i A = E∗ i AI = d � j=0 E∗ i AE∗ j = E∗ i AE∗ i + E∗ i AE∗ i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='14(i) we have AE∗ i = IAE∗ i = d � j=0 E∗ j AE∗ i = E∗ i AE∗ i + E∗ i+1AE∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d, AE∗ i − aiE∗ i = E∗ i+1AE∗ i = E∗ i+1A − ai+1E∗ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The equation on the left follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='21(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The equation on the right follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='21(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We bring in some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define M∗AM∗ = Span{XAY |X, Y ∈ M∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) M∗AM∗ + M∗ = AM∗ + M∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) M∗AM∗ + M∗ = M∗A + M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) To obtain the inclusion ⊆, we use Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='14(i), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='22 to obtain M∗AM∗ = Span{E∗ i AE∗ j |0 ≤ i, j ≤ d} = Span{E∗ i AE∗ j |0 ≤ i, j ≤ d, i − j ∈ {0, 1} (mod n)} = Span{E∗ i AE∗ i |0 ≤ i ≤ d} + Span{E∗ i+1AE∗ i |0 ≤ i ≤ d} ⊆ AM∗ + M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The inclusion ⊇ holds since I ∈ M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' There exists a unique sequence q, α, β, γ of scalars in F such that qAA∗ − A∗A + αA − βA∗ = γI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' First, we show that the sequence q, α, β, γ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='23, we obtain A∗A ∈ M∗A ⊆ M∗A + M∗ = AM∗ + M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 15 Therefore, there exist X, Y ∈ M∗ such that A∗A = AX + Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The elements {(A∗)r}d r=0 form a basis for M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Write X = d � r=0 αr(A∗)r, Y = d � r=0 βr(A∗)r, αr, βr ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We claim that αr = 0 and βr = 0 for 2 ≤ r ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' To prove the claim, we assume that it is false, and get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' There exists an integer r (2 ≤ r ≤ d) such that αr ̸= 0 or βr ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define m = max{r|2 ≤ r ≤ d, αr ̸= 0 or βr ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By construction 2 ≤ m ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Also, αr = 0 and βr = 0 for m + 1 ≤ r ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore X = m � r=0 αr(A∗)r, Y = m � r=0 βr(A∗)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='15(ii) and the construction, we have Em+iXEi = αmEm+i(A∗)mEi, Em+iY Ei = βmEm+i(A∗)mEi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Also, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='14(ii) and m ≥ 2, we have Em+iA∗Ei = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We may now argue 0 = Em+iA∗Eiθi = Em+iA∗AEi = Em+i(AX + Y )Ei = θm+iEm+iXEi + Em+iY Ei = (θm+iαm + βm)Em+i(A∗)mEi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have Em+i(A∗)mEi ̸= 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='15(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By these comments 0 = θm+iαm + βm for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In particular, 0 = θ0αm + βm, 0 = θ1αm + βm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have θ0 ̸= θ1, so αm = 0 and βm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This contradicts the definition of m, so the claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By the claim, X = α0I + α1A∗ and Y = β0I + β1A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Using this to evaluate A∗A = AX + Y , we obtain (16) with q = α1, α = α0, β = −β1, γ = −β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have shown that the sequence q, α, β, γ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This sequence is unique, because the following maps are linearly independent by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='17(ii): AA∗, A, A∗, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The sequence q, α, β, γ from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='24 is called the profile of A, A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 16 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold for 0 ≤ i ≤ d: (i) qθi+1 = θi + β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) θ∗ i+1 = qθ∗ i + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) In the equation (16), multiply each term on the left by Ei+1 and on the right by Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Simplify the result using Ei+1A∗Ei ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) In the equation (16), multiply each term on the left by E∗ i+1 and on the right by E∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Simplify the result using E∗ i+1AE∗ i ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following (i), (ii) hold for 0 ≤ i ≤ d: (i) ai � θ∗ i (q − 1) + α � = βθ∗ i + γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) a∗ i � θi(1 − q) + β � = αθi − γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) In the equation (16), multiply each term on the left and right by E∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Simplify the result using E∗ i AE∗ i = aiE∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) In the equation (16), multiply each term on the left and right by Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Simplify the result using EiA∗Ei = a∗ i Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The scalars α, β satisfy the following inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Assume that q ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then α ̸= (1 − q)θ∗ 0, β ̸= (q − 1)θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then α ̸= 0, β ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The inequality about α is from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='26(ii) with i = 0 and θ∗ 1 ̸= θ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The inequality about β is from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='26(i) with i = 0 and θ1 ̸= θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d we have θi+1 − θ0 θ1 − θ0 = i � ℓ=0 q−ℓ, θ∗ i+1 − θ∗ 0 θ∗ 1 − θ∗ 0 = i � ℓ=0 qℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We verify the equation on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='26(i), θi+1 − q−i−1θ0 = θi+1 − q−1θi + q−1(θi − q−1θi−1) + q−2(θi−1 − q−1θi−2) + · · · + q−i(θ1 − q−1θ0) = (1 + q−1 + q−2 + · · · + q−i)q−1β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Observe that θi+1 − θ0 = θi+1 − q−i−1θ0 + (q−i−1 − 1)θ0 = � 1 + q−1 + q−2 + · · · + q−i�� q−1β + (q−1 − 1)θ0 � = � 1 + q−1 + q−2 + · · · + q−i� (θ1 − θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This verifies the equation on the left in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The equation on the right in (17) is similarly verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 17 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have �d ℓ=0 qℓ = 0, and �i ℓ=0 qℓ ̸= 0 for 0 ≤ i ≤ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have θ∗ d+1 = θ∗ 0, and θ∗ i+1 ̸= θ∗ 0 for 0 ≤ i ≤ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows in view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Recall n = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The scalar q is related to Char(F) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Assume that Char(F) ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then q is a primitive nth root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that Char(F) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' First suppose that q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='30, n = 0 in F and 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , d are nonzero in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore Char(F) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next suppose that q ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='30, qn = 1 and qj ̸= 1 for 1 ≤ j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore q is a primitive nth root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In this case Char(F) ̸= n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' otherwise 0 = qn − 1 = (q − 1)n, forcing q = 1 for a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows from these comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='24, we introduced various parameters that describe the circular bidiagonal pair A, A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we consider how these parameters are affected by an affine transformation of A, A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Pick scalars s, s∗, t, t∗ in F with s, s∗ nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consider the circular bidiagonal pair A∨ = sA + tI, (A∗)∨ = s∗A∗ + t∗I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (18) Note that {Ei}d i=0 and {E∗ i }d i=0 are orderings of the primitive idempotents of A∨ and (A∗)∨, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' These orderings are standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d let θ∨ i (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (θ∗ i )∨) denote the eigenvalue of A∨ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (A∗)∨) for Ei (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' E∗ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Also, define a∨ i = tr(A∨E∗ i ), (a∗ i )∨ = tr � (A∗)∨Ei � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let q∨, α∨, β∨, γ∨ denote the profile of A∨, (A∗)∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We refer to the circular bidiagonal pair A∨, (A∗)∨ from (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In the tables below, we describe various parameters for A∨, (A∗)∨ in terms of the corresponding parameters for A, A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' parameter parameter description θ∨ i sθi + t (θ∗ i )∨ s∗θ∗ i + t∗ a∨ i sai + t (a∗ i )∨ s∗a∗ i + t∗ parameter parameter description q∨ q α∨ s∗α + t∗(1 − q) β∨ sβ + t(q − 1) γ∨ ss∗γ + ts∗α − st∗β + tt∗(1 − q) 18 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The first table is verified using Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10 and the discussion below (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The second table is verified using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='24 and the comment above Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we define what it means for the circular bidiagonal pair A, A∗ to be normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let E (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' E∗) denote a primitive idempotent of A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let θ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' θ∗) denote the corresponding eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pair A, A∗ is said to be normalized with respect to E and E∗ whenever α, β, θ, θ∗ satisfy the requirements in the table below: case α β θ θ∗ Char(F) ̸= n 0 0 1 1 Char(F) = n 1 1 0 0 Next, we put A, A∗ in normalized form by applying an affine transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We normalize the circular bidiagonal pair A, A∗ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Assume that Char(F) ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the circular bidiagonal pair (q − 1)A − βI (q − 1)θ0 − β , (1 − q)A∗ − αI (1 − q)θ∗ 0 − α is normalized with respect to E0 and E∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that Char(F) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the circular bidiagonal pair A − θ0I β , A∗ − θ∗ 0I α is normalized with respect to E0 and E∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This is readily checked using Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='28, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='31, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='32 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that the circular bidiagonal pair A, A∗ is normalized with respect to E0 and E∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Assume that Char(F) ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then qAA∗ − A∗A q − 1 = εI, where ε = γ/(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Moreover θi = q−i, θ∗ i = qi, ai = q−iε, a∗ i = qiε, (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that Char(F) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then AA∗ − A∗A + A − A∗ = γI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Moreover θi = i, θ∗ i = i, ai = i + γ, a∗ i = i − γ, (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Evaluate Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='24 and Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='26, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='27 using Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that the circular bidiagonal pair A, A∗ is normalized with respect to E0 and E∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Assume that Char(F) ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the scalar ε from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35(i) is not among 1, q, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that Char(F) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the scalar γ from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35(ii) is not among 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='20 and the data in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that the circular bidiagonal pair A, A∗ is normalized with respect to E0 and E∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) Assume that Char(F) ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the circular bidiagonal pair A, A∗ is isomorphic to CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε), where q, ε are from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that Char(F) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then the circular bidiagonal pair A, A∗ is isomorphic to CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ), where γ is from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='31(i), q is a primitive nth root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='36(i), ε is not among 1, q, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pair CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) is described in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have θ0 = 1 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore 1 is an eigenvalue of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' let 0 ̸= ξ ∈ V denote a corresponding eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consider the basis {E∗ i ξ}d i=0 of V from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' B∗) denote the matrix in Matd+1(F) that represents A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A∗) with respect to {E∗ i ξ}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19 (with θ = 1) and the data in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35(i), we find that CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) is equal to B, B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='31(ii), Char(F) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='36(ii), γ is not among 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pair CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) is described in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have θ0 = 0 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore 0 is an eigenvalue of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' let 0 ̸= ξ ∈ V denote a corresponding eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consider the basis {E∗ i ξ}d i=0 of V from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Let B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' B∗) denote the matrix in Matd+1(F) that represents A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A∗) with respect to {E∗ i ξ}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19 (with θ = 0) and the data in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='35(ii), we find that CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) is equal to B, B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='18 is immediate from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='34 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 5 The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='13 In this section, we prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='13 (i) Assume that CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q′, ε′) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We will show that q = q′ and ε = ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Write A, A∗ for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) and B, B∗ for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q′, ε′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12, there exists an invertible P ∈ Matd+1(F) such that PA = BP and PA∗ = B∗P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix A∗ is diagonal, and its diagonal entries are mu- tually distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix B∗ is diagonal, and its diagonal entries are mutually distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 20 Examining the entries of PA∗ = B∗P, we find that there exists a permutation p of the set {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d} such that for 0 ≤ i, j ≤ d, j = p(i) ⇔ Pi,j ̸= 0 ⇔ A∗ j,j = B∗ i,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have A∗ 0,0 = 1 = B∗ 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore p(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next we show that P is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrices A and B are circular bidiagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 1 ≤ i ≤ d we examine the � i, p(i − 1) � entry in PA = BP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' this gives Pi,p(i)Ap(i),p(i−1) = Bi,i−1Pi−1,p(i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By construction Bi,i−1 ̸= 0 and Pi−1,p(i−1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore Ap(i),p(i−1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Of course p(i) ̸= p(i − 1), so p(i) = p(i − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Using induction and p(0) = 0, we obtain p(i) = i for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We have shown that P is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Consequently P commutes with A∗, so A∗ = B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore q = q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Also, for 0 ≤ i ≤ d we have Ai,i = Bi,i, which implies that ε = ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We are done in one logical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We now consider the opposite logical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that q = q′ and ε = ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q′, ε′) are the same, and hence isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ✷ 6 The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='17 In this section, we prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We begin with some comments about the matrices A = A(q, ε) and A∗ = A∗(q) from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With the above notation, (A∗ − εI)A(q, qε) = qA(q, ε)(A∗ − εI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This is routinely verified by matrix multiplication, using the data in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The map A∗ − εI from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1 is an isomorphism of circular bidiagonal pairs, from A(q, qε), A∗ to qA(q, ε), A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define the matrix P = A∗ − εI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix P is invertible, since ε is not among 1, q, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1 and the construction, PA(q, qε) = qA(q, ε)P, PA∗ = A∗P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows in view of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pairs CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, qε) are affine equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='9, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='16 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 21 Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following circular bidiagonal pairs are mutually affine equivalent: CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, qiε) i ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3, and since affine equivalence is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we have some comments about the matrices A = A(γ) and A∗ from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With the above notation, (A∗ + γI)A(γ − 1) = � A(γ) − I � (A∗ + γI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This is routinely verified by matrix multiplication, using the data in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The map A∗ + γI from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='5 is an isomorphism of circular bidiagonal pairs, from A(γ − 1), A∗ to A(γ) − I, A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define the matrix P = A∗ + γI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix P is invertible, since γ is not among 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='5 and the construction, PA(γ − 1) = � A(γ) − I � P, PA∗ = A∗P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows in view of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pairs CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ − 1) are affine equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='11, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='16 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following circular bidiagonal pairs are mutually affine equivalent: CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ + i) i ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='7, and since affine equivalence is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='17 (i) Assume that CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q′, ε′) are affine equiva- lent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We will show that q = q′ and ε′ ∈ {ε, qε, q2ε, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qdε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Write A, A∗ for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) and B, B∗ for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q′, ε′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The profile of A, A∗ is q, α, β, γ where α = 0, β = 0, γ = ε(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (19) The profile of B, B∗ is q′, α′, β′, γ′ where α′ = 0, β′ = 0, γ′ = ε′(q′ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (20) By assumption, there exist scalars s, s∗, t, t∗ in F with s, s∗ nonzero such that sA+tI, s∗A∗+ t∗I is isomorphic to B, B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define A∨ = sA + tI and (A∗)∨ = s∗A∗ + t∗I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The profile q∨, α∨, β∨, γ∨ of A∨, (A∗)∨ is described in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pairs A∨, (A∗)∨ and B, B∗ are isomorphic, so they have the same profile: q∨ = q′, α∨ = α′, β∨ = β′, γ∨ = γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 22 Evaluate the above equations using (19), (20) and the second table in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This yields q = q′, t = 0, t∗ = 0, ss∗ε = ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By construction, the circular bidiagonal pairs sA, s∗A∗ and B, B∗ are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore sA has the same eigenvalues as B, and s∗A∗ has the same eigenvalues as B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The scalar 1 is an eigenvalue of A, so the scalar s is an eigenvalue of sA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The eigenvalues of B are 1, q, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By these comments s ∈ {1, q, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The scalar 1 is an eigenvalue of A∗, so the scalar s∗ is an eigenvalue of s∗A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The eigenvalues of B∗ are 1, q, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By these comments s∗ ∈ {1, q, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We may now argue ε′ = ss∗ε ∈ {ε, qε, q2ε, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qdε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we reverse the logical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that q′ = q and ε′ ∈ {ε, qε, q2ε, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , qdε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q′, ε′) are affine equivalent by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Assume that CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ′) are affine equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We will show that γ′ − γ ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Write A, A∗ for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) and B, B∗ for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The profile of A, A∗ is q, α, β, γ where q = 1, α = 1, β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (21) The profile of B, B∗ is q′, α′, β′, γ′ where q′ = 1, α′ = 1, β′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (22) By assumption, there exist scalars s, s∗, t, t∗ in F with s, s∗ nonzero such that sA+tI, s∗A∗+ t∗I is isomorphic to B, B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Define A∨ = sA + tI and (A∗)∨ = s∗A∗ + t∗I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The profile q∨, α∨, β∨, γ∨ of A∨, (A∗)∨ is described in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The circular bidiagonal pairs A∨, (A∗)∨ and B, B∗ are isomorphic, so they have the same profile: q∨ = q′, α∨ = α′, β∨ = β′, γ∨ = γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Evaluate the above equations using (21), (22) and the second table in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This yields s = 1, s∗ = 1, γ′ − γ = t − t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By construction, the circular bidiagonal pairs A + tI, A∗ + t∗I and B, B∗ are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Therefore A+tI has the same eigenvalues as B, and A∗+t∗I has the same eigenvalues as B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The scalar 0 is an eigenvalue of A, so the scalar t is an eigenvalue of A + tI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The eigenvalues of B are 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By these comments t ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The scalar 0 is an eigenvalue of A∗, so the scalar t∗ is an eigenvalue of A∗ + t∗I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The eigenvalues of B∗ are 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By these comments t∗ ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We may now argue γ′ − γ = t − t∗ ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we reverse the logical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Assume that γ′−γ ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=', d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) and CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ′) are affine equivalent by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ✷ 23 7 Isomorphism and duality For circular bidiagonal pairs, the concepts of duality and isomorphism were explained in Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In this section, we use these concepts to interpret the proof of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We refer to the circular bidiagonal pair CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix P(q, ε) from (3) is an isomorphism of circular bidiagonal pairs from A(q−1, ε), A∗(q−1) to A∗(q), A(q, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We showed in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6 that P(q, ε) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows from this along with (4), (5) and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following are dual, up to isomorphism of circular bidiagonal pairs: CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We refer to the circular bidiagonal pair CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The matrix P(γ) from (9) is an isomorphism of circular bidiagonal pairs from A(−γ), A∗ to A∗, A(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We showed in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10 that P(γ) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The result follows from this along with (10), (11) and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The following are dual, up to isomorphism of circular bidiagonal pairs: CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ), CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, −γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 8 The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19 Our goal in this section is to prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Our proof strategy is to display the isomorphism involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We will give a detailed description of this isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Recall the circular bidiagonal pair CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε), define a matrix R = R(q, ε) in Matd+1(F) with entries R0,d = 1 − ε and Ri,i−1 = qi − ε for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' All other entries of R are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We call R the raising matrix for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1, assume that d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then R = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 1 − ε q − ε 0 0 0 0 0 q2 − ε 0 0 0 0 0 q3 − ε 0 0 0 0 0 q4 − ε 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 24 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With reference to Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1, the following (i), (ii) hold: (i) Rd+1 = (−1)d(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)d+1I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) R is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) By matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) By (i) and since (ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q)d+1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we explain how R is related to the matrices A = A(q, ε) and A∗ = A∗(q) from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With the above notation, we have (i) A∗A − εI = R = q(AA∗ − εI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) qAR = RA and q−1A∗R = RA∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) By matrix multiplication, using the data in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6 and Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) Observe that qAR − RA = qA(A∗A − εI) − q(AA∗ − εI)A = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' q−1A∗R − RA∗ = q−1A∗q(AA∗ − εI) − (A∗A − εI)A∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we explain how R is related to the primitive idempotents of A and A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d, let Ei (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' E∗ i ) denote the primitive idempotent of A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A∗) for the eigenvalue q−i (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With the above notation, we have (i) REi = Ei+1R and RE∗ i = E∗ i+1R for 0 ≤ i ≤ d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) REiV = Ei+1V and RE∗ i V = E∗ i+1V for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) To obtain REi = Ei+1R, multiply each side of (14) on the left by R and on the right by R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Evaluate the result using θr = q−r (0 ≤ r ≤ d) along with qA = RAR−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The equation RE∗ i = E∗ i+1R is similar obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) By (i) above and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With the above notation, R is an isomorphism of circular bidiagonal pairs from A, A∗ to qA, q−1A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='4(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We turn our attention to the circular bidiagonal pair CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ), define a matrix R = R(γ) in Matd+1(F) with entries R0,d = γ and Ri,i−1 = i + γ for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' All other entries of R are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' We call R the raising matrix for CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 25 Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='7, assume that d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Then R = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 γ 1 + γ 0 0 0 0 0 2 + γ 0 0 0 0 0 3 + γ 0 0 0 0 0 4 + γ 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With reference to Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='7, the following (i), (ii) hold: (i) Rd+1 = (γ)d+1I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) R is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) By matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) By (i) and since (γ)d+1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we describe how R is related to the matrices A = A(γ) and A∗ from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With the above notation, (i) A∗ − A + γI = R = AA∗ − A∗A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) (A − I)R = RA and (A∗ − I)R = RA∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) By matrix multiplication, using the data in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10 and Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) We have [A, R] = [A, A∗ − A + γI] = [A, A∗] = R, [A∗, R] = [A∗, A∗ − A + γI] = −[A∗, A] = [A, A∗] = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Next, we describe how R is related to the primitive idempotents of A and A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' For 0 ≤ i ≤ d, let Ei (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' E∗ i ) denote the primitive idempotent of A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A∗) for the eigenvalue i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With the above notation, (i) REi = Ei+1R and RE∗ i = E∗ i+1R for 0 ≤ i ≤ d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) REiV = Ei+1V and RE∗ i V = E∗ i+1V for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (i) To obtain REi = Ei+1R, multiply each side of (14) on the left by R and on the right by R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Evaluate the result using θr = r (0 ≤ r ≤ d) along with A − I = RAR−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The equation RE∗ i = E∗ i+1R is similar obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' (ii) By (i) and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='9(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' With the above notation, R is an isomorphism of circular bidiagonal pairs from A, A∗ to A − I, A∗ − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='19 is immediate from Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 26 9 Circular Hessenberg pairs In [25] Jae-ho Lee introduced the concept of a circular Hessenberg pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' A circular bidiagonal pair is a special case of a circular Hessenberg pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10, we gave some examples of a circular bidiagonal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In the present section, we describe these examples using the notation of [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' In [25] it is assumed that d ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' we make the same assumption throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Recall CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, q, ε) from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This corresponds to [25, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1] with parameters a = 0, b = 0, c = 1, a∗ = 0, b∗ = 1, c∗ = 0, y = 1 − ε, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Here are some related parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to [25, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='1], θi = q−i, θ∗ i = qi (0 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' φi = (qi − 1)(qi − ε), ϑi = (qi − 1)(1 − ε) (1 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to [25, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12], bi = 0 (0 ≤ i ≤ d − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ai = q−iε (0 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ci = 1 − q−iε (1 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ξ = 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to [25, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='11], b∗ i = 0 (0 ≤ i ≤ d − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' a∗ i = qiε (0 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' c∗ i = 1 − qiε (1 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ξ∗ = 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Recall CBP(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' d, γ) from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' This corresponds to [25, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2] with parameters a = 0, b = 1, c = 0, a∗ = 0, b∗ = 1, c∗ = 0, y = −γ, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Here are some related parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to [25, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='2], θi = i, θ∗ i = i (0 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' φi = −i(i + γ), ϑi = −iγ (1 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to [25, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='12], bi = 0 (0 ≤ i ≤ d − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ai = i + γ (0 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ci = −i − γ (1 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ξ = −γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Referring to [25, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='11], b∗ i = 0 (0 ≤ i ≤ d − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' a∗ i = i − γ (0 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' c∗ i = γ − i (1 ≤ i ≤ d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' ξ∗ = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 27 10 Acknowledgement The first author thanks Jae-ho Lee for many conversations about circular bidiagonal pairs and circular Hessenberg pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' The authors thank ˇStefko Miklaviˇc for giving this paper a close reading and offering valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Andrews, R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Bidiagonal triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' 521 (2017) 104–134;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content='04882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfUPdn/content/2301.00121v1.pdf'} +page_content=' [12] D.' metadata={'source': 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Anisotropic Layer-by-Layer +Phase Transition in Few-Layer MoTe2 +Chia-Hao Lee,†,‡ Huije Ryu,¶,‡ Gillian Nolan,† Yichao Zhang,† Yangjin Lee,§ +Siwon Oh,∥ Hyeonsik Cheong,∥ Kenji Watanabe,⊥ Takashi Taniguchi,# +Kwanpyo Kim,§ Gwan-Hyoung Lee,∗,¶ and Pinshane Y. Huang∗,†,@ +†Department of Materials Science and Engineering, University of Illinois +Urbana–Champaign, Urbana, Illinois 61801, United States +‡These authors contributed equally to this work +¶Department of Materials Science and Engineering, Seoul National University, Seoul +08826, Korea +§Department of Physics, Yonsei University, Seoul 03722, Korea +∥Department of Physics, Sogang University, Seoul 04107, Korea +⊥Research Center for Functional Materials, National Institute for Materials Science, 1-1 +Namiki, Tsukuba 305-0044, Japan +#International Center for Materials Nanoarchitectonics, National Institute for Materials +Science, 1-1 Namiki, Tsukuba 305-0044, Japan +@Materials Research Laboratory, University of Illinois at Urbana–Champaign, Urbana, +Illinois 61801, United States +* E-mail: gwanlee@snu.ac.kr +* E-mail: pyhuang@illinois.edu +1 +arXiv:2301.02694v1 [cond-mat.mtrl-sci] 6 Jan 2023 + +Abstract +Understanding the phase transition mechanisms in two-dimensional (2D) materials +is a key to precisely tailor their properties at the nanoscale. Molybdenum ditelluride +(MoTe2) exhibits multiple phases at room temperature, making it a promising candi- +date for phase-change applications. Here, we fabricate lateral 2H-Td interfaces with +laser irradiation and probe their phase transitions from micro- to atomic scales with +in situ heating in the transmission electron microscope (TEM). By encapsulating the +MoTe2 with graphene protection layers, we create an in situ reaction cell compatible +with atomic resolution imaging. We find that the Td-to-2H phase transition initiates +at phase boundaries at low temperatures (200–225 ◦C) and propagates anisotropically +along the b-axis in a layer-by-layer fashion. We also demonstrate a fully reversible +2H-Td-2H phase transition cycle, which generates a coherent 2H lattice containing in- +version domain boundaries. Our results provide insights on fabricating 2D hetero-phase +devices with atomically sharp and coherent interfaces. +Keywords +In situ heating, anisotropic phase transition, laser irradiation, molybdenum ditelluride, +transmission electron microscopy, 2D materials +Phase transformations in two-dimensional transition metal dichalcogenides (2D TMDCs) +are an emerging research area due to their polymorphism—the ability to host different phases +of the same chemical composition with distinct crystal structures. Utilizing phases with di- +verse electronic properties, multiple functionalities can be compactly packaged into nanoscale +devices, such as monolithic 2D electronics1–3 and phase-change memories.4,5 Among the +group VI 2D TMDCs, molybdenum ditelluride (MoTe2) has been widely studied because +of the minimal energy difference (40 meV per formula unit)6–8 between the trigonal pris- +matic (2H), monoclinic (1T’), and orthorhombic (Td) phases shown in Figure 1a. While +the honeycomb lattice of the 2H structure is distinct, the 1T’ and Td phase share the same +2 + +monolayer structure, but have different stacking structure in multilayers. The 1T’ phase has +a monoclinic structure with β = 93.9◦, while the Td phase is orthorhombic with β = 90◦.9 +This stacking difference results in the broken inversion symmetry of the Td phase and its +unique quantum properties, including type-II Weyl fermions,10,11 quantum spin Hall effect,12 +giant magnetoresistance,13 and superconductivity.14 +Phase transitions between 2H- and 1T’-MoTe2 have been demonstrated using electric bi- +asing,4,15 strain,16 heat,17,18 ion intercalation,19 and laser irradiation.20–22 However, the phase +transitions between 2H- and Td-MoTe2 remain largely unexplored because the Td phase is +less thermodynamically stable than other phases under ambient conditions, which makes it +difficult to study its room temperature properties and phase-change behaviors. Convention- +ally, the Td phase is obtained by cooling 1T’-MoTe2 crystals down to 250 K23,24 or through +chemically alloying with W substitutions.25–27 Very recently, the 2H-to-Td transition has +been reported with high temperature annealing of hBN-encapsulated MoTe2,28 while this +work focuses on the reverse Td-to-2H transition and its atomic-scale mechanisms. Under- +standing the micro- to atomic scale phase transition mechanisms between the semiconducting +2H and the topological Td phase may open up new possibilities towards low-dissipation 2D +electronics and spintronics. +In situ TEM is a powerful technique to investigate phase transformations in 2D ma- +terials.29,30 However, electron beam irradiation31,32 and vacuum annealing33,34 can cause +major degradation of MoTe2 and other 2D TMDCs due to the significant loss of chalcogen +atoms, making it particularly challenging to probe their phase transitions without altering +the chemical composition. +Here, we combine in situ TEM with graphene encapsulation to study the reversible phase +transitions of MoTe2 from micro- to atomic scales. We first use laser irradiation to locally +convert few-layer MoTe2 flakes from the 2H to a mixture of 1T’ and Td phases, which we +find is primarily Td in the regions examined by our TEM experiments. Then, we apply in +situ pulsed heating to monitor the reverse phase transition from the Td to 2H phase with a +3 + +combination of aberration-corrected scanning transmission electron microscopy (STEM) and +dark-field TEM (DFTEM). We find that the Td-to-2H phase transition initiates at the 2H-Td +interface at around 200–225 ◦C. Between 200–400 ◦C, we observe a highly anisotropic phase +transition: the 2H phase fronts progress along the b-axis of the Td grains, in a layer-by-layer +fashion. The ability to visualize each 2H phase front enables measurements of Td-to-2H +phase transition kinetics of individual MoTe2 layers. Lastly, we demonstrate the reversibility +of phase transitions between 2H and Td phases with cycles of laser irradiation and vacuum +heating. +Figure 1b shows a schematic of the phase conversion process. To create an encapsulated +cell, we fabricate hBN/graphene/MoTe2/graphene/hBN heterostructures using a PDMS- +assisted pick-up technique.35 The MoTe2 flakes are mechanically exfoliated with lateral size +around tens of microns and 4–5 layers in thickness. The MoTe2 is encapsulated by both +monolayer graphene and 10 nm thick hBN on the top and bottom surfaces. The hBN layers +improve adhesion with the polymer film used in the pick-up technique and are removed before +(S)TEM analysis using XeF2 etching36 (see Supporting Information (SI) Section 1 and Figure +S1). The encapsulation is essential because it creates an enclosed reaction cell that acts as +physical and chemical barrier for MoTe2, minimizing the sublimation of Te and interactions +with the atmosphere during further processing. +If the MoTe2 were not encapsulated, it +would be nearly impossible to observe the phase transition without modifying the crystal +stoichiometry through the loss of Te atoms, which has been shown to impact the phase +transition. Using encapsulated samples, we did not observe any Te vacancy formation via +ADF-STEM during heating. Importantly, the graphene contributes minimal background +signal to the TEM images, enabling atomic-resolution imaging.31,37,38 +We then irradiate the encapsulated 2H-MoTe2 with a 532 nm laser to locally initiate +the phase transition from the 2H to a primarily Td phase (SI Section 2). Because it is dif- +ficult to distinguish 1T’ and Td phases, the phases of pristine and laser-irradiated MoTe2 +are characterized by multiple techniques, including aberration-corrected annular dark-field +4 + +STEM (ADF-STEM) images (Figure 1c–d), TEM diffraction, and polarized Raman spec- +troscopy (SI Figure S2). TEM diffraction and polarized Raman measurements indicate that +the resulting materials contain a mixture of 1T’ and Td phases, which is in agreement with +previous reports.39,40 The potential for mixed phases occurs because the calculated energy +difference between 1T’ and Td phase is less than 3 meV per unit cell.8,39 In the TEM samples +analyzed below, however, atomic resolution STEM imaging (Figure 1d) indicates that the +laser-irradiated material is primarily Td (see SI Figure S3 for top-down ADF-STEM image +simulation of 1T’ and Td phases). Therefore, we refer to the transformed phase as Td. +For in situ heating, we transfer the laser-irradiated, encapsulated MoTe2 specimens to a +microelectromechanical system (MEMS)-based heating TEM chip (SI Section 1). Bright-field +TEM (BFTEM) imaging before in situ heating (Figure 2a) shows very little contrast between +the 2H and Td phases, indicating a uniform thickness across the hetero-phase interface. The +selected-area electron diffraction (SAED) patterns in Figure 2b–c exhibit the characteristic +hexagonal and rectangular lattice of the 2H and Td phases. +We use DFTEM to map the real-space location and orientation of the 2H and Td phases +(SI Section 3). +DFTEM has been widely used to determine the crystal orientation and +stacking order of 2D materials41–43 and operates by selecting specific Bragg spots in the +diffraction pattern with an objective aperture, so that only crystal grains that diffract to a +narrow range of k-vectors appear bright in the image. DFTEM images of the 2H and Td +phases in Figure 2d–e are obtained by selecting the (¯1100)2H and (¯210)Td Bragg reflections, +marked with blue and orange circles. We observe Td grains with three orientation directions, +rotated 120◦ from each other. The b-axis of each Td orientation is parallel to one of the three +zig-zag directions of the three-fold symmetric 2H matrix (SI Figure S4). Figure 2f shows a +false-colored DFTEM overlay image mapping the four grains present after laser-irradiation: +the 2H phase (red) and the three Td orientations (green, yellow, and blue). The majority of +the Td region in Figure 2f is oriented in one of the three orientations (green), with needle-like +inclusions of the other two orientations. +5 + +Next, we perform in situ heating to investigate the reverse Td-to-2H phase transition. +We use heat pulses18 instead of continuous heating for three reasons: (1) Pulsing provides +flexibility to “halt” the phase transition at any time, rapidly jump to specific temperatures, +and even hold at different temperatures for more detailed kinetic studies. (2) The ability to +pause between pulses makes it possible to acquire both large field-of-view (FOV) DFTEM and +atomic-resolution ADF-STEM images at several positions between pulses, which provides +both a large-scale view of the phase transition kinetics and atomic scale snapshots at the +interfaces. (3) Heat pulsing minimizes the energy input and potential sublimation during +the phase transition. While the graphene encapsulation minimizes damage and sublimation +to the MoTe2, we find that heating at temperatures above 600 ◦C for 0.5 s can produce small +(5–10 nm) voids (see SI Movie S1). +We use DFTEM imaging to track the propagation of 2H phase between heat pulses, with +pulse durations of 0.5 to 60 s, and temperatures from 200 to 275 ◦C; note that all images are +acquired between pulses, when the sample is at room temperature. DFTEM images of the +2H phase after different heat pulse temperatures (Figure 3a–d and SI Movie S1 ) show that +the 2H region at the phase boundary propagates anisotropically toward the Td grain during +heating, forming a belt-shaped inclusion. +Figure 3e shows a large-FOV DFTEM image +where newly grown 2H regions, marked in red, inherit the orientation of the 2H matrix. We +occasionally observed inversion domains in the 2H phase, which we discuss later in Figure 5. +Contrary to previous reports of high (500–600 ◦C) 1T’-to-2H phase transition temperatures +in bulk samples,44 we observe the Td-to-2H phase transition initiates at temperatures as +low as 200–225 ◦C, from the existing 2H-Td interfaces. There are two reasons for such low +transition temperatures: First, the Td phase is thermodynamically unstable under ambient +condition, so only the kinetic barrier needs to be overcome. Second, the existing 2H-Td +interfaces act as nucleation sites, which further reduce the kinetic barrier of the Td-to-2H +phase transition. +Figure 3f shows a contour overlay of the 2H phase fronts captured between heat pulses +6 + +of 200–275 ◦C in a 4-layer thick sample. We outline the propagating 2H regions that contain +at least a monolayer of 2H phase. This image shows that the 2H phase growth is anisotropic +in-plane, progressing along the [010]Td (b-axis direction) of the Td grain. +This result is +in contrast to previous reports of an isotropic 1T’-to-2H transition, which is an averaged +result from large-scale polycrystalline 1T’ grains.44,45 The preferential b-axis growth of the +2H phase is observed for all Td orientations (SI Figure S5). The anisotropy occurs mainly +at low-temperatures, and we find the phase transition becomes more isotropic above 400 ◦C +(SI Movie S2). +As shown in the atomic models in Figure 3f, 2H-Td phase boundaries can be classified into +two types1 based on their symmetry: Type 1, where the phase boundary is parallel to the +b-axis of Td and Type 2, where the phase boundary is rotated by 120◦ from the b-axis of Td. +The anisotropic propagation of the 2H phase suggests that the propagation (growth) rate of +the type 2 interface is much faster than for type 1, resulting in the formation of belt-shaped +2H grains with mostly type 1 interfaces. This behavior can be described by a kinetic Wulff +construction,46–49 where the final crystal shape is predicted using thermodynamic and kinetic +factors including the interface energy and relative growth rate of different facets. The type 2 +interface energy is estimated to be 70 meV/˚A higher than type 1 interface,50 making type 1 +interfaces more thermodynamically stable. This is consistent with our observations that there +are more kinks and steps in the atomic-resolution ADF-STEM images of type 2 interfaces +than in the type 1 interfaces (Figure 3g,h). The 2H growth preferentially propagates along +the [010]Td direction due to the higher kink formation and expansion rate of type 2 interfaces. +Figure 4a shows that the newly grown 2H region in the 4-layer MoTe2 has multiple phase +fronts (I, II, III, and IV ). Figure 4b schematically shows horizontally staggered 2H phase +fronts and the resulting DFTEM intensities. In the kinematic limit, DFTEM intensities +scale quadratically with the number of 2H layers, making it possible to individually probe +the position of the 2H phase front at each layer. However, we are not able to determine +the depth of each phase front because TEM produces images that are averaged in projection +7 + +(along the direction of the electron beam path). In Figure 4c, we measure the 2H-Td interface +positions of each layer as a function of accumulated heating time. The calculated propagation +rates range from 0.07 to 0.4 nm/s at 225 ◦C and exhibit wide variability. For example, both +interfaces II (orange) and III (green) exhibit a sudden jump in 2H phase front position at +t = 300 sec after the fifteenth 250 ◦C pulse is applied. The non-uniform propagation rates +might be due to strain, defects, and differing surface energies between atomic layers (2H, +Td, and graphene encapsulation), which can locally alter the energy barrier of MoTe2 phase +transitions.51 +Next, we demonstrate that the laser-induced Td phase can be transformed back to the +2H phase via ex situ vacuum annealing (Figure 5 and SI Section 4). We characterize the +pristine, laser-irradiated, and annealed MoTe2 with Raman spectroscopy (Figure 5a–d and +SI Section 5). Pristine (as-stacked) MoTe2 (Figure 5a,b) exhibits the three characteristic +Raman peaks (E2g, A1g, and B2g) of the 2H phase, while laser-irradiated regions (red areas +in Fig 5c) exhibit the A1 and A2 modes of the Td phase. The Raman maps (Figure 5b,c) +show that the Td region transformed from 2H phase is uniform, with 2H-Td boundaries that +are sharp on the micron scale. After annealing at 800 ◦C for 3 hours, Raman mapping +indicates the structure is fully and uniformly converted to 2H phase, as shown in Figure 5d. +This result shows that a reversible phase transition of 2H- and Td-MoTe2 can be achieved +by laser irradiation and vacuum annealing. When we anneal multiple samples at different +temperatures (300–800 ◦C), all of them exhibit the Td-to-2H phase transition. +Finally, we examine the crystal structure of MoTe2 after a full conversion cycle from 2H, +to Td, and back to 2H phase in Figure 5e–h. The six-fold symmetry of the SAED pattern in +Figure 5e indicates that the converted sample has no rotational grain boundaries. However, +the DFTEM image (Figure 5f) shows that for a selected (¯1100)2H Bragg reflection (marked +with a green circle in Figure 5e), one of the two grains appears brighter due to the breaking +of Friedel’s law.43,52 By selecting a neighboring Bragg reflection (orange circle in Figure 5f), +the contrast of the two grains is reversed (Figure 5g). This indicates that the 2H grains have +8 + +inversion symmetric orientations separated by an inversion domain boundary (IDB), a twin +boundary commonly observed in 2D TMDCs.34,43,53 Figure 5h shows an atomic resolution +STEM image of an IDB in the 2H phase region after a full conversion cycle of 2H-Td-2H. +We also observe an IDB running through only 3 of the layers in a 5-layer MoTe2 sample +(SI Figure S6), indicating the IDBs do not necessarily go all the way through the sample. +IDBs are likely generated during the Td-to-2H phase transition because the Td grain has +two equivalent transition pathways (SI Figure S7). As a result, cyclic phase transitions from +2H-Td-2H convert a 2D single crystal to coherent 2H polycrystals stitched with IDBs. Our +work shows that cyclic phase transitions are a promising technique to fabricate the IDBs, +which act as one-dimensional metallic tunnels54,55 embedded in 2D semiconductors. +In conclusion, we have demonstrated that encapsulated, few-layer MoTe2 can be re- +versibly phase engineered between the semiconducting 2H phase and the Td phase using +laser irradiation and thermal annealing. Using in situ pulsed heating and DFTEM, we show +that the Td-to-2H phase transition initiates at the 2H-Td interfaces at temperatures as low +as 200–225 ◦C. Moreover, we observe anisotropic growth of the 2H phase front, which pref- +erentially propagates along the b-axis of the nearby Td grains. Our findings can be applied +to fabrication of coplanar 2D circuitry, including 2D Josephson junctions,56 broadband pho- +todetectors,57 and other hetero-phase devices. Finally, we demonstrate a new approach for +in situ studies of 2D materials using graphene encapsulation and pulsed heating, which can +be applied to other micro- to atomic scale in situ studies of solid state phase transitions. +9 + +Figure 1: Characterization and fabrication of different phases of MoTe2. (a) Atomic structure +models of 2H-, 1T’-, and Td-MoTe2 with top and side views. Monolayer models are made +for top view for clarity. (b) Schematic of the reversible phase transition of MoTe2. The +2H-MoTe2 flakes are encapsulated by graphene and hBN layers. +Local laser-irradiation +induces 2H-to-Td phase transition of MoTe2, while the Td phase reverts back to 2H phase +after thermal annealing. (c,d) Aberration-corrected ADF-STEM images for the 2H and Td +phases, respectively. +10 + +a +2H +1T' +2H +Mo +Te +1T'/T +b +monolayer +b +2H-MoTe2 +hBN +Graphene +SiO2/Si +Laser +Ta-MoTe2 +irradiation +2H +2H-MoTe2 +2 nm +AnnealedFigure 2: Phase and grain orientation mapping of laser-irradiated MoTe2 with DFTEM. (a) +BFTEM image of the suspended, graphene-encapsulated MoTe2 containing both 2H and Td +grains. The Td phase region is delineated by the laser trajectory and outlined by the black +dashed lines. The minimum width of the Td region is determined by the radial laser intensity +profile. (b,c) SAED patterns, (d,e) DFTEM images of 2H and Td phase, respectively. The +diffraction patterns (b,c) are acquired with zone axis perpendicular to the basal planes. The +weaker diffraction spots are generated by the graphene encapsulating layers. The DFTEM +images (d-e) are formed by selecting the (¯1100)2H and (¯210)Td Bragg reflections in (b) and +(c) with the objective aperture position marked with blue and orange circles. The objective +aperture and selected Bragg reflections are centered on the optical axis to reduce image +aberrations. (f) Overlay of false-colored DFTEM images of the 2H matrix (red) and three +different orientations of Td grains (green, yellow, and blue). The DFTEM images (d–f) are +acquired at the region marked by the yellow dashed square in (a). +11 + +b +[0001] +a +2H +(1100) +2H +5 nm.1 +2H +100 nm +C +[001] +e +2H +(210) +2H +200nm +100 nm +T. +5 nm +100nmFigure 3: Anisotropic, low-temperature Td-to-2H phase transition. (a–d) DFTEM images +formed from the (¯1100)2H spot are acquired at room temperature after heat pulses of 0.5 +s from 200 to 275 ◦C. The Td-to-2H transition initiates at the interface, and the 2H phase +front anisotropically propagates into the Td phase region. (e) Overlay of a low magnification +DFTEM image with newly grown 2H regions marked in red. The Td-to-2H phase transition +occurs primarily at 2H-Td interfaces. (f) Contour plot of the 2H phase front in the same +region as (a–d) shows propagation along the b-axis of nearby Td grain. The insets are the +atomic models of 2 different types of interface. The anisotropy arises from the different +interface energy of type 1 and 2 interfaces. ADF-STEM images of (g) type 1 and (h) type 2 +2H-Td interfaces with atomic kinks indicates a step-flow growth model. +12 + +200 °C +b +225 °C +250 °C +d +275 °C +a +C +Propagating +2H +1 layer 2H belt +Initial 2H-Td +interface +50 nm +e +1 +2H phase front I +propagation +2H growth +Type 2 +kink +2H +Type 1 +Type 1 +2H +Td +2H +Type 2 +a +200 nm +50 nm +nm +2.nmFigure 4: Layer-by-layer phase transition and growth kinetics measurement. (a) DFTEM +image that shows the layer-by-layer phase transition. The 2H-Td interface of different layers +are individually identified by their intensity difference. (b) Schematic of the intensity dif- +ferences of 4-layer MoTe2 2H-Td interfaces in DFTEM. The mono-, bi-, tri-, and quad-layer +2H phase fronts are labeled as I, II, III, and IV respectively. Note that the relative positions +(in the z-direction) of each phase front are unknown due to the projection nature of TEM. +(c) Plot of 2H phase front positions of different layers as a function of accumulated heating +time. We perform a series of short heat pulses to capture the phase transition. Each dot +corresponds to a heat pulse. The pulsing time ranges from 0.5 s to 1 min and can be read +from the horizontal spacing between the dots. The pulsing temperatures (200–275 ◦C) are +color-coded by the background shades. The propagation rates are extracted by the slope of +the curves, which have a strong temperature dependence. +13 + +b +a +c +e' beam +2H phase front growth +275 °C +: front position (nm) +2H +Td +2H +二 +200 +=M + 4-layer MoTe, +2H-T. interface +150 +4 +2 +100 +000 +phase t +50 +IV +DFTEM +000000 +川I +(2H selected) +200 °℃ +225 °C +250 °C +275 ° +2H +50 nm +0 +50 +100 +150 +200 +250 +0 +100 +200 +300 +400 +Distance (nm) +Accumulated time (s)Figure 5: Cyclic phase transition and recovery of MoTe2 (2H −→ Td −→ 2H phase) via laser +irradiation and annealing. (a) Raman spectra and (b–d) Raman mapping of pristine (as- +stacked), laser-irradiated, and annealed MoTe2. The Raman maps are visualized with the E2g +(2H) and A1 (Td) peak, respectively. (e) SAED pattern of the 2H phase region. The orange +and green circles denote the objective aperture positions that are used to generate DFTEM +images (f,g) from different (¯1100)2H Bragg reflections. The brighter region corresponds to the +specific 2H orientation that generates the stronger Bragg reflection. The inversion domain +boundary is outlined by the white dashed line. +(h) ADF-STEM image at the inversion +domain boundary of 2H grains with opposite orientations. The atomic models are overlaid +with arrows indicating opposing orientations. +I ASSOCIATED CONTENT +Supporting Information Sample fabrication workflow, 1T’ and Td mixture analysis, sim- +ulated ADF-STEM images, ADF-STEM images and atomic models of inversion domain +boundaries, and in situ movies of MoTe2 phase transition. +I AUTHOR INFORMATION +Corresponding Author +*Email: pyhuang@illinois.edu +*Email: gwanlee@snu.ac.kr +14 + +Pristine +b +a +↑E2g (2H) +2H +e +g +Pristine +A1g (2H) +B2g (2H) +5 μm +Intensity (a.u.) +Laser-irradiated +A (Ta) +2H +Laser-irradiated +3 nm-1 +2 μm +A2(Ta) +d +E2g (2H) +Annealed +Annealed +d +2H +A1g (2H) +B2g (2H) +100 +150 +200 +250 +300 +350 +2 μm +Raman shift (cm-1)Author Contributions +Under supervision by P.Y.H., C.-H.L., G.N. and Y.Z. acquired and analyzed the in situ heat- +ing DFTEM and ADF-STEM images. Under supervision by G.-H.L., H.R. fabricated the +MoTe2 samples, performed ex situ annealing experiments and Raman spectroscopy. Under +supervision by K.K., Y.L. perform TEM analysis of phase-engineered MoTe2. Under super- +vision by H.C., S.O. conducted polarized Raman measurements. K.W. and T.T. synthesized +the hBN flakes. All authors read and contributed to the manuscript. +Notes +The authors declare no competing financial interest. +Acknowledgement +This material is based upon work supported by the U.S. Department of Energy, Office of Sci- +ence, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under +award number DE-SC0020190, which supported the electron microscopy and related data +analysis. This work was carried out in part in the Materials Research Laboratory Central +Facilities at the University of Illinois at Urbana–Champaign. G.-H.L. acknowledges support +by the Creative-Pioneering Researchers Program through Seoul National University (SNU), +the National Research Foundation (NRF) of Korea (NRF-2021R1A2C3014316, SRC pro- +gram: vdWMRC center 2017R1A5A1014862, NRF-2021M3F3A2A01037858), the Research +Institute of Advanced Materials (RIAM), Institute of Engineering Research, and Institute +of Applied Physics at SNU, which supported the sample fabrication and ex situ character- +ization. K.W. and T.T. acknowledge support from the Japan Society for the Promotion of +Science (JSPS) KAKENHI (Grant Numbers 19H05790 and 20H00354) and A3 Foresight by +JSPS, which supported the h-BN synthesis. +15 + +References +(1) Sung, J. H. et al. Coplanar semiconductor–metal circuitry defined on few-layer MoTe2 +via polymorphic heteroepitaxy. Nature Nanotechnology 2017, 12, 1064–1070. +(2) Ma, R.; Zhang, H.; Yoo, Y.; Degregorio, Z. P.; Jin, L.; Golani, P.; Ghasemi Azadani, J.; +Low, T.; Johns, J. E.; Bendersky, L. 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Advanced Materials 2018, 30, 1707152. +23 + +Graphical TOC Entry +24 + +View direction +2H +2H +50 nm +200 °C +- +2H +2H +AL +275 °C +IVSupporting Information +In situ Imaging of an Anisotropic Layer-by-Layer +Phase Transition in Few-Layer MoTe2 +Chia-Hao Lee, Huije Ryu, Gillian Nolan, Yichao Zhang, Yangjin Lee, Siwon Oh, +Hyeonsik Cheong, Kenji Watanabe, Takashi Taniguchi, Kwanpyo Kim, +Gwan-Hyoung Lee,∗ and Pinshane Y. Huang∗ +* E-mail: gwanlee@snu.ac.kr +* E-mail: pyhuang@illinois.edu +1 +arXiv:2301.02694v1 [cond-mat.mtrl-sci] 6 Jan 2023 + +1. Sample preparation for in and ex situ experiments +To fabricate the hBN/Gr/MoTe2/Gr/hBN heterostructures, we mechanically exfoliated thin +layers of 2D materials (MoTe2, hBN, graphene) from bulk crystals (MoTe2: HQ graphene, +graphene: NGS Naturgraphite GmbH, hBN: NIMS) onto SiO2/Si substrate. We then used +the pick-up transfer technique1 with a poly(bisphenol A carbonate, Sigma Aldrich) (PC)- +coated poly (dimethyl siloxane) (PDMS) lens mounted on a microscope slide to pick-up and +released the constituent flakes on the substrate. The PC/PDMS/glass slide was held in a +3-axis micromanipulator to control the position of the contact area with the 2D materials. +The substrate was placed on a heating stage. By controlling the temperature of the heating +stage (80–130 ◦C), the 2D flakes were picked up by the PC with minimal cracking or folding, +leaving the substrate on the heating stage. The hBN/Gr/MoTe2/Gr/hBN heterostructures +were fabricated by repeating the above steps, and then transferred onto a clean SiO2/Si +substrate by releasing the PC film from the PDMS lens at a temperature above 180 ◦C. +Lastly, we placed the entire sample in chloroform for 30 min to remove the PC film. +To image the MoTe2 at atomic resolution and reduce multiple scattering from thick hBN +layers, we removed the hBN layers with XeF2 dry etching.2 The sample fabrication process +was similar with the one used for ex situ experiments but with some modifications, see +Figure S1 for the schematic. The bottom hBN layer was etched away by the XeF2 exposure, +and the etching process was self-limited at the graphene layer. We transferred the stack +onto a clean SiO2/Si substrate and exposed it with chloroform, oxygen plasma, and XeF2 +again to remove the top hBN layers. After these steps, the stack was encapsulated with +fluorinated graphene and ready for transferring onto an in situ heating TEM chip (E-FHDC- +VO-10, Protochips). We used the conventional polymer transfer technique with poly(methyl +methacrylate) (PMMA) and KOH to transfer the stack3 from SiO2/Si substrate.4 After +transferring, the PMMA film was removed by placing the samples in acetone for 12 hours. +2 + +2. Laser irradiation parameters for phase transition +To initiate the 2H-to-Td phase transition of MoTe2, we irradiated the encapsulated samples +using continuous wave (CW) 532 nm laser and power of 21 mW at ambient conditions. The +laser was focused by a 100× objective lens (N.A. = 0.9) and the resulting spot size on the +substrate was around 1µm. The laser-irradiated area was patterned by rastering the laser +spot with 200 nm point-to-point distance and 0.1 s exposure time per step. +3. S/TEM measurement +In situ TEM experiments were done in a Thermo Fisher Scientific Themis-Z aberration- +corrected S/TEM operated at 80 kV. For atomic-resolution ADF-STEM imaging, the point +resolution was about 1 ˚A with 25 mrad convergence semi angle, 35 pA probe current, 63 +to 200 mrad collection semi angles, 20 pm pixel size and a total dwell time of 20 µs/pixel +using 10-frame averages. For BFTEM, SAED, and DFTEM, the data were acquired with +a Ceta 16M camera at parallel illumination using the three-condenser TEM mode. The +electron dose rate was around 103 e-/nm2/s and the exposure times for SAED and DFTEM +were 2 to 5 s. Note that sparse dark pits were observed in DFTEM at the end of the in +situ imaging after an accumulated total dose around 4 × 105 e-/nm2. While the graphene +encapsulation was unlikely to form holes under this condition, these dark pits were likely to +be crystallographic defects such as voids formed by displacing atoms of the MoTe2 flakes. +3 + +4. Ex situ Td-to-2H phase transition with annealing +The ex situ Td-to-2H phase transition of MoTe2 (Figure 5 of the main text) was performed +by an annealing process in a vacuum furnace. We annealed the laser-irradiated sample in +a vacuum chamber (10-4 Torr) and slowly ramped up to targeted temperatures in 3 hours +and held for another 3 hours. The targeted temperatures were set from 300 to 800 ◦C. The +furnace was naturally cooled to room temperature. +5. Raman spectroscopy measurement +The linearly polarized Raman measurements (SI Figure S2) were carried out in the backscat- +tering geometry using 514.5 nm laser excitation. The input laser beam was focused onto the +samples by a 50× microscope objective lens (0.8 NA), and the scattered light was collected +and collimated by the same objective lens. +To access the low-frequency range below 50 +cm–1, volume holographic filters (OptiGrate) were used to clean the laser lines and reject the +Rayleigh-scattered light. A laser with a low power of 300 µW was used to avoid local heat- +ing. The Raman scattering signals were dispersed by a Jobin–Yvon iHR550 spectrometer +with a 2400 grooves/mm grating (400 nm blaze) and detected by a liquid-nitrogen-cooled, +back-illuminated CCD detector. An achromatic half-wave plate was used to rotate the polar- +ization of the linearly polarized laser beam to the desired direction. The analyzer angle was +set such that photons with polarization parallel to the incident polarization passed through. +Another achromatic half-wave plate was placed in front of the spectrometer to keep the +polarization direction of the signal entering the spectrometer constant with respect to the +groove direction of the grating. The Raman spectra (Figure 5 of the main text) were ac- +quired using a HORIBA LabRAM HR Evolution with the laser wavelength at 532 nm. To +minimize the irradiation damage, the laser power was set below 5 mW with an acquisition +time of 60 s. All measurements were conducted at ambient conditions. +4 + +Figure S1: Sample preparation for in situ TEM experiments. (a-1) The schematic illustration +of hBN/Gr/MoTe2/Gr/hBN structure on SiO2 (285 nm)/Si++ substrate. (a-2,3) Pick-up +the structure by polycarbonate (PC) film on polydimethylsiloxane (PDMS). (b-1,2) Etching +the bottom side of the hBN by exposing to XeF2 gas. (b-3) After exposing to XeF2 gas, +the bottom hBN is completely etched, while the graphene layers and the encapsulated layers +remain. In addition, the PC exposed by XeF2 is also chemically modified, which can not be +dissolved by chloroform. (c-1,2) One-side-etched sample is transferred to another SiO2(285 +nm)/Si++ substrate at about 180 ◦C and separated from the PDMS lens. The Si substrate +was treated with O2 plasma to increase the adhesion energy of SiO2. (c-3,4) Remove the +PC film by chloroform bath. Note that the fluorinated PC was not dissolved in chloroform. +(c-5,6) Etch off the fluorinated PC layer by O2 plasma. Since the etch rate of hBN is much +slower than fluorinated PC layer using O2 plasma, the fluorinated PC layer is removed while +the Gr/MoTe2/fluorinated Gr structure remains. +(c-7,8) Remove the top hBN by XeF2 +gas etching. Finally, we transferred the heterostructures on MEMS TEM chips using the +PMMA-assisted, wet-transfer method. +5 + +a. Pick-up +a-1 +a-2 +a-3 +Glass +PDMS Lens +hBN +1L Gr +Pick-up +PC +MoTe2 +SiO, +b. 1st XeF² gas etching +b-1 +b-2 +b-3 +F-treated PC layer +Fluorinated Gr +XeF2 gas etching +c. Transfer on SiO & 2nd XeF2 gas etching +c-1 +c-3 Chloroform +C-2 +c-4 +XeF2 gas etching +C-6 +C-7 +C-8 +C-5 +O, plasmaFigure S2: Mixture of 1T’ and Td phase characterized by TEM diffraction and polarized +Raman spectroscopy. (a) Selected area electron diffraction pattern (SAED) acquired at a +region with mixed 1T’ and Td phases. The red circles are Bragg peaks of Td phase that +would be absent if it were 1T’ phase, however, the intensities are too weak for the region to +be pure Td phase, indicating a mixture of 1T’ and Td phases. The Bragg peaks inside the +blue circles are also characteristic peaks of Td phase that are absent in the 1T’ phase. (b) +Polarized Raman spectra of Td + 1T’- (purple) and 2H-MoTe2 (orange). The 1T’ and Td +phase are typically characterized by the peak splitting around 128 cm-1: those with a split +peak were identified as the Td phase, and those with a single peak as the 1T’ phase.5 The +blue peak in the inset indicates the presence of Td phase. However, considering the SAED +result in (a), our specimen shows a spatial inhomogeneity of mixture of 1T’ and Td phases. +6 + +b +a +Ta+1T +2H +O +Intensity (a.u.) +125130 +135 +. +50 +100 +150 +200250 +300 +0.5 A-1 +Raman Shift (cm-1)Figure S3: ADF-STEM image simulation of 1T’- and Td-MoTe2. (a–d) Simulated ADF- +STEM images of 1T’ and Td phase at different orientations using semi-quantitative image +simulation package6 (e–h) Atomic models of 1T’ and Td phase at different orientations. The +3.9◦ tilt angle is chosen to match the β angle of 1T’ phase. +7 + +1T' +along c-axis +along (001) normal +along c-axis +3.90 tiltFigure S4: Crystallographic relation between the Td and the 2H matrix. (a,b) Atomic models +of top-view, monolayer 2H and Td phases. (c,d) Simulated diffraction patterns of 2H and Td +phase. (e) Overlay of the simulated diffraction patterns of 2H and Td phase. The Td phase +can be derived by shifting the chalcogen layers in 2H phase along one of the three arm-chair +directions followed by some metal atom dimerization. Therefore, the b-axis of the derived +Td variants are parallel to one of the three zig-zag directions of the 2H matrix. +8 + +2HFigure S5: Anisotropic phase transition for all 3 Td orientations. +(a) Overlay of false- +colored DFTEM images of the 2H matrix (red) and three different orientations of Td grains. +Reproduced from Figure 2f of the main text. (b–d) DFTEM images of 2H-Td interfaces +with different Td phase orientations. (e–g) DFTEM images of 2H-Td interfaces after heat +pulses. The white arrows indicate the growth directions of the 2H phase front. The growth +directions are parallel to the b-axis directions of the nearby Td grains. (h–j) Atomic models +of 2H-Td interface with three different orientations. The b-axis directions of the Td phases +are marked by the black arrows. +9 + +d +a +00 + nm +e +g +100 nm +2HFigure S6: ADF-STEM images of an inversion domain boundary (IDB) at (a) lower and +(b) higher magnifications. +The field-of-view is marked by the yellow square. +The IDBs +are marked by the blue arrows. In this specific region, the IDB is only observed at the +4-layer region, indicating the IDB does not go all-the-way-through this 5-layer sample and +suggesting an independent layer-by-layer phase transition mechanism. The layer number is +determined by quantitative ADF-STEM intensities. +10 + +a +5L +IDB +4L +4L +2L +2L +Gr +50 nm +5 nmFigure S7: Formation mechanism of the inversion domain boundary. Atomic models of two +potential pathways of Td-to-2H phase transition in (a) side-view and (b,c) top-views. By +sliding either the top or bottom chalcogen layers along the a-axis direction, 2H grains with +opposite orientations can be derived from a single crystalline Td grain. Therefore, shifting +opposite layers of the chalcogen atoms in a Td grain will generate an inversion domain +boundary. The dark (Pathway 1) and light (Pathway 2) blue arrows indicate the sliding +directions of each chalcogen layer. +11 + +a +Pathway 2 +Pathway 1 +C +Shift the bottom chalcogen atoms +Shift the top chalcogen atoms +Mutually invertedMovie S1: DFTEM video acquired using the (¯1100)2H spot at room temperature after each +heat pulse from 200 to 275 ◦C, showing the in-plane, layer-by-layer, and anisotropic Td-to-2H +phase transition. The heat pulses range from 0.5 s to 1 min. +12 + +200°℃ +100nmMovie S2: DFTEM video acquired using the (¯1100)2H spot at room temperature after each +heat pulse from 200 to 700 ◦C, showing the anisotropic Td-to-2H phase transition at lower +temperatures. The phase transition then become more isotropic at temperatures above 400 +◦C. We applied two rounds of heating, the first round is 200–400 ◦C, while the second round +is 200–700 ◦C. The temperature intervals are all 25 ◦C and the heat pulses are all 0.5 s. +References +(1) Purdie, D. G.; Pugno, N. M.; Taniguchi, T.; Watanabe, K.; Ferrari, A. C.; Lombardo, A. +Cleaning interfaces in layered materials heterostructures. Nature Communications 2018, +9, 5387. +(2) Son, J.; Kwon, J.; Kim, S.; Lv, Y.; Yu, J.; Lee, J.-Y.; Ryu, H.; Watanabe, K.; +Taniguchi, T.; Garrido-Menacho, R.; Mason, N.; Ertekin, E.; Huang, P. Y.; Lee, G.-H.; +M. van der Zande, A. Atomically precise graphene etch stops for three dimensional inte- +grated systems from two dimensional material heterostructures. Nature Communications +2018, 9, 3988. +13 + +200 +500 nm(3) Reina, A.; Jia, X.; Ho, J.; Nezich, D.; Son, H.; Bulovic, V.; Dresselhaus, M. S.; Kong, J. +Large Area, Few-Layer Graphene Films on Arbitrary Substrates by Chemical Vapor +Deposition. Nano Letters 2009, 9, 30–35. +(4) van der Zande, A. M.; Huang, P. Y.; Chenet, D. A.; Berkelbach, T. C.; You, Y.; Lee, G.- +H.; Heinz, T. F.; Reichman, D. 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Computem. 2013; http://sourceforge.net/projects/computem. +14 + diff --git a/PtE0T4oBgHgl3EQf1AKn/content/tmp_files/load_file.txt b/PtE0T4oBgHgl3EQf1AKn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb5b2f8d7f19aa7cb6dc32cd04698b132e9e1d7b --- /dev/null +++ b/PtE0T4oBgHgl3EQf1AKn/content/tmp_files/load_file.txt @@ -0,0 +1,1502 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf,len=1501 +page_content='In situ Imaging of an Anisotropic Layer-by-Layer Phase Transition in Few-Layer MoTe2 Chia-Hao Lee,†,‡ Huije Ryu,¶,‡ Gillian Nolan,† Yichao Zhang,† Yangjin Lee,§ Siwon Oh,∥ Hyeonsik Cheong,∥ Kenji Watanabe,⊥ Takashi Taniguchi,# Kwanpyo Kim,§ Gwan-Hyoung Lee,∗,¶ and Pinshane Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Huang∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='@ †Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' University of Illinois Urbana–Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Urbana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Illinois 61801,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' United States ‡These authors contributed equally to this work ¶Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Seoul National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Seoul 08826,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Korea §Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Yonsei University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Seoul 03722,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Korea ∥Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Sogang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Seoul 04107,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Korea ⊥Research Center for Functional Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Tsukuba 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Japan #International Center for Materials Nanoarchitectonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Tsukuba 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Japan @Materials Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' University of Illinois at Urbana–Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Urbana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Illinois 61801,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' United States E-mail: gwanlee@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='kr E-mail: pyhuang@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='02694v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='mtrl-sci] 6 Jan 2023 Abstract Understanding the phase transition mechanisms in two-dimensional (2D) materials is a key to precisely tailor their properties at the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Molybdenum ditelluride (MoTe2) exhibits multiple phases at room temperature, making it a promising candi- date for phase-change applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Here, we fabricate lateral 2H-Td interfaces with laser irradiation and probe their phase transitions from micro- to atomic scales with in situ heating in the transmission electron microscope (TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' By encapsulating the MoTe2 with graphene protection layers, we create an in situ reaction cell compatible with atomic resolution imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We find that the Td-to-2H phase transition initiates at phase boundaries at low temperatures (200–225 ◦C) and propagates anisotropically along the b-axis in a layer-by-layer fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We also demonstrate a fully reversible 2H-Td-2H phase transition cycle, which generates a coherent 2H lattice containing in- version domain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Our results provide insights on fabricating 2D hetero-phase devices with atomically sharp and coherent interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Keywords In situ heating, anisotropic phase transition, laser irradiation, molybdenum ditelluride, transmission electron microscopy, 2D materials Phase transformations in two-dimensional transition metal dichalcogenides (2D TMDCs) are an emerging research area due to their polymorphism—the ability to host different phases of the same chemical composition with distinct crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Utilizing phases with di- verse electronic properties, multiple functionalities can be compactly packaged into nanoscale devices, such as monolithic 2D electronics1–3 and phase-change memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='4,5 Among the group VI 2D TMDCs, molybdenum ditelluride (MoTe2) has been widely studied because of the minimal energy difference (40 meV per formula unit)6–8 between the trigonal pris- matic (2H), monoclinic (1T’), and orthorhombic (Td) phases shown in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' While the honeycomb lattice of the 2H structure is distinct, the 1T’ and Td phase share the same 2 monolayer structure, but have different stacking structure in multilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The 1T’ phase has a monoclinic structure with β = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='9◦, while the Td phase is orthorhombic with β = 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='9 This stacking difference results in the broken inversion symmetry of the Td phase and its unique quantum properties, including type-II Weyl fermions,10,11 quantum spin Hall effect,12 giant magnetoresistance,13 and superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='14 Phase transitions between 2H- and 1T’-MoTe2 have been demonstrated using electric bi- asing,4,15 strain,16 heat,17,18 ion intercalation,19 and laser irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='20–22 However, the phase transitions between 2H- and Td-MoTe2 remain largely unexplored because the Td phase is less thermodynamically stable than other phases under ambient conditions, which makes it difficult to study its room temperature properties and phase-change behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Convention- ally, the Td phase is obtained by cooling 1T’-MoTe2 crystals down to 250 K23,24 or through chemically alloying with W substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='25–27 Very recently, the 2H-to-Td transition has been reported with high temperature annealing of hBN-encapsulated MoTe2,28 while this work focuses on the reverse Td-to-2H transition and its atomic-scale mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Under- standing the micro- to atomic scale phase transition mechanisms between the semiconducting 2H and the topological Td phase may open up new possibilities towards low-dissipation 2D electronics and spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' In situ TEM is a powerful technique to investigate phase transformations in 2D ma- terials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='29,30 However, electron beam irradiation31,32 and vacuum annealing33,34 can cause major degradation of MoTe2 and other 2D TMDCs due to the significant loss of chalcogen atoms, making it particularly challenging to probe their phase transitions without altering the chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Here, we combine in situ TEM with graphene encapsulation to study the reversible phase transitions of MoTe2 from micro- to atomic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We first use laser irradiation to locally convert few-layer MoTe2 flakes from the 2H to a mixture of 1T’ and Td phases, which we find is primarily Td in the regions examined by our TEM experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Then, we apply in situ pulsed heating to monitor the reverse phase transition from the Td to 2H phase with a 3 combination of aberration-corrected scanning transmission electron microscopy (STEM) and dark-field TEM (DFTEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We find that the Td-to-2H phase transition initiates at the 2H-Td interface at around 200–225 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Between 200–400 ◦C, we observe a highly anisotropic phase transition: the 2H phase fronts progress along the b-axis of the Td grains, in a layer-by-layer fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The ability to visualize each 2H phase front enables measurements of Td-to-2H phase transition kinetics of individual MoTe2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Lastly, we demonstrate the reversibility of phase transitions between 2H and Td phases with cycles of laser irradiation and vacuum heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Figure 1b shows a schematic of the phase conversion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' To create an encapsulated cell, we fabricate hBN/graphene/MoTe2/graphene/hBN heterostructures using a PDMS- assisted pick-up technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='35 The MoTe2 flakes are mechanically exfoliated with lateral size around tens of microns and 4–5 layers in thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The MoTe2 is encapsulated by both monolayer graphene and 10 nm thick hBN on the top and bottom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The hBN layers improve adhesion with the polymer film used in the pick-up technique and are removed before (S)TEM analysis using XeF2 etching36 (see Supporting Information (SI) Section 1 and Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The encapsulation is essential because it creates an enclosed reaction cell that acts as physical and chemical barrier for MoTe2, minimizing the sublimation of Te and interactions with the atmosphere during further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' If the MoTe2 were not encapsulated, it would be nearly impossible to observe the phase transition without modifying the crystal stoichiometry through the loss of Te atoms, which has been shown to impact the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Using encapsulated samples, we did not observe any Te vacancy formation via ADF-STEM during heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Importantly, the graphene contributes minimal background signal to the TEM images, enabling atomic-resolution imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='31,37,38 We then irradiate the encapsulated 2H-MoTe2 with a 532 nm laser to locally initiate the phase transition from the 2H to a primarily Td phase (SI Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Because it is dif- ficult to distinguish 1T’ and Td phases, the phases of pristine and laser-irradiated MoTe2 are characterized by multiple techniques, including aberration-corrected annular dark-field 4 STEM (ADF-STEM) images (Figure 1c–d), TEM diffraction, and polarized Raman spec- troscopy (SI Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' TEM diffraction and polarized Raman measurements indicate that the resulting materials contain a mixture of 1T’ and Td phases, which is in agreement with previous reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='39,40 The potential for mixed phases occurs because the calculated energy difference between 1T’ and Td phase is less than 3 meV per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='8,39 In the TEM samples analyzed below, however, atomic resolution STEM imaging (Figure 1d) indicates that the laser-irradiated material is primarily Td (see SI Figure S3 for top-down ADF-STEM image simulation of 1T’ and Td phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Therefore, we refer to the transformed phase as Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' For in situ heating, we transfer the laser-irradiated, encapsulated MoTe2 specimens to a microelectromechanical system (MEMS)-based heating TEM chip (SI Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Bright-field TEM (BFTEM) imaging before in situ heating (Figure 2a) shows very little contrast between the 2H and Td phases, indicating a uniform thickness across the hetero-phase interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The selected-area electron diffraction (SAED) patterns in Figure 2b–c exhibit the characteristic hexagonal and rectangular lattice of the 2H and Td phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We use DFTEM to map the real-space location and orientation of the 2H and Td phases (SI Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' DFTEM has been widely used to determine the crystal orientation and stacking order of 2D materials41–43 and operates by selecting specific Bragg spots in the diffraction pattern with an objective aperture, so that only crystal grains that diffract to a narrow range of k-vectors appear bright in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' DFTEM images of the 2H and Td phases in Figure 2d–e are obtained by selecting the (¯1100)2H and (¯210)Td Bragg reflections, marked with blue and orange circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We observe Td grains with three orientation directions, rotated 120◦ from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The b-axis of each Td orientation is parallel to one of the three zig-zag directions of the three-fold symmetric 2H matrix (SI Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Figure 2f shows a false-colored DFTEM overlay image mapping the four grains present after laser-irradiation: the 2H phase (red) and the three Td orientations (green, yellow, and blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The majority of the Td region in Figure 2f is oriented in one of the three orientations (green), with needle-like inclusions of the other two orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 5 Next, we perform in situ heating to investigate the reverse Td-to-2H phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We use heat pulses18 instead of continuous heating for three reasons: (1) Pulsing provides flexibility to “halt” the phase transition at any time, rapidly jump to specific temperatures, and even hold at different temperatures for more detailed kinetic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (2) The ability to pause between pulses makes it possible to acquire both large field-of-view (FOV) DFTEM and atomic-resolution ADF-STEM images at several positions between pulses, which provides both a large-scale view of the phase transition kinetics and atomic scale snapshots at the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (3) Heat pulsing minimizes the energy input and potential sublimation during the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' While the graphene encapsulation minimizes damage and sublimation to the MoTe2, we find that heating at temperatures above 600 ◦C for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='5 s can produce small (5–10 nm) voids (see SI Movie S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We use DFTEM imaging to track the propagation of 2H phase between heat pulses, with pulse durations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='5 to 60 s, and temperatures from 200 to 275 ◦C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' note that all images are acquired between pulses, when the sample is at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' DFTEM images of the 2H phase after different heat pulse temperatures (Figure 3a–d and SI Movie S1 ) show that the 2H region at the phase boundary propagates anisotropically toward the Td grain during heating, forming a belt-shaped inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Figure 3e shows a large-FOV DFTEM image where newly grown 2H regions, marked in red, inherit the orientation of the 2H matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We occasionally observed inversion domains in the 2H phase, which we discuss later in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Contrary to previous reports of high (500–600 ◦C) 1T’-to-2H phase transition temperatures in bulk samples,44 we observe the Td-to-2H phase transition initiates at temperatures as low as 200–225 ◦C, from the existing 2H-Td interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' There are two reasons for such low transition temperatures: First, the Td phase is thermodynamically unstable under ambient condition, so only the kinetic barrier needs to be overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Second, the existing 2H-Td interfaces act as nucleation sites, which further reduce the kinetic barrier of the Td-to-2H phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Figure 3f shows a contour overlay of the 2H phase fronts captured between heat pulses 6 of 200–275 ◦C in a 4-layer thick sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We outline the propagating 2H regions that contain at least a monolayer of 2H phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' This image shows that the 2H phase growth is anisotropic in-plane, progressing along the [010]Td (b-axis direction) of the Td grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' This result is in contrast to previous reports of an isotropic 1T’-to-2H transition, which is an averaged result from large-scale polycrystalline 1T’ grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='44,45 The preferential b-axis growth of the 2H phase is observed for all Td orientations (SI Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The anisotropy occurs mainly at low-temperatures, and we find the phase transition becomes more isotropic above 400 ◦C (SI Movie S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' As shown in the atomic models in Figure 3f, 2H-Td phase boundaries can be classified into two types1 based on their symmetry: Type 1, where the phase boundary is parallel to the b-axis of Td and Type 2, where the phase boundary is rotated by 120◦ from the b-axis of Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The anisotropic propagation of the 2H phase suggests that the propagation (growth) rate of the type 2 interface is much faster than for type 1, resulting in the formation of belt-shaped 2H grains with mostly type 1 interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' This behavior can be described by a kinetic Wulff construction,46–49 where the final crystal shape is predicted using thermodynamic and kinetic factors including the interface energy and relative growth rate of different facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The type 2 interface energy is estimated to be 70 meV/˚A higher than type 1 interface,50 making type 1 interfaces more thermodynamically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' This is consistent with our observations that there are more kinks and steps in the atomic-resolution ADF-STEM images of type 2 interfaces than in the type 1 interfaces (Figure 3g,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The 2H growth preferentially propagates along the [010]Td direction due to the higher kink formation and expansion rate of type 2 interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Figure 4a shows that the newly grown 2H region in the 4-layer MoTe2 has multiple phase fronts (I, II, III, and IV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Figure 4b schematically shows horizontally staggered 2H phase fronts and the resulting DFTEM intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' In the kinematic limit, DFTEM intensities scale quadratically with the number of 2H layers, making it possible to individually probe the position of the 2H phase front at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' However, we are not able to determine the depth of each phase front because TEM produces images that are averaged in projection 7 (along the direction of the electron beam path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' In Figure 4c, we measure the 2H-Td interface positions of each layer as a function of accumulated heating time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The calculated propagation rates range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='07 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='4 nm/s at 225 ◦C and exhibit wide variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' For example, both interfaces II (orange) and III (green) exhibit a sudden jump in 2H phase front position at t = 300 sec after the fifteenth 250 ◦C pulse is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The non-uniform propagation rates might be due to strain, defects, and differing surface energies between atomic layers (2H, Td, and graphene encapsulation), which can locally alter the energy barrier of MoTe2 phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='51 Next, we demonstrate that the laser-induced Td phase can be transformed back to the 2H phase via ex situ vacuum annealing (Figure 5 and SI Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We characterize the pristine, laser-irradiated, and annealed MoTe2 with Raman spectroscopy (Figure 5a–d and SI Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Pristine (as-stacked) MoTe2 (Figure 5a,b) exhibits the three characteristic Raman peaks (E2g, A1g, and B2g) of the 2H phase, while laser-irradiated regions (red areas in Fig 5c) exhibit the A1 and A2 modes of the Td phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Raman maps (Figure 5b,c) show that the Td region transformed from 2H phase is uniform, with 2H-Td boundaries that are sharp on the micron scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' After annealing at 800 ◦C for 3 hours, Raman mapping indicates the structure is fully and uniformly converted to 2H phase, as shown in Figure 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' This result shows that a reversible phase transition of 2H- and Td-MoTe2 can be achieved by laser irradiation and vacuum annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' When we anneal multiple samples at different temperatures (300–800 ◦C), all of them exhibit the Td-to-2H phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Finally, we examine the crystal structure of MoTe2 after a full conversion cycle from 2H, to Td, and back to 2H phase in Figure 5e–h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The six-fold symmetry of the SAED pattern in Figure 5e indicates that the converted sample has no rotational grain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' However, the DFTEM image (Figure 5f) shows that for a selected (¯1100)2H Bragg reflection (marked with a green circle in Figure 5e), one of the two grains appears brighter due to the breaking of Friedel’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='43,52 By selecting a neighboring Bragg reflection (orange circle in Figure 5f), the contrast of the two grains is reversed (Figure 5g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' This indicates that the 2H grains have 8 inversion symmetric orientations separated by an inversion domain boundary (IDB), a twin boundary commonly observed in 2D TMDCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='34,43,53 Figure 5h shows an atomic resolution STEM image of an IDB in the 2H phase region after a full conversion cycle of 2H-Td-2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We also observe an IDB running through only 3 of the layers in a 5-layer MoTe2 sample (SI Figure S6), indicating the IDBs do not necessarily go all the way through the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' IDBs are likely generated during the Td-to-2H phase transition because the Td grain has two equivalent transition pathways (SI Figure S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' As a result, cyclic phase transitions from 2H-Td-2H convert a 2D single crystal to coherent 2H polycrystals stitched with IDBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Our work shows that cyclic phase transitions are a promising technique to fabricate the IDBs, which act as one-dimensional metallic tunnels54,55 embedded in 2D semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' In conclusion, we have demonstrated that encapsulated, few-layer MoTe2 can be re- versibly phase engineered between the semiconducting 2H phase and the Td phase using laser irradiation and thermal annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Using in situ pulsed heating and DFTEM, we show that the Td-to-2H phase transition initiates at the 2H-Td interfaces at temperatures as low as 200–225 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Moreover, we observe anisotropic growth of the 2H phase front, which pref- erentially propagates along the b-axis of the nearby Td grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Our findings can be applied to fabrication of coplanar 2D circuitry, including 2D Josephson junctions,56 broadband pho- todetectors,57 and other hetero-phase devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Finally, we demonstrate a new approach for in situ studies of 2D materials using graphene encapsulation and pulsed heating, which can be applied to other micro- to atomic scale in situ studies of solid state phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 9 Figure 1: Characterization and fabrication of different phases of MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a) Atomic structure models of 2H-, 1T’-, and Td-MoTe2 with top and side views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Monolayer models are made for top view for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (b) Schematic of the reversible phase transition of MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The 2H-MoTe2 flakes are encapsulated by graphene and hBN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Local laser-irradiation induces 2H-to-Td phase transition of MoTe2, while the Td phase reverts back to 2H phase after thermal annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (c,d) Aberration-corrected ADF-STEM images for the 2H and Td phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=" 10 a 2H 1T' 2H Mo Te 1T'/T b monolayer b 2H-MoTe2 hBN Graphene SiO2/Si Laser Ta-MoTe2 irradiation 2H 2H-MoTe2 2 nm AnnealedFigure 2: Phase and grain orientation mapping of laser-irradiated MoTe2 with DFTEM." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a) BFTEM image of the suspended, graphene-encapsulated MoTe2 containing both 2H and Td grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Td phase region is delineated by the laser trajectory and outlined by the black dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The minimum width of the Td region is determined by the radial laser intensity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (b,c) SAED patterns, (d,e) DFTEM images of 2H and Td phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The diffraction patterns (b,c) are acquired with zone axis perpendicular to the basal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The weaker diffraction spots are generated by the graphene encapsulating layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The DFTEM images (d-e) are formed by selecting the (¯1100)2H and (¯210)Td Bragg reflections in (b) and (c) with the objective aperture position marked with blue and orange circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The objective aperture and selected Bragg reflections are centered on the optical axis to reduce image aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (f) Overlay of false-colored DFTEM images of the 2H matrix (red) and three different orientations of Td grains (green, yellow, and blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The DFTEM images (d–f) are acquired at the region marked by the yellow dashed square in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 11 b [0001] a 2H (1100) 2H 5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='1 2H 100 nm C [001] e 2H (210) 2H 200nm 100 nm T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 5 nm 100nmFigure 3: Anisotropic, low-temperature Td-to-2H phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a–d) DFTEM images formed from the (¯1100)2H spot are acquired at room temperature after heat pulses of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='5 s from 200 to 275 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Td-to-2H transition initiates at the interface, and the 2H phase front anisotropically propagates into the Td phase region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (e) Overlay of a low magnification DFTEM image with newly grown 2H regions marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Td-to-2H phase transition occurs primarily at 2H-Td interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (f) Contour plot of the 2H phase front in the same region as (a–d) shows propagation along the b-axis of nearby Td grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The insets are the atomic models of 2 different types of interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The anisotropy arises from the different interface energy of type 1 and 2 interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' ADF-STEM images of (g) type 1 and (h) type 2 2H-Td interfaces with atomic kinks indicates a step-flow growth model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 12 200 °C b 225 °C 250 °C d 275 °C a C Propagating 2H 1 layer 2H belt Initial 2H-Td interface 50 nm e 1 2H phase front I propagation 2H growth Type 2 kink 2H Type 1 Type 1 2H Td 2H Type 2 a 200 nm 50 nm nm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='nmFigure 4: Layer-by-layer phase transition and growth kinetics measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a) DFTEM image that shows the layer-by-layer phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The 2H-Td interface of different layers are individually identified by their intensity difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (b) Schematic of the intensity dif- ferences of 4-layer MoTe2 2H-Td interfaces in DFTEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The mono-, bi-, tri-, and quad-layer 2H phase fronts are labeled as I, II, III, and IV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Note that the relative positions (in the z-direction) of each phase front are unknown due to the projection nature of TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (c) Plot of 2H phase front positions of different layers as a function of accumulated heating time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We perform a series of short heat pulses to capture the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Each dot corresponds to a heat pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The pulsing time ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='5 s to 1 min and can be read from the horizontal spacing between the dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The pulsing temperatures (200–275 ◦C) are color-coded by the background shades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The propagation rates are extracted by the slope of the curves, which have a strong temperature dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=" 13 b a c e' beam 2H phase front growth 275 °C : front position (nm) 2H Td 2H 二 200 =M 4-layer MoTe, 2H-T." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' interface 150 4 2 100 000 phase t 50 IV DFTEM 000000 川I (2H selected) 200 °℃ 225 °C 250 °C 275 ° 2H 50 nm 0 50 100 150 200 250 0 100 200 300 400 Distance (nm) Accumulated time (s)Figure 5: Cyclic phase transition and recovery of MoTe2 (2H −→ Td −→ 2H phase) via laser irradiation and annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a) Raman spectra and (b–d) Raman mapping of pristine (as- stacked), laser-irradiated, and annealed MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Raman maps are visualized with the E2g (2H) and A1 (Td) peak, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (e) SAED pattern of the 2H phase region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The orange and green circles denote the objective aperture positions that are used to generate DFTEM images (f,g) from different (¯1100)2H Bragg reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The brighter region corresponds to the specific 2H orientation that generates the stronger Bragg reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The inversion domain boundary is outlined by the white dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (h) ADF-STEM image at the inversion domain boundary of 2H grains with opposite orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The atomic models are overlaid with arrows indicating opposing orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' I ASSOCIATED CONTENT Supporting Information Sample fabrication workflow, 1T’ and Td mixture analysis, sim- ulated ADF-STEM images, ADF-STEM images and atomic models of inversion domain boundaries, and in situ movies of MoTe2 phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' I AUTHOR INFORMATION Corresponding Author Email: pyhuang@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='edu Email: gwanlee@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='kr 14 Pristine b a ↑E2g (2H) 2H e g Pristine A1g (2H) B2g (2H) 5 μm Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=') Laser-irradiated A (Ta) 2H Laser-irradiated 3 nm-1 2 μm A2(Ta) d E2g (2H) Annealed Annealed d 2H A1g (2H) B2g (2H) 100 150 200 250 300 350 2 μm Raman shift (cm-1)Author Contributions Under supervision by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' acquired and analyzed the in situ heat- ing DFTEM and ADF-STEM images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Under supervision by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' fabricated the MoTe2 samples, performed ex situ annealing experiments and Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Under supervision by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' perform TEM analysis of phase-engineered MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Under super- vision by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' conducted polarized Raman measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' synthesized the hBN flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' All authors read and contributed to the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Notes The authors declare no competing financial interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Acknowledgement This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Department of Energy, Office of Sci- ence, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award number DE-SC0020190, which supported the electron microscopy and related data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' This work was carried out in part in the Materials Research Laboratory Central Facilities at the University of Illinois at Urbana–Champaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' acknowledges support by the Creative-Pioneering Researchers Program through Seoul National University (SNU), the National Research Foundation (NRF) of Korea (NRF-2021R1A2C3014316, SRC pro- gram: vdWMRC center 2017R1A5A1014862, NRF-2021M3F3A2A01037858), the Research Institute of Advanced Materials (RIAM), Institute of Engineering Research, and Institute of Applied Physics at SNU, which supported the sample fabrication and ex situ character- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' acknowledge support from the Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant Numbers 19H05790 and 20H00354) and A3 Foresight by JSPS, which supported the h-BN synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 15 References (1) Sung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Ren, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Wang, G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Ji, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Inversion Domain Boundary Induced Stacking and Band- structure Diversity in Bilayer MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Nano Letters 2017, 17, 6653–6660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (54) Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Jiao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Ho, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Gao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Jia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Fan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Yao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Xie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Dense Network of One-Dimensional Midgap Metallic Modes in Monolayer MoSe2 and Their Spatial Undulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Physical Review Letters 2014, 113, 066105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (55) Lehtinen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Komsa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Pulkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Whitwick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Lehnert, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Mohn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Yazyev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Kis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Kaiser, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Krasheninnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Atomic Scale Mi- crostructure and Properties of Se-Deficient Two-Dimensional MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' ACS Nano 2015, 9, 3274–3283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 22 (56) Madsen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Bergholtz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Brouwer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Josephson effect in a Weyl SNS junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Physical Review B 2017, 95, 064511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (57) Lai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Anisotropic Broadband Photoresponse of Layered Type-II Weyl Semimetal MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Advanced Materials 2018, 30, 1707152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 23 Graphical TOC Entry 24 View direction 2H 2H 50 nm 200 °C 2H 2H AL 275 °C IVSupporting Information In situ Imaging of an Anisotropic Layer-by-Layer Phase Transition in Few-Layer MoTe2 Chia-Hao Lee, Huije Ryu, Gillian Nolan, Yichao Zhang, Yangjin Lee, Siwon Oh, Hyeonsik Cheong, Kenji Watanabe, Takashi Taniguchi, Kwanpyo Kim, Gwan-Hyoung Lee,∗ and Pinshane Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Huang∗ E-mail: gwanlee@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='kr E-mail: pyhuang@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='02694v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='mtrl-sci] 6 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Sample preparation for in and ex situ experiments To fabricate the hBN/Gr/MoTe2/Gr/hBN heterostructures, we mechanically exfoliated thin layers of 2D materials (MoTe2, hBN, graphene) from bulk crystals (MoTe2: HQ graphene, graphene: NGS Naturgraphite GmbH, hBN: NIMS) onto SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We then used the pick-up transfer technique1 with a poly(bisphenol A carbonate, Sigma Aldrich) (PC)- coated poly (dimethyl siloxane) (PDMS) lens mounted on a microscope slide to pick-up and released the constituent flakes on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The PC/PDMS/glass slide was held in a 3-axis micromanipulator to control the position of the contact area with the 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The substrate was placed on a heating stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' By controlling the temperature of the heating stage (80–130 ◦C), the 2D flakes were picked up by the PC with minimal cracking or folding, leaving the substrate on the heating stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The hBN/Gr/MoTe2/Gr/hBN heterostructures were fabricated by repeating the above steps, and then transferred onto a clean SiO2/Si substrate by releasing the PC film from the PDMS lens at a temperature above 180 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Lastly, we placed the entire sample in chloroform for 30 min to remove the PC film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' To image the MoTe2 at atomic resolution and reduce multiple scattering from thick hBN layers, we removed the hBN layers with XeF2 dry etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='2 The sample fabrication process was similar with the one used for ex situ experiments but with some modifications, see Figure S1 for the schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The bottom hBN layer was etched away by the XeF2 exposure, and the etching process was self-limited at the graphene layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We transferred the stack onto a clean SiO2/Si substrate and exposed it with chloroform, oxygen plasma, and XeF2 again to remove the top hBN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' After these steps, the stack was encapsulated with fluorinated graphene and ready for transferring onto an in situ heating TEM chip (E-FHDC- VO-10, Protochips).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We used the conventional polymer transfer technique with poly(methyl methacrylate) (PMMA) and KOH to transfer the stack3 from SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='4 After transferring, the PMMA film was removed by placing the samples in acetone for 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Laser irradiation parameters for phase transition To initiate the 2H-to-Td phase transition of MoTe2, we irradiated the encapsulated samples using continuous wave (CW) 532 nm laser and power of 21 mW at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The laser was focused by a 100× objective lens (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='9) and the resulting spot size on the substrate was around 1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The laser-irradiated area was patterned by rastering the laser spot with 200 nm point-to-point distance and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='1 s exposure time per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' S/TEM measurement In situ TEM experiments were done in a Thermo Fisher Scientific Themis-Z aberration- corrected S/TEM operated at 80 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' For atomic-resolution ADF-STEM imaging, the point resolution was about 1 ˚A with 25 mrad convergence semi angle, 35 pA probe current, 63 to 200 mrad collection semi angles, 20 pm pixel size and a total dwell time of 20 µs/pixel using 10-frame averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' For BFTEM, SAED, and DFTEM, the data were acquired with a Ceta 16M camera at parallel illumination using the three-condenser TEM mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The electron dose rate was around 103 e-/nm2/s and the exposure times for SAED and DFTEM were 2 to 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Note that sparse dark pits were observed in DFTEM at the end of the in situ imaging after an accumulated total dose around 4 × 105 e-/nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' While the graphene encapsulation was unlikely to form holes under this condition, these dark pits were likely to be crystallographic defects such as voids formed by displacing atoms of the MoTe2 flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Ex situ Td-to-2H phase transition with annealing The ex situ Td-to-2H phase transition of MoTe2 (Figure 5 of the main text) was performed by an annealing process in a vacuum furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We annealed the laser-irradiated sample in a vacuum chamber (10-4 Torr) and slowly ramped up to targeted temperatures in 3 hours and held for another 3 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The targeted temperatures were set from 300 to 800 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The furnace was naturally cooled to room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Raman spectroscopy measurement The linearly polarized Raman measurements (SI Figure S2) were carried out in the backscat- tering geometry using 514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='5 nm laser excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The input laser beam was focused onto the samples by a 50× microscope objective lens (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='8 NA), and the scattered light was collected and collimated by the same objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' To access the low-frequency range below 50 cm–1, volume holographic filters (OptiGrate) were used to clean the laser lines and reject the Rayleigh-scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' A laser with a low power of 300 µW was used to avoid local heat- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Raman scattering signals were dispersed by a Jobin–Yvon iHR550 spectrometer with a 2400 grooves/mm grating (400 nm blaze) and detected by a liquid-nitrogen-cooled, back-illuminated CCD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' An achromatic half-wave plate was used to rotate the polar- ization of the linearly polarized laser beam to the desired direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The analyzer angle was set such that photons with polarization parallel to the incident polarization passed through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Another achromatic half-wave plate was placed in front of the spectrometer to keep the polarization direction of the signal entering the spectrometer constant with respect to the groove direction of the grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Raman spectra (Figure 5 of the main text) were ac- quired using a HORIBA LabRAM HR Evolution with the laser wavelength at 532 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' To minimize the irradiation damage, the laser power was set below 5 mW with an acquisition time of 60 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' All measurements were conducted at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 4 Figure S1: Sample preparation for in situ TEM experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a-1) The schematic illustration of hBN/Gr/MoTe2/Gr/hBN structure on SiO2 (285 nm)/Si++ substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a-2,3) Pick-up the structure by polycarbonate (PC) film on polydimethylsiloxane (PDMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (b-1,2) Etching the bottom side of the hBN by exposing to XeF2 gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (b-3) After exposing to XeF2 gas, the bottom hBN is completely etched, while the graphene layers and the encapsulated layers remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' In addition, the PC exposed by XeF2 is also chemically modified, which can not be dissolved by chloroform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (c-1,2) One-side-etched sample is transferred to another SiO2(285 nm)/Si++ substrate at about 180 ◦C and separated from the PDMS lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Si substrate was treated with O2 plasma to increase the adhesion energy of SiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (c-3,4) Remove the PC film by chloroform bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Note that the fluorinated PC was not dissolved in chloroform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (c-5,6) Etch off the fluorinated PC layer by O2 plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Since the etch rate of hBN is much slower than fluorinated PC layer using O2 plasma, the fluorinated PC layer is removed while the Gr/MoTe2/fluorinated Gr structure remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (c-7,8) Remove the top hBN by XeF2 gas etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Finally, we transferred the heterostructures on MEMS TEM chips using the PMMA-assisted, wet-transfer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Pick-up a-1 a-2 a-3 Glass PDMS Lens hBN 1L Gr Pick-up PC MoTe2 SiO, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 1st XeF² gas etching b-1 b-2 b-3 F-treated PC layer Fluorinated Gr XeF2 gas etching c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Transfer on SiO & 2nd XeF2 gas etching c-1 c-3 Chloroform C-2 c-4 XeF2 gas etching C-6 C-7 C-8 C-5 O, plasmaFigure S2: Mixture of 1T’ and Td phase characterized by TEM diffraction and polarized Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a) Selected area electron diffraction pattern (SAED) acquired at a region with mixed 1T’ and Td phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The red circles are Bragg peaks of Td phase that would be absent if it were 1T’ phase, however, the intensities are too weak for the region to be pure Td phase, indicating a mixture of 1T’ and Td phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Bragg peaks inside the blue circles are also characteristic peaks of Td phase that are absent in the 1T’ phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (b) Polarized Raman spectra of Td + 1T’- (purple) and 2H-MoTe2 (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The 1T’ and Td phase are typically characterized by the peak splitting around 128 cm-1: those with a split peak were identified as the Td phase, and those with a single peak as the 1T’ phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='5 The blue peak in the inset indicates the presence of Td phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' However, considering the SAED result in (a), our specimen shows a spatial inhomogeneity of mixture of 1T’ and Td phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 6 b a Ta+1T 2H O Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=') 125130 135 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 50 100 150 200250 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='5 A-1 Raman Shift (cm-1)Figure S3: ADF-STEM image simulation of 1T’- and Td-MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a–d) Simulated ADF- STEM images of 1T’ and Td phase at different orientations using semi-quantitative image simulation package6 (e–h) Atomic models of 1T’ and Td phase at different orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='9◦ tilt angle is chosen to match the β angle of 1T’ phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=" 7 1T' along c-axis along (001) normal along c-axis 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='90 tiltFigure S4: Crystallographic relation between the Td and the 2H matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a,b) Atomic models of top-view, monolayer 2H and Td phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (c,d) Simulated diffraction patterns of 2H and Td phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (e) Overlay of the simulated diffraction patterns of 2H and Td phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The Td phase can be derived by shifting the chalcogen layers in 2H phase along one of the three arm-chair directions followed by some metal atom dimerization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Therefore, the b-axis of the derived Td variants are parallel to one of the three zig-zag directions of the 2H matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 8 2HFigure S5: Anisotropic phase transition for all 3 Td orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (a) Overlay of false- colored DFTEM images of the 2H matrix (red) and three different orientations of Td grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Reproduced from Figure 2f of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (b–d) DFTEM images of 2H-Td interfaces with different Td phase orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (e–g) DFTEM images of 2H-Td interfaces after heat pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The white arrows indicate the growth directions of the 2H phase front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The growth directions are parallel to the b-axis directions of the nearby Td grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (h–j) Atomic models of 2H-Td interface with three different orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The b-axis directions of the Td phases are marked by the black arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 9 d a 00 nm e g 100 nm 2HFigure S6: ADF-STEM images of an inversion domain boundary (IDB) at (a) lower and (b) higher magnifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The field-of-view is marked by the yellow square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The IDBs are marked by the blue arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' In this specific region, the IDB is only observed at the 4-layer region, indicating the IDB does not go all-the-way-through this 5-layer sample and suggesting an independent layer-by-layer phase transition mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The layer number is determined by quantitative ADF-STEM intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 10 a 5L IDB 4L 4L 2L 2L Gr 50 nm 5 nmFigure S7: Formation mechanism of the inversion domain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Atomic models of two potential pathways of Td-to-2H phase transition in (a) side-view and (b,c) top-views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' By sliding either the top or bottom chalcogen layers along the a-axis direction, 2H grains with opposite orientations can be derived from a single crystalline Td grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Therefore, shifting opposite layers of the chalcogen atoms in a Td grain will generate an inversion domain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The dark (Pathway 1) and light (Pathway 2) blue arrows indicate the sliding directions of each chalcogen layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 11 a Pathway 2 Pathway 1 C Shift the bottom chalcogen atoms Shift the top chalcogen atoms Mutually invertedMovie S1: DFTEM video acquired using the (¯1100)2H spot at room temperature after each heat pulse from 200 to 275 ◦C, showing the in-plane, layer-by-layer, and anisotropic Td-to-2H phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The heat pulses range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='5 s to 1 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 12 200°℃ 100nmMovie S2: DFTEM video acquired using the (¯1100)2H spot at room temperature after each heat pulse from 200 to 700 ◦C, showing the anisotropic Td-to-2H phase transition at lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The phase transition then become more isotropic at temperatures above 400 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' We applied two rounds of heating, the first round is 200–400 ◦C, while the second round is 200–700 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' The temperature intervals are all 25 ◦C and the heat pulses are all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' References (1) Purdie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Pugno, N.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Nature Materials 2013, 12, 554– 561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (5) Cheon, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Lim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Cheong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Structural Phase Transition and Interlayer Coupling in Few-Layer 1T’ and Td MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' ACS Nano 2021, 15, 2962–2970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' (6) Kirkland, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' Computem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' http://sourceforge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content='net/projects/computem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE0T4oBgHgl3EQf1AKn/content/2301.02694v1.pdf'} diff --git a/Q9E0T4oBgHgl3EQfUQAL/content/2301.02246v1.pdf b/Q9E0T4oBgHgl3EQfUQAL/content/2301.02246v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..13f3daabb40b0727eefd1b4272241ca5b34992ea --- /dev/null +++ b/Q9E0T4oBgHgl3EQfUQAL/content/2301.02246v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fe4f2aba629f2a8c77e219194f402f046e2451bd2efe597a38b8c4f372031fae +size 672181 diff --git a/QtAzT4oBgHgl3EQfI_sd/content/tmp_files/2301.01070v1.pdf.txt b/QtAzT4oBgHgl3EQfI_sd/content/tmp_files/2301.01070v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2f9ab069d5b9298aaa655bf77f815c7c4053b48 --- /dev/null +++ b/QtAzT4oBgHgl3EQfI_sd/content/tmp_files/2301.01070v1.pdf.txt @@ -0,0 +1,1574 @@ +arXiv:2301.01070v1 [gr-qc] 3 Jan 2023 +Nonlocal-in-time effective one body Hamiltonian in scalar-tensor gravity at third +post-Newtonian order +Tamanna Jain∗ +Department of Applied Mathematics and Theoretical Physics, +University of Cambridge,Wilberforce Road CB3 0WA Cambridge, United Kingdom. +(Dated: January 4, 2023) +We complete the nonlocal-in-time effective-one-body (EOB) formalism of conservative dynamics +for massless Scalar-Tensor (ST) theories at third post-Newtonian (PN) order. +The nonlocal-in- +time EOB Hamiltonian is obtained by mapping the order-reduced Hamiltonian corresponding to +the nonlocal-in-time Lagrangian derived in [Phys. Rev. D 99, 044047 (2019)]. To transcribe the +dynamics within EOB formalism, we use a strategy of order-reduction of nonlocal dynamics to local +ordinary action-angle Hamiltonian. We then map this onto the EOB Hamiltonian to determine the +nonlocal-in-time ST corrections to the EOB potentials (A, B, Qe) at 3PN order. +I. +INTRODUCTION +In 2015, the direct detection of gravitational waves +(GW) by the LIGO-Virgo Collaboration [1] emitted by +inspiralling compact binary, opened new avenues for +probing the dynamics in strong gravity regime [2–8]. It +is expected that future GW detectors, like the Einstein +Telescope [9] and Cosmic Explorer [10] will shed more +light on alternative theories of gravity by constraining +the parameters of such theories. +The simplest theory amongst the alternative theories +of gravity is the addition of a massless scalar field to +GR, scalar-tensor (ST) theories, which are are exten- +sively studied [11–16]. The motivation for ST theories +is to explain both the accelerated expansion of the uni- +verse as f(R)-theories [17] as well as UV complete alter- +nate theories of GR. The two-body PN formalism for ST +theories has been extensively studied [18–26]. +The important violations for ST theories arise through +the non-perturbative strong field effects in neutron-stars +such as spontaneous scalarisation [13]. Although the cur- +rent constraints come from the binary pulsar observa- +tions, the future GW detections can better constraint +the parameters using strong-field information and addi- +tional terms in radiation, i.e. +dipolar radiation which +is not present in GR, due to the scalar extension of +GR [15, 27, 28]. +The EOB formalism was introduced to construct ana- +lytical waveform templates for GR [29–35]. Recently, the +two-body PN dynamics has also been mapped within the +EOB formalism to construct waveform templates for ST +theories [36–38]. +In our previous work [38], we deter- +mined the EOB potentials for the local part of dynamics +at 3PN order. The aim of this paper is to determine the +complete nonlocal-in-time EOB potentials following our +results of local part in Ref. +[38] starting from the 3PN +nonlocal-in-time Lagrangian of Ref. [20, 21]. Hereafter, +the companion paper [38] will be referred as Paper I. +∗ tj317@cam.ac.uk +The paper is organised as follows. +In Sec. II, we +give a summary of results obtained in Paper I. Then, +in Sec. III we derive the conserved energy for nonlocal- +in-time part using two methods, (i) non-order-reduced +nonlocal Hamiltonian using nonlocal phase shift, and (ii) +order-reduction of nonlocal dynamics to local ordinary +action-angle Hamiltonian. +Finally, in Sec. IV we map +the nonlocal-in-time ordinary Hamiltonian into an EOB +Hamiltonian at 3PN order. +II. +SUMMARY OF PREVIOUS RESULTS +We consider mono-scalar massless ST theories de- +scribed by the following action in the Einstein Frame +(the scalar field minimally couples to the metric), +S = +c4 +16πG +� +d4x√−g(R − 2gµν∂µϕ∂νϕ) ++ Sm[Ψ, A(ϕ)2gµν] , +(2.1) +where gµν is the Einstein metric, R is the Ricci scalar, ϕ +is the scalar field, Ψ collectively denotes the matter fields, +g ≡ det(gµν) and G is the bare Newton’s constant [38]. +As Paper I (see, Table I), we adopt the conventions and +notations of Refs. [11, 13]. In Einstein Frame, the dy- +namics of the scalar field arises from its coupling to the +matter fields Ψ, and the field equations can be found in +Ref. [11] where the parameter +α(ϕ) = ∂ ln A +∂ϕ +, +(2.2) +measures the coupling between the matter and the scalar +field. The scalar field is non-minimally coupled to the +metric in Jordan Frame (physical frame) +˜gµν = A(ϕ)2gµν , +(2.3) +where ˜gµν is the metric in Jordan frame. +We follow the approach suggested by [39] to “skele- +tonize” the compact, self-gravitating objects in ST the- +ories as point particles, i.e. the total mass of each body + +2 +is dependent on the local value of the scalar field. The +skeletonized matter action with the scalar field depen- +dent mass ˜mI(ϕ) is then given by +Sm = − +� +J=A,B +� � +−˜gµν +dxµ +dλ +dxν +dλ ˜mJ(ϕ) , +(2.4) +where λ is the affine parameter. Since ˜gµν = A(ϕ)2gµν, +the Einstein-frame mass is defined as +m(ϕ) = A(ϕ) ˜m(ϕ) . +(2.5) +In Paper I, we first derive the ordinary Hamiltonian +(dependent only on the positions and momenta) using +the contact transformation at 3PN order starting from +the Lagrangian of Ref. [21] only for the local-in-time +part of the dynamics. The Jordan-Frame parameters of +Ref. [21] that encompass the scalar field effect are con- +verted to the dimensionless Einstein-Frame parameters +(see, Table I). The mass function m(ϕ) is used to define +these dimensionless body-dependent parameters follow- +ing Refs. [11, 13, 37] i.e. +αI = d ln m(ϕ)I +dϕ +, +(2.6) +βI = dαI +dϕ , +(2.7) +β′ +I = dβI +dϕ , +(2.8) +β′′ +I = dβ′ +I +dϕ . +(2.9) +Here, we follow the notations of Paper I for the binary +parameters and use the same notation as [20, 21] to de- +note weak-field and strong-field parameters. +Finally, we then determine the ST corrections to the +EOB metric potential (A, B, Qe) at 3PN order for the +local in time (instantaneous) part of the dynamics by +mapping the EOB Hamiltonian in DJS gauge [31] +ˆHeff = Heff +µ += +� +� +� +�A(ˆr) +� +1 + +ˆp2r +B(ˆr) + +ˆp2 +φ +ˆr2 + q3 +ˆp4r +ˆr2 +� +, +(2.10) +where ˆpr, ˆpφ are the dimensionless radial and angular mo- +menta, and ˆr(= r/(GABM) is the dimensionless radial +separation, to the ordinary two-body Hamiltonian (here- +after the superscript hat is used to denote the dimension- +less variables). +The three EOB potentials at 3PN are +A(ˆr) = 1 − 2 +ˆr + a2 +ˆr2 + a3 +ˆr3 + a4 +ˆr4 , +(2.11) +B(r) = 1 + b1 +ˆr + b2 +ˆr2 + b3 +ˆr3 , +(2.12) +Qe(ˆr) = q3 +ˆp4 +r +ˆr2 . +(2.13) +The GR and ST corrections in coefficients (ai, bi) are +separated as +ai = aGR +i ++ δaST +i +, +(2.14) +bi = bGR +i ++ δbST +i , +(2.15) +q3 = qGR +3 ++ δqST +3 +. +(2.16) +Since there are also nonlocal-in-time and tidal contri- +butions at 3PN order in ST theory, all the 3PN ST coef- +ficients can thus be decomposed as Eq. (5.23) of Paper I. +The complete expressions of local-in-time ST corrections +at 3PN can be found in Eqs.(5.14)-(5.16) of Paper I. +In Paper I, we also derive the nonlocal-in-time (tail) +and tidal corrections only for the circular orbits using +the gauge invariant energy for circular orbits given in +Ref. [21, 22]. The complete expression for these coeffi- +cients can be found in Eqs. (5.25)-(5.27) of Paper I. +III. +TAIL CONTRIBUTION TO THE 3PN +DYNAMICS +The nonlocal-in-time two-body 3PN Lagrangian for +massless ST theory obtained in Ref. [21] is in harmonic +coordinates, i.e. it depends (linearly) on the acceleration +of the two bodies. In this section, we will use two different +methods to derive the Noetherian conserved energy for +the tail contributions. First, we will remove the accelera- +tion dependence from the Lagrangian (hence, the Hamil- +tonian) and stay within the non-order-reduced nonlocal +framework (as done in Refs. [40, 41] for GR). Second, we +will derive the order-reduced, local Hamiltonian using the +action-angle variables (see, Ref. [34] for GR). +A. +Non-order-reduced Ordinary Hamiltonian +In Paper I, we derived the ordinary (dependent only +on positions and momenta) Hamiltonian for local-in- +time contribution using contact transformation (see, Ap- +pendix A of Paper I for the contact transformation). +Now, concerning the nonlocal-in-time part we need to +find the nonlocal shift that removes the acceleration de- +pendence from the tail part of the Lagrangian of Ref. [21] +(see, Refs. [40, 41] for GR). Corresponding to this ordi- +nary Lagrangian, we can then derive the ordinary Hamil- +tonian. +The tail part of the Lagrangian at 3PN order reads +[21], +Ltail = 2G2M +3c6 +(3 + 2ω0) Pf2rAB/c +� ∞ +−∞ +dτ +|τ| I(2) +s,i (t)I(2) +s,i (t + τ), +(3.1) +where +Pf +is +the +Hadamard +partie +finie +function, +Hadamard scale rAB(= r) is the relative separation of +two bodies, and I(2) +s,i is the second time derivative of the +dipole moment. Here, we find the shift that transforms + +3 +this Lagrangian into the same expression but with the +derivatives of the dipole moment evaluated using the +Newtonian equations of motion. In the centre of mass +(COM) frame in notations of Ref. [20, 21] it is, +´I(2) +s,i = 2Mν(sA − sB) +φ0(3 + 2w0) +� +−GABM +r2 +ni +AB +� +, +(3.2) +where sA, sB are the sensitivity of two bodies. +As the nonlocal contribution starts at 3PN order, the +ordinary Lagrangian is +Ltail +ord = Ltail + +� +J=A,B +mJ + +−ai +J − +� +J̸=K +GAB mK +r2 +ni +JK + + ξJ,i , +(3.3) +where Ltail +ord is given by the same expression as Eq. (3.1) +but with second time derivative of the dipole moment +replaced by its on-shell value given in Eq. (3.2), and the +nonlocal shift, ξJ,j, +ξJ,j = +1 +mJ +2G2M +3c6 +(3 + 2w0) +� +−mJ(1 − 2sJ) +φ0(3 + 2w0) +� +δi +j +Pf2r/c +� ∞ +−∞ +dτ +|τ| +´I(2) +s,i (t + τ) . +(3.4) +The ordinary Hamiltonian is then derived using the +ordinary Legendre transformation, Hord = � +A pAvA − +Lord which reads Hord = Hloc +ord + Htail +ord, where the local +contribution Hloc +ord is derived in Paper I (see, Appendix +C) and the tail contribution is +Htail +ord = −2G2M +3c6 +(3 + 2w0) Pf2r/c +� ∞ +−∞ +dτ +|τ| +´I(2) +s,i (t)´I(2) +s,i (t + τ). +(3.5) +The tail part of the Hamiltonian is just opposite to tail +part of Lagrangian. +As shown in Ref. [35, 41] for the non-order-reduced, +nonlocal framework the Noetherian conserved energy +(Econs) is not given by the Hamiltonian but is given by, +Econs = Htail +ord + δH. This additional term δH consists +of purely a constant term (DC type) and time oscillating +term with zero average value (AC type) and is same as +given in Eq. (4.10) of Ref. [21]. +B. +Order-reduced Ordinary Hamiltonian +The second method to derive the conserved energy for +tail part is to work in the order-reduced, local framework +as given in Ref. [34, 35] for GR. +The tail part of the Hamiltonian in ST theory is, +Htail = −2G2M +3c6 +(3 + 2ω0) +� +Pf2r/c +� ∞ +−∞ +dτ +|τ| I(2) +s,i (t)I(2) +s,i (t + τ) +−2 ln +� ˆr +a +� +I(2) +s,i (t)2 +� +. +(3.6) +As mentioned in Ref. [41], in the action-angle form there +should be an additional term (second term in Eq. (3.6)) +which is local and accounts for dependence of Hadamard +Partie finie function on the radial separation (r) at time +t i.e., ˆr = a(1 − e cos(u)) in action-angle variables. +The basic methodology we use to order-reduce the non- +local dynamics of the above form is based on Refs. [34, 35] +for GR, and consists of four main steps: (i) Re-express +the Hamiltonian in terms of action angle variables, +(ii)“order-reduce” the nonolocal dependence on action +angle variable, (iii) expand it in powers of eccentricity, +and (iv) eliminate the periodic terms in order-reduced +Hamiltonian by a canonical transformation. All of these +steps lead to the order-reduced ordinary local Hamilto- +nian for the tail part in terms of action-angle variables. +Let us consider the expression of nonlocal-in-time piece +of Eq. (3.6), i.e. +K(t, τ) = ¨Is,i(t)¨Is,i(t + τ) . +(3.7) +To order reduce the nonlocal piece, we use the equations +of motion to express the phase-space variables at shifted +time t+τ in terms of the phase-space variables at time t. +As the zeroth order equations are Newtonian equations, +it will be convenient to use the action-angle form of the +Newtonian equations of motion, +∂l +∂ˆt = ∂H0 +∂L = 1 +L3 = Ω(L) , +∂L +∂ˆt = ∂H0 +∂l += 0 , +∂G +∂ˆt = ∂H0 +∂g += 0 , +∂g +∂ˆt = ∂H0 +∂G = 0 , +(3.8) +where ˆt = t/(GABM) is the dimensionless time variable, +(L, l, G, g) are the action-angle variables. +The zeroth- +order (Newtonian) Hamiltonian in action-angle variable +is H0 = −1/(2L2). +Here, +the variable L is conjugate to the “mean +anamoly” l and G is conjugate to argument of perias- +tron g. In terms of the Keplerian variables, semi-major +axis a, and eccentricity e, these are +L = √a, +G = +� +a(1 − e2) . +(3.9) +From Eq. (3.8), the variables L, G and g are indepen- +dent of time, and l varies linearly with time, hence it will +be sufficient to use +l(t + τ) = l(t) + Ω ˆτ , +(3.10) +where ˆτ = τ/(GABM). The order-reduced non-local in +time expression of Eq. (3.7) becomes +K(t, τ) = +� +1 +GABM +�4 +K(ˆt, ˆτ) += +� +Ω +GABM +�4 d2 +dl2 Is,i(l) d2 +dl2 Is,i(l + Ωˆτ) . (3.11) +Using the Fourier decomposition of dipole moment +given in Eq. (A11), we find the structure of nonlocal-in- +time expression K(t, ˆτ) and hence the Hamiltonian. As + +4 +shown in [34] for GR, all the periodically varying terms +can be eliminated by a suitable canonical transforma- +tion. Hence, the order-reduced Hamiltonian can be fur- +ther simplified by replacing Htail with its l-average value +¯Htail = +� 2π +0 +dlHtail . +(3.12) +Using the result +PfT +� ∞ +0 +dv +v cos(ωv) = −(γE + ln(ω T )) +∀ (ω > 0) , +(3.13) +where γE is the Euler’s constant, and inserting the ex- +pression of r from Eq. (A7), the Hamiltonian, Eq. (3.12), +reads +¯Htail = 8G2M +3 +� +Ω +GABM +�4 +(3 + 2w0) +∞ +� +p=1 +p4��Is,i(p) +��2 +ln +� +eγE 2p a Ω +c +� +. +(3.14) +Now, inserting the Fourier-Bessel expansion of scalar +dipole moment from Eqs. (A13)-(A14) (see, Appendix A +for derivation) in Eq. (3.14), the real two-body nonlocal- +in-time Hamiltonian in order-reduced, local framework is +(in notations of Paper I) +ˆ¯Htail ≡ +¯Htail +µ += 2ν +3a4 +� +2δ+ + ¯γAB(¯γAB + 2) +2 +� ∞ +� +p=1 +p2 +e2 +� +4e2J2 +p−1(pe) + (8 − 4e2)J2 +p(pe) − 8eJp−1(pe)Jp(pe) +� + +γE + ln +� +2p a−1/2 +c +� + . +(3.15) +Expanding the result in powers of eccentricity, the Hamil- +tonian as an expansion in eccentricity upto order of e4 +reads +ˆ¯Htail = 2ν +3a4 +� +2δ+ + ¯γAB(¯γAB + 2) +2 +� � +2 ln(2) − ln(a) + 2γE + e2 � +14 ln(2) + 6γE − 3 ln(a) +� ++e4 +�45 +4 γE − 3 +4 ln(2) + 729 +32 ln(3) − 45 +8 ln(a) +� ++ O(e6) +� +. +(3.16) +IV. +SCALAR TENSOR CORRECTIONS TO +EFFECTIVE ONE BODY AT 3PN:TAIL +In this section, we will derive the complete tail cor- +rections to the EOB metric potentials (A, B, Qe) for ST +theories at 3PN order. +Similar to the decomposition of complete 3PN coeffi- +cient δaST +4 +in Eq. (5.23) of Paper I, we decompose the +complete 3PN ST coefficients δbST +3 , δqST +3 +as +δbST +3 += δbST +3,loc + δbST +3,nonloc + δbST +3,tidal , +(4.1) +δqST +3 += δqST +3,loc + δqST +3,nonloc + δqST +3,tidal , +(4.2) +where the local contributions (δbST +3,loc, δqST +3,loc) are derived +in Paper I (see, Eqs. (5.14)-(5.15)), (δbST +3,nonloc, δqST +3,nonloc) +are the nonlocal contributions, and (δbST +3,tidal, δqST +3,tidal) are +the tidal contributions. The nonlocal contributions can +be further decomposed similar to Eq. (5.24) of Paper as +δaST +4,nonloc = δaST +4,nonloc,0 + δaST +4,nonloc,log ln(ˆr) , +(4.3) +δbST +3,nonloc = δbST +3,nonloc,0 + δbST +3,nonloc,log ln(ˆr) , +(4.4) +δqST +3,nonloc = δqST +3,nonloc,0 + δqST +3,nonloc,log ln(ˆr) . +(4.5) +Inserting the split of the EOB functions (A, B, q3) +using Eqs. (4.1)-(4.2) and Eq. (5.23) of Paper I in the +effective Hamiltonian of Eq. (2.10), and then after ex- +panding the right-side into a Taylor series of 1/c2, we +obtain +ˆHeff = ˆHloc +eff + ˆHnonloc +eff +, +(4.6) +where +ˆHloc +eff +is +computed +only +by +the +local +con- +tributions (δaST +4,loc, δbST +3,loc, δqST +3,loc) and +ˆHnonloc +eff +is the +nonlocal contribution of +Hamiltonian +computed +by + +5 +(δaST +4,nonloc, δbST +3,nonloc, δqST +3,nonloc). The nonlocal contribu- +tion ˆHnonloc +eff +reads +ˆHnonloc +eff += 1 +2 +� +δaST +4,nonloc +1 +ˆr4 − δbST +3,nonloc +ˆp2 +r +ˆr3 + δqST +3,nonloc +ˆp4 +r +ˆr2 +� +. +(4.7) +To map the real two-body dynamics to EOB, we ex- +press the nonlocal effective Hamiltonian, +ˆHnonloc +eff +, in +action-angle variables L, l, G, and g (hence the Keple- +rian variables a and e) and compute its l-averaged value, +ˆ¯Hnonloc +eff += 1 +2π +� 2π +0 +dl ˆHnonloc +eff +. +(4.8) +The explicit expression of ˆHnonloc +eff +depends on l-average +monomials involving powers of 1/ˆr and ˆpr (and also ln(ˆr) +from Eqs. (4.3), (4.4), and (4.5)). These computations +can be performed by expanding Eq. (4.7) in terms of +eccentricity upto e5 using the Newtonian equations of +motion in action-angle form recalled in Sec. III B. The +l-averaged value we obtain is +ˆ¯HII +eff = +1 +2a4 +� +δaST +4,nonloc,0 + δaST +4,nonloc,log ln(a) ++ +� +3δaST +4,nonloc,0 − 7 +4δaST +4,nonloc,log − 1 +2δbST +3,nonloc,0 + 3δaST +4,nonloc,log ln(a) − 1 +2δbST +3,nonloc,log ln(a) +� +e2 ++ +�45 +8 +� +δaST +4,nonloc,0 + δaST +4,nonloc,log ln(a) +� +− 5 +4 +� +δbST +3,nonloc,0 + δbST +3,nonloc,log ln(a) +� ++ 3 +8 +� +δqST +3,nonloc,0 + δqST +3,nonloc,log ln(a) +� +−171 +32 δaST +4,nonloc,log + 9 +16δbST +3,nonloc,log +� +e4 + O(e6) +� +. +(4.9) +The final step is then to map the real two-body dy- +namics to EOB metric by the nontrivial map, +ˆHreal = Hreal +µ += 1 +ν +� +1 + 2ν( ˆHeff − 1) , +(4.10) +between the EOB Hamiltonian ( ˆHeff) and real two-body +Hamiltonian ( ˆHreal). The quadratic map relating the two +Hamiltonians is proven at all PN orders in GR and ST +within the Post-Minkowskian scheme in Ref. [42]. How- +ever, it can be seen that only for the nonlocal contribu- +tions at 3PN order, the map relating the two nonlocal +Hamiltonians is +ˆ¯Hnonloc +eff += ˆ¯HII +real,nonloc . +(4.11) +The unique nonlocal ST contributions at 3PN from +this matching are +δaST +4,nonloc,0 = 4 +3ν +� +2δ+ + ¯γAB(¯γAB + 2) +2 +� +(2 ln 2 + 2γE), +(4.12) +δaST +4,nonloc,log = −4 +3ν +� +2δ+ + ¯γAB(¯γAB + 2) +2 +� +, +(4.13) +δbST +3,nonloc,0 = 4 +3ν +� +2δ+ + ¯γAB(¯γAB + 2) +2 +� �21 +2 − 16 ln 2 +� +, +(4.14) +δbST +3,nonloc,log = 0, +(4.15) +δqST +3,nonloc,0 = 4 +3ν +� +2δ+ + ¯γAB(¯γAB + 2) +2 +� � +−31 +4 − 256 +3 ln 2 + 243 +4 ln 3 +� +, +(4.16) +δqST +3,nonloc,log = 0 . +(4.17) +The ST tensor correction δaST +4,nonloc for the circular or- +bit case, Eqs. (4.12)-(4.13), matches with the results ob- +tained in Paper I (see, Eqs.(5.25)-(5.26)) except a nega- +tive sign in Eq. (4.13). The negative sign is due to the + +6 +difference in the definition of δaST +4,nonloc in Eq. (4.3) used +in this work with the Eq. (5.24) of Paper I. +V. +CONCLUSIONS +In Paper I, building upon the results of [21] for massless +scalar-tensor theory, we determined the EOB coefficients +at 3PN order though restricting ourselves to local-in-time +part of the dynamics and nonlocal-in-time and tail con- +tributions only for the circular case. In the present pa- +per, we derived the complete nonlocal-in-time EOB co- +efficients starting from the nonlocal-in-time Lagrangian +of Ref. [21]. First, we derived the two-body conserved +ordinary Hamiltonian (dependent only on positions and +momenta) for nonlocal-in-time part by two methods: (i) +non-order-reduced nonlocal Hamiltonian using nonlocal +phase shift (see, Ref. [40, 41] for GR), and (ii) order- +reduction of nonlocal dynamics to local ordinary action- +angle Hamiltonian [34]. We then expressed the effective +Hamiltonian in Delaunay variables to recast the order- +reduced ordinary action-angle Hamiltonian into equiv- +alent, 3PN-accurate, nonlocal part of EOB potentials +(A, B, Qe), see Eqs. (4.12)-(4.17). +By combining the results of Paper I and the present +work, we could transcribe the two-body Hamiltonian into +equivalent 3PN-accurate EOB potentials (A, B, Qe) for +both local-in-time and nonlocal-in-time part of dynamics. +Note: During the preparation of the final manuscript of +this work, the author became aware of the independent +effort which recently arrived on arXiv [43]. +ACKNOWLEDGMENTS +The author is grateful to P. Rettegno, M. Agathos +and A. Nagar for useful discussions and suggestions dur- +ing the preparation of this work. The author is jointly +funded by the University of Cambridge Trust, Depart- +ment of Applied Mathematics and Theoretical Physics +(DAMTP), and Centre for Doctoral Training, University +of Cambridge. +Appendix A: Fourier Coefficients of dipole moment +in ST theory +In this appendix, we will determine the explicit expres- +sions of Newtonian dipole moment in ST theory using +the known Fourier decomposition of the Keplerian mo- +tion (see, Refs. [44, 45] for GR). +The dipole moment, Is,i(t), in COM frame is +Is,i(t) = 2Mν(sA − sB) +φ0(3 + 2w0) +xi , +(A1) +where xi = (ZA − ZB)i is the relative separation vector +and ZA,B indicate the positions of the two bodies. +Since the motion is planar, we can choose the coor- +dinate system (x, y, z) such that it coincides with the +xy-plane. Using the polar coordinates (ˆr, φa), +x = ˆr cos(φa), y = ˆr sin(φa). +(A2) +The coordinates (x, y) are the coordinates of the dimen- +sionless relative separation, ˆr = xA − xB with xJ = +xJ/(GABM) denoting the position of two bodies. +As mentioned in Ref. [34, 44, 45] for GR, for leading +order contributions it is convenient to use the Delaunay +(action-angle) form of the Newtonian equations of mo- +tion. In terms of the action-angle variables (L, l, G, g), +the Cartesian coordinates (x, y) are given by (Here, we +follow the notations of [46]) +x = x0 cos(g) − y0 sin(g) , +(A3) +y = x0 cos(g) + y0 sin(g) , +(A4) +x0 = ˆr cos(f) = a(cos(u) − e) , +(A5) +y0 = ˆr sin(f) = a +� +1 − e2 sin(u) , +(A6) +ˆr = a(1 − e cos(u)) , +(A7) +where a is the semi-major axis, e is the eccentricity, f is +the “true anamoly” and the “eccenteric anamoly” u in +terms of Bessels functions is given by +u = l + +∞ +� +n=1 +2 +nJn(ne) sin(nl) . +(A8) +The Bessel-Fourier expansion of cos(u) and sin(u), which +directly enters x0, y0 are: +cos(u) = −e +2 + +∞ +� +n=1 +1 +n +� +Jn−1(ne) − Jn+1(ne) +� +cos(nl) , +(A9) +sin(u) = +∞ +� +n=1 +1 +n +� +Jn−1(ne) + Jn+1(ne) +� +sin(nl) . +(A10) +From Eqs. (A3)-(A8), the dipole moment Is,i is a pe- +riodic function of l (and hence time) at the Newtonian +order. Thus it can be decomposed into Fourier series +Is,i(l) = +∞ +� +p=−∞ +Ii,s(p) eipl , +(A11) +with +Is,i(p) = 1 +2π +� 2π +0 +dl Is,i e−ipl . +(A12) +The Fourier coefficients of the scalar dipole moment +at the Newtonian order are derived using Eq. (A12) in +terms of combinations of Bessel Functions. +Inserting the expression of Cartesian coordinates in +terms of action-angle variables using Eqs. (A3)-(A10), +we find the Fourier-Bessel coefficients of the scalar dipole +moment are + +7 +Is,x(p) = GABM +�2Mν(sA − sB) +φ0(3 + 2w0) +a +2p +�� +Jp−1(pe) − Jp+1(pe) +� +cos(g) + i +� +1 − e2 � +Jp−1(pe) + Jp+1(pe) +� +sin(g) +�� +, +(A13) +Is,y(p) = GABM +�2Mν(sA − sB) +φ0(3 + 2w0) +a +2p +�� +Jp−1(pe) − Jp+1(pe) +� +sin(g) + i +� +1 − e2 � +Jp−1(pe) + Jp+1(pe) +� +cos(g) +�� +. +(A14) +[1] B. P. Abbott et al. (Virgo, LIGO Scientific), Obser- +vation of Gravitational Waves from a Binary Black +Hole +Merger, +Phys. Rev. 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Clemence, Methods of Celestial +Mechanics (Academic Press, 1961). + diff --git a/QtAzT4oBgHgl3EQfI_sd/content/tmp_files/load_file.txt b/QtAzT4oBgHgl3EQfI_sd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5afbb69a79c27deae734e1e285fb3cae34d568e1 --- /dev/null +++ b/QtAzT4oBgHgl3EQfI_sd/content/tmp_files/load_file.txt @@ -0,0 +1,647 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf,len=646 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='01070v1 [gr-qc] 3 Jan 2023 Nonlocal-in-time effective one body Hamiltonian in scalar-tensor gravity at third post-Newtonian order Tamanna Jain∗ Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Wilberforce Road CB3 0WA Cambridge, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (Dated: January 4, 2023) We complete the nonlocal-in-time effective-one-body (EOB) formalism of conservative dynamics for massless Scalar-Tensor (ST) theories at third post-Newtonian (PN) order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The nonlocal-in- time EOB Hamiltonian is obtained by mapping the order-reduced Hamiltonian corresponding to the nonlocal-in-time Lagrangian derived in [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' D 99, 044047 (2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' To transcribe the dynamics within EOB formalism, we use a strategy of order-reduction of nonlocal dynamics to local ordinary action-angle Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' We then map this onto the EOB Hamiltonian to determine the nonlocal-in-time ST corrections to the EOB potentials (A, B, Qe) at 3PN order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' INTRODUCTION In 2015, the direct detection of gravitational waves (GW) by the LIGO-Virgo Collaboration [1] emitted by inspiralling compact binary, opened new avenues for probing the dynamics in strong gravity regime [2–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' It is expected that future GW detectors, like the Einstein Telescope [9] and Cosmic Explorer [10] will shed more light on alternative theories of gravity by constraining the parameters of such theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The simplest theory amongst the alternative theories of gravity is the addition of a massless scalar field to GR, scalar-tensor (ST) theories, which are are exten- sively studied [11–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The motivation for ST theories is to explain both the accelerated expansion of the uni- verse as f(R)-theories [17] as well as UV complete alter- nate theories of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The two-body PN formalism for ST theories has been extensively studied [18–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The important violations for ST theories arise through the non-perturbative strong field effects in neutron-stars such as spontaneous scalarisation [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Although the cur- rent constraints come from the binary pulsar observa- tions, the future GW detections can better constraint the parameters using strong-field information and addi- tional terms in radiation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' dipolar radiation which is not present in GR, due to the scalar extension of GR [15, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The EOB formalism was introduced to construct ana- lytical waveform templates for GR [29–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Recently, the two-body PN dynamics has also been mapped within the EOB formalism to construct waveform templates for ST theories [36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' In our previous work [38], we deter- mined the EOB potentials for the local part of dynamics at 3PN order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The aim of this paper is to determine the complete nonlocal-in-time EOB potentials following our results of local part in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [38] starting from the 3PN nonlocal-in-time Lagrangian of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Hereafter, the companion paper [38] will be referred as Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' ∗ tj317@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='uk The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' II, we give a summary of results obtained in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' III we derive the conserved energy for nonlocal- in-time part using two methods, (i) non-order-reduced nonlocal Hamiltonian using nonlocal phase shift, and (ii) order-reduction of nonlocal dynamics to local ordinary action-angle Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' IV we map the nonlocal-in-time ordinary Hamiltonian into an EOB Hamiltonian at 3PN order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' SUMMARY OF PREVIOUS RESULTS We consider mono-scalar massless ST theories de- scribed by the following action in the Einstein Frame (the scalar field minimally couples to the metric), S = c4 16πG � d4x√−g(R − 2gµν∂µϕ∂νϕ) + Sm[Ψ, A(ϕ)2gµν] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='1) where gµν is the Einstein metric, R is the Ricci scalar, ϕ is the scalar field, Ψ collectively denotes the matter fields, g ≡ det(gµν) and G is the bare Newton’s constant [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' As Paper I (see, Table I), we adopt the conventions and notations of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' In Einstein Frame, the dy- namics of the scalar field arises from its coupling to the matter fields Ψ, and the field equations can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [11] where the parameter α(ϕ) = ∂ ln A ∂ϕ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='2) measures the coupling between the matter and the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The scalar field is non-minimally coupled to the metric in Jordan Frame (physical frame) ˜gµν = A(ϕ)2gµν , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='3) where ˜gµν is the metric in Jordan frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' We follow the approach suggested by [39] to “skele- tonize” the compact, self-gravitating objects in ST the- ories as point particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' the total mass of each body 2 is dependent on the local value of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The skeletonized matter action with the scalar field depen- dent mass ˜mI(ϕ) is then given by Sm = − � J=A,B � � −˜gµν dxµ dλ dxν dλ ˜mJ(ϕ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='4) where λ is the affine parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Since ˜gµν = A(ϕ)2gµν, the Einstein-frame mass is defined as m(ϕ) = A(ϕ) ˜m(ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='5) In Paper I, we first derive the ordinary Hamiltonian (dependent only on the positions and momenta) using the contact transformation at 3PN order starting from the Lagrangian of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [21] only for the local-in-time part of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The Jordan-Frame parameters of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [21] that encompass the scalar field effect are con- verted to the dimensionless Einstein-Frame parameters (see, Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The mass function m(ϕ) is used to define these dimensionless body-dependent parameters follow- ing Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [11, 13, 37] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' αI = d ln m(ϕ)I dϕ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='6) βI = dαI dϕ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='7) β′ I = dβI dϕ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='8) β′′ I = dβ′ I dϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='9) Here, we follow the notations of Paper I for the binary parameters and use the same notation as [20, 21] to de- note weak-field and strong-field parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Finally, we then determine the ST corrections to the EOB metric potential (A, B, Qe) at 3PN order for the local in time (instantaneous) part of the dynamics by mapping the EOB Hamiltonian in DJS gauge [31] ˆHeff = Heff µ = � � � �A(ˆr) � 1 + ˆp2r B(ˆr) + ˆp2 φ ˆr2 + q3 ˆp4r ˆr2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='10) where ˆpr, ˆpφ are the dimensionless radial and angular mo- menta, and ˆr(= r/(GABM) is the dimensionless radial separation, to the ordinary two-body Hamiltonian (here- after the superscript hat is used to denote the dimension- less variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The three EOB potentials at 3PN are A(ˆr) = 1 − 2 ˆr + a2 ˆr2 + a3 ˆr3 + a4 ˆr4 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='11) B(r) = 1 + b1 ˆr + b2 ˆr2 + b3 ˆr3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='12) Qe(ˆr) = q3 ˆp4 r ˆr2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='13) The GR and ST corrections in coefficients (ai, bi) are separated as ai = aGR i + δaST i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='14) bi = bGR i + δbST i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='15) q3 = qGR 3 + δqST 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='16) Since there are also nonlocal-in-time and tidal contri- butions at 3PN order in ST theory, all the 3PN ST coef- ficients can thus be decomposed as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='23) of Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The complete expressions of local-in-time ST corrections at 3PN can be found in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='14)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='16) of Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' In Paper I, we also derive the nonlocal-in-time (tail) and tidal corrections only for the circular orbits using the gauge invariant energy for circular orbits given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The complete expression for these coeffi- cients can be found in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='25)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='27) of Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' TAIL CONTRIBUTION TO THE 3PN DYNAMICS The nonlocal-in-time two-body 3PN Lagrangian for massless ST theory obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [21] is in harmonic coordinates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' it depends (linearly) on the acceleration of the two bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' In this section, we will use two different methods to derive the Noetherian conserved energy for the tail contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' First, we will remove the accelera- tion dependence from the Lagrangian (hence, the Hamil- tonian) and stay within the non-order-reduced nonlocal framework (as done in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [40, 41] for GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Second, we will derive the order-reduced, local Hamiltonian using the action-angle variables (see, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [34] for GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Non-order-reduced Ordinary Hamiltonian In Paper I, we derived the ordinary (dependent only on positions and momenta) Hamiltonian for local-in- time contribution using contact transformation (see, Ap- pendix A of Paper I for the contact transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Now, concerning the nonlocal-in-time part we need to find the nonlocal shift that removes the acceleration de- pendence from the tail part of the Lagrangian of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [21] (see, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [40, 41] for GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Corresponding to this ordi- nary Lagrangian, we can then derive the ordinary Hamil- tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The tail part of the Lagrangian at 3PN order reads [21], Ltail = 2G2M 3c6 (3 + 2ω0) Pf2rAB/c � ∞ −∞ dτ |τ| I(2) s,i (t)I(2) s,i (t + τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='1) where Pf is the Hadamard partie finie function, Hadamard scale rAB(= r) is the relative separation of two bodies, and I(2) s,i is the second time derivative of the dipole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Here, we find the shift that transforms 3 this Lagrangian into the same expression but with the derivatives of the dipole moment evaluated using the Newtonian equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' In the centre of mass (COM) frame in notations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [20, 21] it is, ´I(2) s,i = 2Mν(sA − sB) φ0(3 + 2w0) � −GABM r2 ni AB � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='2) where sA, sB are the sensitivity of two bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' As the nonlocal contribution starts at 3PN order, the ordinary Lagrangian is Ltail ord = Ltail + � J=A,B mJ \uf8eb \uf8ed−ai J − � J̸=K GAB mK r2 ni JK \uf8f6 \uf8f8 ξJ,i , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='3) where Ltail ord is given by the same expression as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='1) but with second time derivative of the dipole moment replaced by its on-shell value given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='2), and the nonlocal shift, ξJ,j, ξJ,j = 1 mJ 2G2M 3c6 (3 + 2w0) � −mJ(1 − 2sJ) φ0(3 + 2w0) � δi j Pf2r/c � ∞ −∞ dτ |τ| ´I(2) s,i (t + τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='4) The ordinary Hamiltonian is then derived using the ordinary Legendre transformation, Hord = � A pAvA − Lord which reads Hord = Hloc ord + Htail ord, where the local contribution Hloc ord is derived in Paper I (see, Appendix C) and the tail contribution is Htail ord = −2G2M 3c6 (3 + 2w0) Pf2r/c � ∞ −∞ dτ |τ| ´I(2) s,i (t)´I(2) s,i (t + τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='5) The tail part of the Hamiltonian is just opposite to tail part of Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [35, 41] for the non-order-reduced, nonlocal framework the Noetherian conserved energy (Econs) is not given by the Hamiltonian but is given by, Econs = Htail ord + δH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' This additional term δH consists of purely a constant term (DC type) and time oscillating term with zero average value (AC type) and is same as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='10) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Order-reduced Ordinary Hamiltonian The second method to derive the conserved energy for tail part is to work in the order-reduced, local framework as given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [34, 35] for GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The tail part of the Hamiltonian in ST theory is, Htail = −2G2M 3c6 (3 + 2ω0) � Pf2r/c � ∞ −∞ dτ |τ| I(2) s,i (t)I(2) s,i (t + τ) −2 ln � ˆr a � I(2) s,i (t)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='6) As mentioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [41], in the action-angle form there should be an additional term (second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='6)) which is local and accounts for dependence of Hadamard Partie finie function on the radial separation (r) at time t i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=', ˆr = a(1 − e cos(u)) in action-angle variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The basic methodology we use to order-reduce the non- local dynamics of the above form is based on Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [34, 35] for GR, and consists of four main steps: (i) Re-express the Hamiltonian in terms of action angle variables, (ii)“order-reduce” the nonolocal dependence on action angle variable, (iii) expand it in powers of eccentricity, and (iv) eliminate the periodic terms in order-reduced Hamiltonian by a canonical transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' All of these steps lead to the order-reduced ordinary local Hamilto- nian for the tail part in terms of action-angle variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Let us consider the expression of nonlocal-in-time piece of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' K(t, τ) = ¨Is,i(t)¨Is,i(t + τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='7) To order reduce the nonlocal piece, we use the equations of motion to express the phase-space variables at shifted time t+τ in terms of the phase-space variables at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' As the zeroth order equations are Newtonian equations, it will be convenient to use the action-angle form of the Newtonian equations of motion, ∂l ∂ˆt = ∂H0 ∂L = 1 L3 = Ω(L) , ∂L ∂ˆt = ∂H0 ∂l = 0 , ∂G ∂ˆt = ∂H0 ∂g = 0 , ∂g ∂ˆt = ∂H0 ∂G = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='8) where ˆt = t/(GABM) is the dimensionless time variable, (L, l, G, g) are the action-angle variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The zeroth- order (Newtonian) Hamiltonian in action-angle variable is H0 = −1/(2L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Here, the variable L is conjugate to the “mean anamoly” l and G is conjugate to argument of perias- tron g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' In terms of the Keplerian variables, semi-major axis a, and eccentricity e, these are L = √a, G = � a(1 − e2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='9) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='8), the variables L, G and g are indepen- dent of time, and l varies linearly with time, hence it will be sufficient to use l(t + τ) = l(t) + Ω ˆτ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='10) where ˆτ = τ/(GABM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The order-reduced non-local in time expression of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='7) becomes K(t, τ) = � 1 GABM �4 K(ˆt, ˆτ) = � Ω GABM �4 d2 dl2 Is,i(l) d2 dl2 Is,i(l + Ωˆτ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='11) Using the Fourier decomposition of dipole moment given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A11), we find the structure of nonlocal-in- time expression K(t, ˆτ) and hence the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' As 4 shown in [34] for GR, all the periodically varying terms can be eliminated by a suitable canonical transforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Hence, the order-reduced Hamiltonian can be fur- ther simplified by replacing Htail with its l-average value ¯Htail = � 2π 0 dlHtail .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='12) Using the result PfT � ∞ 0 dv v cos(ωv) = −(γE + ln(ω T )) ∀ (ω > 0) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='13) where γE is the Euler’s constant, and inserting the ex- pression of r from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A7), the Hamiltonian, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='12), reads ¯Htail = 8G2M 3 � Ω GABM �4 (3 + 2w0) ∞ � p=1 p4��Is,i(p) ��2 ln � eγE 2p a Ω c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='14) Now, inserting the Fourier-Bessel expansion of scalar dipole moment from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A13)-(A14) (see, Appendix A for derivation) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='14), the real two-body nonlocal- in-time Hamiltonian in order-reduced, local framework is (in notations of Paper I) ˆ¯Htail ≡ ¯Htail µ = 2ν 3a4 � 2δ+ + ¯γAB(¯γAB + 2) 2 � ∞ � p=1 p2 e2 � 4e2J2 p−1(pe) + (8 − 4e2)J2 p(pe) − 8eJp−1(pe)Jp(pe) � \uf8ee \uf8f0γE + ln � 2p a−1/2 c �\uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='15) Expanding the result in powers of eccentricity, the Hamil- tonian as an expansion in eccentricity upto order of e4 reads ˆ¯Htail = 2ν 3a4 � 2δ+ + ¯γAB(¯γAB + 2) 2 � � 2 ln(2) − ln(a) + 2γE + e2 � 14 ln(2) + 6γE − 3 ln(a) � +e4 �45 4 γE − 3 4 ln(2) + 729 32 ln(3) − 45 8 ln(a) � + O(e6) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='16) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' SCALAR TENSOR CORRECTIONS TO EFFECTIVE ONE BODY AT 3PN:TAIL In this section, we will derive the complete tail cor- rections to the EOB metric potentials (A, B, Qe) for ST theories at 3PN order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Similar to the decomposition of complete 3PN coeffi- cient δaST 4 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='23) of Paper I, we decompose the complete 3PN ST coefficients δbST 3 , δqST 3 as δbST 3 = δbST 3,loc + δbST 3,nonloc + δbST 3,tidal , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='1) δqST 3 = δqST 3,loc + δqST 3,nonloc + δqST 3,tidal , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='2) where the local contributions (δbST 3,loc, δqST 3,loc) are derived in Paper I (see, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='14)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='15)), (δbST 3,nonloc, δqST 3,nonloc) are the nonlocal contributions, and (δbST 3,tidal, δqST 3,tidal) are the tidal contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The nonlocal contributions can be further decomposed similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='24) of Paper as δaST 4,nonloc = δaST 4,nonloc,0 + δaST 4,nonloc,log ln(ˆr) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='3) δbST 3,nonloc = δbST 3,nonloc,0 + δbST 3,nonloc,log ln(ˆr) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='4) δqST 3,nonloc = δqST 3,nonloc,0 + δqST 3,nonloc,log ln(ˆr) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='5) Inserting the split of the EOB functions (A, B, q3) using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='1)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='2) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='23) of Paper I in the effective Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='10), and then after ex- panding the right-side into a Taylor series of 1/c2, we obtain ˆHeff = ˆHloc eff + ˆHnonloc eff , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='6) where ˆHloc eff is computed only by the local con- tributions (δaST 4,loc, δbST 3,loc, δqST 3,loc) and ˆHnonloc eff is the nonlocal contribution of Hamiltonian computed by 5 (δaST 4,nonloc, δbST 3,nonloc, δqST 3,nonloc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The nonlocal contribu- tion ˆHnonloc eff reads ˆHnonloc eff = 1 2 � δaST 4,nonloc 1 ˆr4 − δbST 3,nonloc ˆp2 r ˆr3 + δqST 3,nonloc ˆp4 r ˆr2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='7) To map the real two-body dynamics to EOB, we ex- press the nonlocal effective Hamiltonian, ˆHnonloc eff , in action-angle variables L, l, G, and g (hence the Keple- rian variables a and e) and compute its l-averaged value, ˆ¯Hnonloc eff = 1 2π � 2π 0 dl ˆHnonloc eff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='8) The explicit expression of ˆHnonloc eff depends on l-average monomials involving powers of 1/ˆr and ˆpr (and also ln(ˆr) from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='4), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' These computations can be performed by expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='7) in terms of eccentricity upto e5 using the Newtonian equations of motion in action-angle form recalled in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The l-averaged value we obtain is ˆ¯HII eff = 1 2a4 � δaST 4,nonloc,0 + δaST 4,nonloc,log ln(a) + � 3δaST 4,nonloc,0 − 7 4δaST 4,nonloc,log − 1 2δbST 3,nonloc,0 + 3δaST 4,nonloc,log ln(a) − 1 2δbST 3,nonloc,log ln(a) � e2 + �45 8 � δaST 4,nonloc,0 + δaST 4,nonloc,log ln(a) � − 5 4 � δbST 3,nonloc,0 + δbST 3,nonloc,log ln(a) � + 3 8 � δqST 3,nonloc,0 + δqST 3,nonloc,log ln(a) � −171 32 δaST 4,nonloc,log + 9 16δbST 3,nonloc,log � e4 + O(e6) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='9) The final step is then to map the real two-body dy- namics to EOB metric by the nontrivial map, ˆHreal = Hreal µ = 1 ν � 1 + 2ν( ˆHeff − 1) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='10) between the EOB Hamiltonian ( ˆHeff) and real two-body Hamiltonian ( ˆHreal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The quadratic map relating the two Hamiltonians is proven at all PN orders in GR and ST within the Post-Minkowskian scheme in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' How- ever, it can be seen that only for the nonlocal contribu- tions at 3PN order, the map relating the two nonlocal Hamiltonians is ˆ¯Hnonloc eff = ˆ¯HII real,nonloc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='11) The unique nonlocal ST contributions at 3PN from this matching are δaST 4,nonloc,0 = 4 3ν � 2δ+ + ¯γAB(¯γAB + 2) 2 � (2 ln 2 + 2γE), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='12) δaST 4,nonloc,log = −4 3ν � 2δ+ + ¯γAB(¯γAB + 2) 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='13) δbST 3,nonloc,0 = 4 3ν � 2δ+ + ¯γAB(¯γAB + 2) 2 � �21 2 − 16 ln 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='14) δbST 3,nonloc,log = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='15) δqST 3,nonloc,0 = 4 3ν � 2δ+ + ¯γAB(¯γAB + 2) 2 � � −31 4 − 256 3 ln 2 + 243 4 ln 3 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='16) δqST 3,nonloc,log = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='17) The ST tensor correction δaST 4,nonloc for the circular or- bit case, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='12)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='13), matches with the results ob- tained in Paper I (see, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='25)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='26)) except a nega- tive sign in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The negative sign is due to the 6 difference in the definition of δaST 4,nonloc in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='3) used in this work with the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='24) of Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' CONCLUSIONS In Paper I, building upon the results of [21] for massless scalar-tensor theory, we determined the EOB coefficients at 3PN order though restricting ourselves to local-in-time part of the dynamics and nonlocal-in-time and tail con- tributions only for the circular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' In the present pa- per, we derived the complete nonlocal-in-time EOB co- efficients starting from the nonlocal-in-time Lagrangian of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' First, we derived the two-body conserved ordinary Hamiltonian (dependent only on positions and momenta) for nonlocal-in-time part by two methods: (i) non-order-reduced nonlocal Hamiltonian using nonlocal phase shift (see, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [40, 41] for GR), and (ii) order- reduction of nonlocal dynamics to local ordinary action- angle Hamiltonian [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' We then expressed the effective Hamiltonian in Delaunay variables to recast the order- reduced ordinary action-angle Hamiltonian into equiv- alent, 3PN-accurate, nonlocal part of EOB potentials (A, B, Qe), see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='12)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' By combining the results of Paper I and the present work, we could transcribe the two-body Hamiltonian into equivalent 3PN-accurate EOB potentials (A, B, Qe) for both local-in-time and nonlocal-in-time part of dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Note: During the preparation of the final manuscript of this work, the author became aware of the independent effort which recently arrived on arXiv [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' ACKNOWLEDGMENTS The author is grateful to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Rettegno, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Agathos and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Nagar for useful discussions and suggestions dur- ing the preparation of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The author is jointly funded by the University of Cambridge Trust, Depart- ment of Applied Mathematics and Theoretical Physics (DAMTP), and Centre for Doctoral Training, University of Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Appendix A: Fourier Coefficients of dipole moment in ST theory In this appendix, we will determine the explicit expres- sions of Newtonian dipole moment in ST theory using the known Fourier decomposition of the Keplerian mo- tion (see, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [44, 45] for GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' The dipole moment, Is,i(t), in COM frame is Is,i(t) = 2Mν(sA − sB) φ0(3 + 2w0) xi , (A1) where xi = (ZA − ZB)i is the relative separation vector and ZA,B indicate the positions of the two bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Since the motion is planar, we can choose the coor- dinate system (x, y, z) such that it coincides with the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Using the polar coordinates (ˆr, φa), x = ˆr cos(φa), y = ˆr sin(φa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A2) The coordinates (x, y) are the coordinates of the dimen- sionless relative separation, ˆr = xA − xB with xJ = xJ/(GABM) denoting the position of two bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' As mentioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' [34, 44, 45] for GR, for leading order contributions it is convenient to use the Delaunay (action-angle) form of the Newtonian equations of mo- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' In terms of the action-angle variables (L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' the Cartesian coordinates (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' y) are given by (Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' we follow the notations of [46]) x = x0 cos(g) − y0 sin(g) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A3) y = x0 cos(g) + y0 sin(g) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A4) x0 = ˆr cos(f) = a(cos(u) − e) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A5) y0 = ˆr sin(f) = a � 1 − e2 sin(u) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A6) ˆr = a(1 − e cos(u)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A7) where a is the semi-major axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' e is the eccentricity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' f is the “true anamoly” and the “eccenteric anamoly” u in terms of Bessels functions is given by u = l + ∞ � n=1 2 nJn(ne) sin(nl) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A8) The Bessel-Fourier expansion of cos(u) and sin(u), which directly enters x0, y0 are: cos(u) = −e 2 + ∞ � n=1 1 n � Jn−1(ne) − Jn+1(ne) � cos(nl) , (A9) sin(u) = ∞ � n=1 1 n � Jn−1(ne) + Jn+1(ne) � sin(nl) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A10) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A3)-(A8), the dipole moment Is,i is a pe- riodic function of l (and hence time) at the Newtonian order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Thus it can be decomposed into Fourier series Is,i(l) = ∞ � p=−∞ Ii,s(p) eipl , (A11) with Is,i(p) = 1 2π � 2π 0 dl Is,i e−ipl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A12) The Fourier coefficients of the scalar dipole moment at the Newtonian order are derived using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A12) in terms of combinations of Bessel Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' Inserting the expression of Cartesian coordinates in terms of action-angle variables using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} +page_content=' (A3)-(A10), we find the Fourier-Bessel coefficients of the scalar dipole moment are 7 Is,x(p) = GABM �2Mν(sA − sB) φ0(3 + 2w0) a 2p �� Jp−1(pe) − Jp+1(pe) � cos(g) + i � 1 − e2 � Jp−1(pe) + Jp+1(pe) � sin(g) �� , (A13) Is,y(p) = GABM �2Mν(sA − sB) φ0(3 + 2w0) a 2p �� Jp−1(pe) − Jp+1(pe) � sin(g) + i � 1 − e2 � Jp−1(pe) + Jp+1(pe) � cos(g) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} 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Clemence, Methods of Celestial Mechanics (Academic Press, 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfI_sd/content/2301.01070v1.pdf'} diff --git a/RNFRT4oBgHgl3EQf7zg5/content/tmp_files/2301.13681v1.pdf.txt b/RNFRT4oBgHgl3EQf7zg5/content/tmp_files/2301.13681v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd5cdff949cffc7e2ed47a299263a0990d1908b2 --- /dev/null +++ b/RNFRT4oBgHgl3EQf7zg5/content/tmp_files/2301.13681v1.pdf.txt @@ -0,0 +1,622 @@ +Potential of monolayer charge +Heigo Ers and Ritums Cepitis +Institute of Chemistry, University of Tartu, Tartu 50411, Estonia +Vladislav B. Ivaniˇstˇsev∗ +Department of Chemistry, Center for High Entropy Catalysis, +University of Copenhagen, Copenhagen 2100, Denmark +(Dated: February 1, 2023) +Investigation of charged interfaces is often limited by their complexity. Here, we introduce the +potential of monolayer charge as a new milepost in electric double layer studies. The potential of +monolayer charge signifies a point with the simplest possible interfacial structure. We use density +functional theory calculations to predict the potential of monolayer charge for a range of polyatomic +heterohydrocarbons. +The results suggest that increasing the lateral size of the ions allows the +formation of saturated monolayers at experimentally achievable potentials. Further investigation +of the potential of monolayer charge combined with suggestions for experimental verification will +enable new developments in electric double layer studies. +Introduction +Surface charge screening at interfaces is +a central concept in numerous research fields from plasma +physics to electrochemistry[1–3]. The screening happens +within the electrical double layer (EDL) – a nm-wide in- +terfacial region – and affects all the interfacial processes +and reactions. Therefore, due to its importance in appli- +cations, the development of a consistent and accurate +EDL paradigm is one of the grand challenges in fun- +damental research. [4, 5] The potential of zero charge +(PZC) has long served as a cornerstone in understand- +ing the EDL [6], but the estimation of PZC, especially in +experimental measurements of concentrated electrolytes, +has proven to be a non-trivial task [7–9]. +To address +this conundrum, we propose a new fundamental potential +in electrochemistry: the potential of monolayer charge +(PMC). We expect that establishing the PMC to open +new research directions and potentially even move to- +ward solving the Galvani problem. +One of the main complications in EDL studies is its +structural complexity, with 3D layers of ionic charges +screening the surface charge [4]. +This has been most +evident in the case of concentrated electrolytes such as +ionic liquids, where both long- and short-range interac- +tions are of essence [10]. The importance of short-range +Coulomb interactions, long-range charge networks as well +as finite size of ions, paves the way to the overscreening +and crowding phenomenons [11]. Traditional methods for +addressing this complexity, such as increasingly detailed +and precise models, have not yielded consistent results. +For instance, numerous theories have been developed for +the description of electrode–ionic liquid interfaces, which +have limitations in describing the charge screening and +ion oscillations [11–14]. +Meanwhile, coarse-grained as +well as atomic-scale computational models has allowed +a more accurate description of the interactions between +∗ http://www.doublelayer.eu; vliv@chem.ku.dk +the ions and their interfacial properties, while being hin- +dered by their computational cost and sampling issues +[15–18]. +We suggest a different approach: overcoming +this complexity by reaching the PMC, where the EDL +structure takes the simplest form of a monolayer of ions +exactly screening the surface charge. The reduction in +dimensionality offers a simpler way to obtain an accu- +rate and coherent assessment of the EDL, which can be +achieved by the self-assembly of the organic cations on +the electrode-electrolyte interface [19–22]. +In this letter, we provide the definition of the poten- +tial of monolayer charge and demonstrate that the PMC +value decreases with increasing lateral size of flat hetero- +atom polyaromatic hydrocarbons resembling a frisbee +disk. Based on that we suggest a setup for experimental +PMC concept verification: hypothetically stable inter- +faces made of frisbee shape ions in ionic liquid-based elec- +trolytes. To guide the verification, we provide an analyt- +ical relation between lateral size, electrode-ion distance, +ion charge, and PMC values, estimated at the density +functional level of theory. +Concept +Based on our previous Molecular Dynamics +simulations [23, 24], we conclude that PMC is the unique +point that distinguishes between overscreening to crowd- +ing mechanisms of the surface charge screening (see Fig. +1). The most remarkable feature of the EDL at PMC is +its dynamical and structural simplicity – it forms a sin- +gle 2D static monolayer of ions. This monolayer of max- +imum packing density θmax, exactly screens the surface: +σPMC = −θmax. According to previous Molecular Dy- +namics simulations, the contributions of the ions above +that monolayer to the potential drop across the EDL are +negligible so the PMC can be defined as +φPMC = σPMC +l +ε0 +, +(1) +where l is the distance between the electrode and ionic +layer charge planes and ε0 is the electric constant (see +arXiv:2301.13681v1 [cond-mat.mtrl-sci] 31 Jan 2023 + +2 +⊖ ⊕ ⊕ +37.009 ++ ++ +− +− ++ ++ ++ +− +− +⊕ +− +− +− +− +− +− ++ ++ ++ +− +− +− +− +− +− +− ++ +− +− +− +− +− +− +− +− +− +− +⊖ +crowding +overscreening +PZC +PMC +U +(a) +(b) +(c) +(d) +⎫ +⎪ +⎪ +⎪ +⎪ +⎪ +⎪ +⎪ +⎪ +⎬ +⎪ +⎪ +⎪ +⎪ +⎪ +⎪ +⎪ +⎪ +⎭ +⎫ +⎪ +⎪ +⎪ +⎪ +⎪ +⎪ +⎪ +⎪ +⎬ +⎪ +⎪ +⎪ +FIG. 1. +Schematic illustration of the interfacial structure at +(a) potential of zero charge (PZC); (b) overscreening poten- +tial range; (c) potential of monolayer charge (PMC); and (d) +crowding potential range; on a relative potential scale (U). +Wavecurve indicates the electrode surface; anions and cations +are marked with “+” and “−” signs. +Method below). +Below, we will use these Galvani potentials (φ), yet, +in general, the PMC values can be defined using any +potential scale (U), if it is referred to the PZC, as +φPMC − φPZC = UPMC − UPZC. In that sense, the PMC +is an additional reference point that in combination with +the PZC gives a consistent physics-based potential scale +along with an asymptotic scaling relation between σ and +φ: +σ +σPMC += +� +φ − φPZC +φPMC − φPZC +�α +. +(2) +Here, we present a Density Functional Theory-based es- +timation of the PMC values for frisbee-shaped ions at +Au(111) electrode and leave all other aspects of the PMC +concept out of the scope. +Method +The interfaces between Au(111) slab and a +monolayer of frisbee ions were constructed with the ASE +program [25]. The Au(111) slab was set to be four atomic +layers, where the two bottom layers were fixed in the +bulk positions. The studied frisbee ions were pyridinium +(Py+), sesquixanthylium (TOTA+), triazatriangulenium +(TATA+), and cations based on (N-doped) azacorene (N- +Cor+), azacircumcorene (N-CCor+), and azacircumcir- +cumcorene (N-CCCor+). +We used the PBE exchange-correlation functional [26] +along with D4 dispersion correction [27] and dipole cor- +rection. All calculations were carried out using GPAW +software [28, 29] with a 500 eV cut-off plane-wave basis in +spin-paired electron configuration, 12 ˚A above the ionic +layer (at least 18 ˚A between periodic images), and the +grid-spacing of 0.182 ˚A. The Brillouin zone was sampled +using a Monkhorst–Pack grid [30], where the number of +k-points (k) in periodic directions was chosen so that the +product ka was greater than 70 k-point·˚A, where a is +the length of the basis vector in a given direction. Pre- +liminary tests showed that the adsorption energy con- +verges into ±2.5 meV at the cut-off of 440 eV, ka of 50 +k-points·˚A, and a vacuum of 10 ˚A above the ionic layer. +The atomic regions were treated with the PAW formal- +ism, and 11, 4, 5, 6, and 1 valence electrons are included +for each Au, C, N, O, and H atom, respectively. +All optimizations were done with the quasi-Newton op- +timization algorithm. The interface model was relaxed to +a maximum force of 0.05 eV·˚A−1. The lattice parame- +ter of bulk gold was optimized (0.002 eV·˚A−1) to 4.118 +˚A, using the exponential cell filter. Separate single-point +calculations were conducted for the Au(111) slab and fris- +bee ions to evaluate the electron density difference. +For the estimation of electrodes’ surface charge den- +sity (σ), distance between the charge planes (l), and the +potential drop at Au(111)–ion interface (φ), the charge +density (ρq), was estimated as +ρq = −(ρe +Au|Ion − ρe +Au − ρe +Ion), +(3) +where ρe +Au and ρe +Ion correspond to the electron densities +of Au(111) slab and lone ion, respectively. The locations +of the electrode and the ion charge planes, illustrated +in Fig. +2, were estimated as weighted averages, using +the charge densities as weights. For that purpose, the +estimated ρq(z) profile was divided into two parts – one +associated with the electrode, and the other with the ion +– using the plane z1/2 where ρq ≈ 0 between the electrode +and the ion, as a boundary. The σ was evaluated as +σ = +� z1/2 +0 +ρq(z)dz. +(4) +The electrostatic potential (V ) in direction perpendic- +ular to the electrode’s surface (z-direction) was obtained +by Poisson’s equation: +V (z) = − 1 +ε0 +� z +0 +(z − z′)ρq(z′)dz′. +(5) +The potential drop across the interface (φ) was esti- +mated as the difference between the electrostatic poten- +tial at the boundaries of the simulation cell in z-direction. +PMC as the milepost between crowding and overscreen- +ing +The PMC concept originates from our Molecular +Dynamics simulations of ionic liquid–electrode interfaces. +We found that at the PMC, a single monolayer of ions +can exactly screen the surface charge. +This is a spe- +cific point between overscreening and crowding regimes, +which can also be detected in Molecular Dynamics sim- +ulations. Our previous work has shown that the PMC +values for 40 common ionic liquid ions are outside the +experimentally measurable stability window of the cor- +responding ionic liquids. We also found that a specific +shape – lateral large and flat – is needed to reach mea- +surable PMC values. +In this study, we focus on two families of ions that +meet this criterion. The first is the triangulenium family, +which consists of four representatives of similar size but +different polarizability. [31] These ions are well known + +3 +electrode +charge +plane +ionic layer +charge +plane +0 +1 +2 +distance, z / nm +charge density, ρq / e·nm−3 +8 +4 +4 +8 +l +FIG. 2. +Charge density vs. distance dependence (solid line), +illustrating interfacial charge density fluctuations in the direc- +tion perpendicular to the surface plane. Dashed vertical lines +indicate the positions of the electrode and ionic layer charge +planes. +for their surface self-assembly and are considered as can- +didates for being a discovery platform in molecular elec- +tronics. [32] The second is the azacoronenes family, which +consists of four ions of significantly different size but sim- +ilar polarizability. The smallest compound, pyridinium, +is similar to the core of most common ionic liquid cations, +such as alkylpyridinium and alkylimidazolium ions. +We confirmed that as the size of ions increases, the +potential drop decreases according to the equation: +φ = σ l +ε0 +, +(6) +as shown in Fig. 3. Here, σ accounts for the variable ionic +charge that decreases from +1 to a lower value through a +charge transfer to the surface. For azacoronene ions, the +partial charge transfer becomes negligible with increasing +their size. However, in the case of angulenium ions, it +depends on the ion composition and is responsible for +the potential variation in Fig. 3. +As noted in Fig. 3, the PMC values for all ions, except +for pyridinium, are actually from −2 to −1 V, which is +within the stability range of ionic liquids at such elec- +trodes as Au(111). That means interfaces of the model +ions could be experimentally studied. +One promising +technique for studying these interfaces is electrochemi- +cal in situ scanning tunneling microscopy (STM). The +STM technique allows for the direct imaging of the ionic +liquid-electrode interface at the atomic scale and pro- +vides valuable information about the structure and dy- +namics of the EDL. [33–35] The simulated STM image +in Fig. 4 illustrates the kind of image that could be ex- +pected for azacircumcoronen cation around −1.3 V vs. +PZC. From the image, we can see the clear monolayer +of ions exactly screening the surface charge at the PMC, +N+ +N+ +H +N+ +N+ +potential drop, ϕ / V +surface dipole, σl / Å·μC·cm−2 +φ = σl / ε0 +C+ +N +N +N +H +H +H +C+ +O +O +O +FIG. 3. +Potential drop vs. surface dipole dependence for +frisbee cations. +and the electrode–ionic liquid interface is clearly visible. +This kind of imaging will provide valuable information +about the structure and dynamics of the EDL and will +be important in further understanding the potential of +monolayer charge (PMC) as an additional fundamental +potential in electrochemistry. +x / nm +y / nm +0 +2 +2 +4 +4 +6 +6 +8 +8 +10 +10 +0 +FIG. 4. +Calculated scanning tunneling microscopy image of +the N-CCor+ monolayer on the Au(111) surface. +In conclusion, we show that in Density Functional +Theory calculations polyatomic heterohydrocarbon (an- +gulenium and azacorone-based) ions can form saturated +monolayers without significant partial charge transfer. +That means they can exactly screen the surface charge +at the potential of monolayer charge (PMC) within the + +4 +experimentally measurable potential window of common +ionic liquids.[36, 37] Thus, the PMC concept – as a +milepost potential between overscreening and crowding +regimes in concentrated electrolytes – could be experi- +mentally verified using electrochemical techniques such +as in situ scanning tunneling microscopy. +This work +paves the way for further research on the PMC and its +role in interfacial processes. +References +[1] J. 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Pikma, J. P. Perdew, +and E. Borguet, Potential-Induced High-Conductance +Transport Pathways through Single-Molecule Junctions, +J. Am. Chem. Soc 141, 10109 (2019). +[35] R. Atkin, N. Borisenko, M. Dr¨uschler, S. Z. El Abe- +din, F. Endres, R. Hayes, B. Huber, and B. Roling, An +in situ STM/AFM and impedance spectroscopy study +of the extremely pure 1-butyl-1-methylpyrrolidinium +tris(pentafluoroethyl)trifluorophosphate/Au(111) +inter- +face: potential dependent solvation layers and the her- +ringbone reconstruction, Phys. Chem. Chem. Phys. 13, +6849 (2011). +[36] Y.-Z. Su, Y.-C. Fu, J.-W. Yan, Z.-B. Chen, and B.-W. +Mao, Double Layer of Au(100)/Ionic Liquid Interface and +Its Stability in Imidazolium-Based Ionic Liquids, Angew. +Chem. Int. Ed. 48, 5148 (2009). +[37] M. P. S. Mousavi, A. J. Dittmer, B. E. Wilson, J. Hu, +A. Stein, and P. B¨uhlmann, Unbiased Quantification of +the Electrochemical Stability Limits of Electrolytes and +Ionic Liquids, J. Electrochem. Soc. 162, A2250 (2015). +ACKNOWLEDGMENTS +V.I. receives funding from the European Union’s +Horizon 2020 research and innovation program un- +der the Marie Sk�lodowska–Curie grant agreement no. +101031656. This research was also supported by the Es- +tonian Research Council grants PSG250 and PSG249; +and by the EU through the European Regional Devel- +opment Fund (TK141, “Advanced materials and high- +technology devices for energy recuperation systems”). + diff --git a/RNFRT4oBgHgl3EQf7zg5/content/tmp_files/load_file.txt b/RNFRT4oBgHgl3EQf7zg5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..67771b490a667d9383243fccc713bbccb0a911b5 --- /dev/null +++ b/RNFRT4oBgHgl3EQf7zg5/content/tmp_files/load_file.txt @@ -0,0 +1,564 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf,len=563 +page_content='Potential of monolayer charge Heigo Ers and Ritums Cepitis Institute of Chemistry, University of Tartu, Tartu 50411, Estonia Vladislav B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Ivaniˇstˇsev∗ Department of Chemistry, Center for High Entropy Catalysis, University of Copenhagen, Copenhagen 2100, Denmark (Dated: February 1, 2023) Investigation of charged interfaces is often limited by their complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Here, we introduce the potential of monolayer charge as a new milepost in electric double layer studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The potential of monolayer charge signifies a point with the simplest possible interfacial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' We use density functional theory calculations to predict the potential of monolayer charge for a range of polyatomic heterohydrocarbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The results suggest that increasing the lateral size of the ions allows the formation of saturated monolayers at experimentally achievable potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Further investigation of the potential of monolayer charge combined with suggestions for experimental verification will enable new developments in electric double layer studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Introduction Surface charge screening at interfaces is a central concept in numerous research fields from plasma physics to electrochemistry[1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The screening happens within the electrical double layer (EDL) – a nm-wide in- terfacial region – and affects all the interfacial processes and reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Therefore, due to its importance in appli- cations, the development of a consistent and accurate EDL paradigm is one of the grand challenges in fun- damental research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' [4, 5] The potential of zero charge (PZC) has long served as a cornerstone in understand- ing the EDL [6], but the estimation of PZC, especially in experimental measurements of concentrated electrolytes, has proven to be a non-trivial task [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' To address this conundrum, we propose a new fundamental potential in electrochemistry: the potential of monolayer charge (PMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' We expect that establishing the PMC to open new research directions and potentially even move to- ward solving the Galvani problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' One of the main complications in EDL studies is its structural complexity, with 3D layers of ionic charges screening the surface charge [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' This has been most evident in the case of concentrated electrolytes such as ionic liquids, where both long- and short-range interac- tions are of essence [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The importance of short-range Coulomb interactions, long-range charge networks as well as finite size of ions, paves the way to the overscreening and crowding phenomenons [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Traditional methods for addressing this complexity, such as increasingly detailed and precise models, have not yielded consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' For instance, numerous theories have been developed for the description of electrode–ionic liquid interfaces, which have limitations in describing the charge screening and ion oscillations [11–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Meanwhile, coarse-grained as well as atomic-scale computational models has allowed a more accurate description of the interactions between ∗ http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='doublelayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='eu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' vliv@chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='dk the ions and their interfacial properties, while being hin- dered by their computational cost and sampling issues [15–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' We suggest a different approach: overcoming this complexity by reaching the PMC, where the EDL structure takes the simplest form of a monolayer of ions exactly screening the surface charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The reduction in dimensionality offers a simpler way to obtain an accu- rate and coherent assessment of the EDL, which can be achieved by the self-assembly of the organic cations on the electrode-electrolyte interface [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' In this letter, we provide the definition of the poten- tial of monolayer charge and demonstrate that the PMC value decreases with increasing lateral size of flat hetero- atom polyaromatic hydrocarbons resembling a frisbee disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Based on that we suggest a setup for experimental PMC concept verification: hypothetically stable inter- faces made of frisbee shape ions in ionic liquid-based elec- trolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' To guide the verification, we provide an analyt- ical relation between lateral size, electrode-ion distance, ion charge, and PMC values, estimated at the density functional level of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Concept Based on our previous Molecular Dynamics simulations [23, 24], we conclude that PMC is the unique point that distinguishes between overscreening to crowd- ing mechanisms of the surface charge screening (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The most remarkable feature of the EDL at PMC is its dynamical and structural simplicity – it forms a sin- gle 2D static monolayer of ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' This monolayer of max- imum packing density θmax, exactly screens the surface: σPMC = −θmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' According to previous Molecular Dy- namics simulations, the contributions of the ions above that monolayer to the potential drop across the EDL are negligible so the PMC can be defined as φPMC = σPMC l ε0 , (1) where l is the distance between the electrode and ionic layer charge planes and ε0 is the electric constant (see arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='13681v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='mtrl-sci] 31 Jan 2023 2 ⊖ ⊕ ⊕ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='009 + + − − + + + − − ⊕ − − − − − − + + + − − − − − − − + − − − − − − − − − − ⊖ crowding overscreening PZC PMC U (a) (b) (c) (d) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Schematic illustration of the interfacial structure at (a) potential of zero charge (PZC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' (b) overscreening poten- tial range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' (c) potential of monolayer charge (PMC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' and (d) crowding potential range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' on a relative potential scale (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Wavecurve indicates the electrode surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' anions and cations are marked with “+” and “−” signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Method below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Below, we will use these Galvani potentials (φ), yet, in general, the PMC values can be defined using any potential scale (U), if it is referred to the PZC, as φPMC − φPZC = UPMC − UPZC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' In that sense, the PMC is an additional reference point that in combination with the PZC gives a consistent physics-based potential scale along with an asymptotic scaling relation between σ and φ: σ σPMC = � φ − φPZC φPMC − φPZC �α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' (2) Here, we present a Density Functional Theory-based es- timation of the PMC values for frisbee-shaped ions at Au(111) electrode and leave all other aspects of the PMC concept out of the scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Method The interfaces between Au(111) slab and a monolayer of frisbee ions were constructed with the ASE program [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The Au(111) slab was set to be four atomic layers, where the two bottom layers were fixed in the bulk positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The studied frisbee ions were pyridinium (Py+), sesquixanthylium (TOTA+), triazatriangulenium (TATA+), and cations based on (N-doped) azacorene (N- Cor+), azacircumcorene (N-CCor+), and azacircumcir- cumcorene (N-CCCor+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' We used the PBE exchange-correlation functional [26] along with D4 dispersion correction [27] and dipole cor- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' All calculations were carried out using GPAW software [28, 29] with a 500 eV cut-off plane-wave basis in spin-paired electron configuration, 12 ˚A above the ionic layer (at least 18 ˚A between periodic images), and the grid-spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='182 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The Brillouin zone was sampled using a Monkhorst–Pack grid [30], where the number of k-points (k) in periodic directions was chosen so that the product ka was greater than 70 k-point·˚A, where a is the length of the basis vector in a given direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Pre- liminary tests showed that the adsorption energy con- verges into ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='5 meV at the cut-off of 440 eV, ka of 50 k-points·˚A, and a vacuum of 10 ˚A above the ionic layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The atomic regions were treated with the PAW formal- ism, and 11, 4, 5, 6, and 1 valence electrons are included for each Au, C, N, O, and H atom, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' All optimizations were done with the quasi-Newton op- timization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The interface model was relaxed to a maximum force of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='05 eV·˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The lattice parame- ter of bulk gold was optimized (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='002 eV·˚A−1) to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='118 ˚A, using the exponential cell filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Separate single-point calculations were conducted for the Au(111) slab and fris- bee ions to evaluate the electron density difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' For the estimation of electrodes’ surface charge den- sity (σ), distance between the charge planes (l), and the potential drop at Au(111)–ion interface (φ), the charge density (ρq), was estimated as ρq = −(ρe Au|Ion − ρe Au − ρe Ion), (3) where ρe Au and ρe Ion correspond to the electron densities of Au(111) slab and lone ion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The locations of the electrode and the ion charge planes, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 2, were estimated as weighted averages, using the charge densities as weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' For that purpose, the estimated ρq(z) profile was divided into two parts – one associated with the electrode, and the other with the ion – using the plane z1/2 where ρq ≈ 0 between the electrode and the ion, as a boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The σ was evaluated as σ = � z1/2 0 ρq(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' (4) The electrostatic potential (V ) in direction perpendic- ular to the electrode’s surface (z-direction) was obtained by Poisson’s equation: V (z) = − 1 ε0 � z 0 (z − z′)ρq(z′)dz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' (5) The potential drop across the interface (φ) was esti- mated as the difference between the electrostatic poten- tial at the boundaries of the simulation cell in z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' PMC as the milepost between crowding and overscreen- ing The PMC concept originates from our Molecular Dynamics simulations of ionic liquid–electrode interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' We found that at the PMC, a single monolayer of ions can exactly screen the surface charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' This is a spe- cific point between overscreening and crowding regimes, which can also be detected in Molecular Dynamics sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Our previous work has shown that the PMC values for 40 common ionic liquid ions are outside the experimentally measurable stability window of the cor- responding ionic liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' We also found that a specific shape – lateral large and flat – is needed to reach mea- surable PMC values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' In this study, we focus on two families of ions that meet this criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The first is the triangulenium family, which consists of four representatives of similar size but different polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' [31] These ions are well known 3 electrode charge plane ionic layer charge plane 0 1 2 distance, z / nm charge density, ρq / e·nm−3 8 4 4 8 l FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Charge density vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' distance dependence (solid line), illustrating interfacial charge density fluctuations in the direc- tion perpendicular to the surface plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Dashed vertical lines indicate the positions of the electrode and ionic layer charge planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' for their surface self-assembly and are considered as can- didates for being a discovery platform in molecular elec- tronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' [32] The second is the azacoronenes family, which consists of four ions of significantly different size but sim- ilar polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The smallest compound, pyridinium, is similar to the core of most common ionic liquid cations, such as alkylpyridinium and alkylimidazolium ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' We confirmed that as the size of ions increases, the potential drop decreases according to the equation: φ = σ l ε0 , (6) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Here, σ accounts for the variable ionic charge that decreases from +1 to a lower value through a charge transfer to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' For azacoronene ions, the partial charge transfer becomes negligible with increasing their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' However, in the case of angulenium ions, it depends on the ion composition and is responsible for the potential variation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' As noted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 3, the PMC values for all ions, except for pyridinium, are actually from −2 to −1 V, which is within the stability range of ionic liquids at such elec- trodes as Au(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' That means interfaces of the model ions could be experimentally studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' One promising technique for studying these interfaces is electrochemi- cal in situ scanning tunneling microscopy (STM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' The STM technique allows for the direct imaging of the ionic liquid-electrode interface at the atomic scale and pro- vides valuable information about the structure and dy- namics of the EDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' [33–35] The simulated STM image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 4 illustrates the kind of image that could be ex- pected for azacircumcoronen cation around −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content='3 V vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' PZC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' From the image, we can see the clear monolayer of ions exactly screening the surface charge at the PMC, N+ N+ H N+ N+ potential drop, ϕ / V surface dipole, σl / Å·μC·cm−2 φ = σl / ε0 C+ N N N H H H C+ O O O FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Potential drop vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' surface dipole dependence for frisbee cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' and the electrode–ionic liquid interface is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' This kind of imaging will provide valuable information about the structure and dynamics of the EDL and will be important in further understanding the potential of monolayer charge (PMC) as an additional fundamental potential in electrochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' x / nm y / nm 0 2 2 4 4 6 6 8 8 10 10 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Calculated scanning tunneling microscopy image of the N-CCor+ monolayer on the Au(111) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' In conclusion, we show that in Density Functional Theory calculations polyatomic heterohydrocarbon (an- gulenium and azacorone-based) ions can form saturated monolayers without significant partial charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' That means they can exactly screen the surface charge at the potential of monolayer charge (PMC) within the 4 experimentally measurable potential window of common ionic liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' [36, 37] Thus, the PMC concept – as a milepost potential between overscreening and crowding regimes in concentrated electrolytes – could be experi- mentally verified using electrochemical techniques such as in situ scanning tunneling microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' This work paves the way for further research on the PMC and its role in interfacial processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' de Souza, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' Goodwin, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' receives funding from the European Union’s Horizon 2020 research and innovation program un- der the Marie Sk�lodowska–Curie grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' 101031656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' This research was also supported by the Es- tonian Research Council grants PSG250 and PSG249;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} +page_content=' and by the EU through the European Regional Devel- opment Fund (TK141, “Advanced materials and high- technology devices for energy recuperation systems”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFRT4oBgHgl3EQf7zg5/content/2301.13681v1.pdf'} diff --git a/SNE5T4oBgHgl3EQfaQ8G/content/tmp_files/2301.05586v1.pdf.txt b/SNE5T4oBgHgl3EQfaQ8G/content/tmp_files/2301.05586v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..00ce60f7cf8aeba88cad2f18786d6d98edea4cc6 --- /dev/null +++ b/SNE5T4oBgHgl3EQfaQ8G/content/tmp_files/2301.05586v1.pdf.txt @@ -0,0 +1,1305 @@ +YOLOv6 v3.0: A Full-Scale Reloading +Chuyi Li∗ +Lulu Li∗ +Yifei Geng∗ Hongliang Jiang∗ +MengCheng +Bo Zhang +Zaidan Ke +Xiaoming Xu† +Xiangxiang Chu +Meituan Inc. +{lichuyi, lilulu05, gengyifei, jianghongliang02, +chengmeng05, zhangbo97, kezaidan, xuxiaoming04, chuxiangxiang}@meituan.com +0 +200 +400 +600 +800 +1000 +1200 +Tesla T4 TensorRT FP16 Throughput (FPS), BS=32 +25 +30 +35 +40 +45 +50 +55 +60 +COCO AP (%) +YOLOv6-N +YOLOv6-S +YOLOv6-M +YOLOv6-L +YOLOv6-M6 +YOLOv6-L6 +YOLOv6 +YOLOv5 +YOLOv7 +YOLOv8 +YOLOX +PP-YOLOE +0 +10 +20 +30 +40 +50 +60 +Tesla T4 TensorRT FP16 Latency (ms), BS=1 +25 +30 +35 +40 +45 +50 +55 +60 +COCO AP (%) +YOLOv6-N +YOLOv6-S +YOLOv6-M +YOLOv6-L +YOLOv6-M6 +YOLOv6-L6 +YOLOv6 +YOLOv5 +YOLOv7 +YOLOv8 +YOLOX +PP-YOLOE +Figure 1: Comparison of state-of-the-art efficient object detectors. Both latency and throughput (at a batch size of 32) are +given for a handy reference. All models are test with TensorRT 7. +Abstract +The YOLO community has been in high spirits since our +first two releases! By the advent of Chinese New Year 2023, +which sees the Year of the Rabbit, we refurnish YOLOv6 +with numerous novel enhancements on the network archi- +tecture and the training scheme. This release is identified as +YOLOv6 v3.0. For a glimpse of performance, our YOLOv6- +N hits 37.5% AP on the COCO dataset at a throughput of +1187 FPS tested with an NVIDIA Tesla T4 GPU. YOLOv6- +S strikes 45.0% AP at 484 FPS, outperforming other main- +stream detectors at the same scale (YOLOv5-S, YOLOv8- +S, YOLOX-S and PPYOLOE-S). Whereas, YOLOv6-M/L +also achieve better accuracy performance (50.0%/52.8% +respectively) than other detectors at a similar inference +speed. Additionally, with an extended backbone and neck +design, our YOLOv6-L6 achieves the state-of-the-art ac- +curacy in real-time. +Extensive experiments are carefully +conducted to validate the effectiveness of each improving +component. +Our code is made available at https:// +* Equal contributions. +† Corresponding author. +github.com/meituan/YOLOv6. +1. Introduction +The YOLO series has been the most popular detection +frameworks in industrial applications, for its excellent bal- +ance between speed and accuracy. +Pioneering works of +YOLO series are YOLOv1-3 [12–14], which blaze a new +trail of one-stage detectors along with the later substan- +tial improvements. YOLOv4 [1] reorganized the detection +framework into several separate parts (backbone, neck and +head), and verified bag-of-freebies and bag-of-specials at +the time to design a framework suitable for training on a +single GPU. At present, YOLOv5 [5], YOLOX [3], PPY- +OLOE [17], YOLOv7 [16] and most recently YOLOv8 [6] +are all the competing candidates for efficient detectors to +deploy. +In this release, we strenuously renovate the network de- +sign and the training strategy. We show the comparison of +YOLOv6 with other peers at a similar scale in Fig. 1. The +new features of YOLOv6 are summarized as follows: +• We renew the neck of the detector with a Bi-directional +1 +arXiv:2301.05586v1 [cs.CV] 13 Jan 2023 + +Figure 2: (a) The neck of YOLOv6 (N and S are shown). Note for M/L, RepBlocks is replaced with CSPStackRep. (b) The +structure of a BiC module. (c) A SimCSPSPPF block. +Concatenation (BiC) module to provide more accurate +localization signals. SPPF [5] is simplified to form the +SimCSPSPPF Block, which brings performance gains +with negligible speed degradation. +• We propose an anchor-aided training (AAT) strat- +egy to enjoy the advantages of both anchor-based and +anchor-free paradigms without touching inference ef- +ficiency. +• We deepen YOLOv6 to have another stage in the back- +bone and the neck, which reinforces it to hit a new +state-of-the-art performance on the COCO dataset at +a high-resolution input. +• We involve a new self-distillation strategy to boost the +performance of small models of YOLOv6, in which +the heavier branch for DFL [8] is taken as an enhanced +auxiliary regression branch during training and is re- +moved at inference to avoid the marked speed decline. +2. Method +2.1. Network Design +In practice, feature integration at multiple scales has +been proven to be a critical and effective component of +object detection. Feature Pyramid Network (FPN) [9] is +proposed to aggregate the high-level semantic features and +low-level features via a top-down pathway, which provides +more accurate localization. Subsequently, there have been +several works [2, 4, 10, 15] on Bi-directional FPN in or- +der to enhance the ability of hierarchical feature represen- +tation. PANet [10] adds an extra bottom-up pathway on +top of FPN to shorten the information path of low-level +and top-level features, which facilitates the propagation of +accurate signals from low-level features. BiFPN [15] in- +troduces learnable weights for different input features and +simplifies PAN to achieve better performance with high ef- +ficiency. PRB-FPN [2] is proposed to retain high-quality +features for accurate localization by a parallel FP structure +with bi-directional fusion and associated improvements. +Motivated by the above works, we design an enhanced- +PAN as our detection neck. In order to augment localiza- +tion signals without bringing in excessive computation bur- +den, we propose a Bi-directional Concatenation(BiC) mod- +ule to integrate feature maps of three adjacent layers, which +fuses an extra low-level feature from backbone Ci−1 into +Pi (Fig. 2). In this case, more accurate localization signals +can be preserved, which is significant for the localization of +small objects. +Moreover, we simplify the SPPF block [5] to have +a CSP-like version called SimCSPSPPF Block, which +strengthens the representational ability. Particularly, we re- +vise the SimSPPCSPC Block in [16] by shrinking the chan- +nels of hidden layers and retouching space pyramid pool- +ing. In addition, we upgrade the CSPBlock with RepBlock +(for small models) or CSPStackRep Block (for large mod- +els) and accordingly adjust the width and depth. The neck +of YOLOv6 is denoted as RepBi-PAN, the framework of +which is shown in Fig. 2. +2.2. Anchor-Aided Training +YOLOv6 is an anchor-free detector to pursue a higher +inference speed. However, we experimentally find that the +anchor-based paradigm brings additional performance gains +on YOLOv6-N under the same settings when compared +with the anchor-free one, as shown in Table 1. Moreover, +anchor-based ATSS [18] is adopted as the warm-up label as- +signment strategy in the early versions of YOLOv6, which +stabilizes the training. +Paradigm +APval +APs +APm +APl +Anchor-free +35.5% +16.0% +39.5% +51.0% +Anchor-based +35.6% +17.2% +39.8% +52.1% +Table 1: Comparisons of the anchor-free and the anchor- +based paradigms on YOLOv6-N. +In light of this, we propose anchor-aided training (AAT), +in which the anchor-based auxiliary branches are intro- +2 + +Pi+1 +RepBi-PAN +: Concatenation over channel dimension +--- +1x1 Conv +P5 +SimCSPSPPF +RepBlock +N5 +3x3 Conv + Up-sample +1x1 Conv +Conv +Conv +Ci +1x1 Conv +1x1 +1x1 +(C +C4 +BiC +N4 + RepBlock +RepBlock +Conv +Conv +-- +Conv +Conv + Down-sample +- +1x1 Conv +C3 +RepBlock +N3 +BiC +1x1 +Conv +3x3 Conv +C +Ci-1 +1x1 Conv +C2 +(a) RepBi-PAN +(b) BiC Module +(c) SimCSPSPPF Blockduced to combine the advantages of anchor-based and +anchor-free paradigms. And they are applied both in the +classification and the regression head. Fig. 3 shows the de- +tection head with the auxiliaries. +Figure 3: The detection head with anchor-based auxiliary +branches during training. The auxiliary branches are re- +moved at inference. ‘af’ and ‘ab’ are short for ‘anchor-free’ +and ‘anchor-based’. +During the training stage, the auxiliary branches and the +anchor-free branches learn from independent losses while +signals are propagated altogether. Therefore, additional em- +bedded guidance information from auxiliary branches is in- +tegrated into the main anchor-free heads. Worth mentioning +that the auxiliary branches is removed at inference, which +boosts the accuracy performance without decreasing speed. +2.3. Self-distillation +In early versions of YOLOv6, the self-distillation is only +introduced in large models (i.e., YOLOv6-M/L), which ap- +plies the vanilla knowledge distillation technique by mini- +mizing the KL-divergence between the class prediction of +the teacher and the student. Meanwhile DFL [8] is adopted +as regression loss to perform self-distillation on box regres- +sion similar to [19]. +The knowledge distillation loss is formulated as: +LKD = KL(pcls +t ||pcls +s ) + KL(preg +t +||preg +s +), +(1) +where pcls +t +and pcls +s +are class prediction of the teacher model +and the student model respectively, and accordingly preg +t +and preg +s +are box regression predictions. The overall loss +function is now formulated as: +Ltotal = Ldet + αLKD, +(2) +where Ldet is the detection loss computed with predictions +and labels. The hyperparameter α is introduced to balance +two losses. In the early stage of training, the soft labels from +the teacher are easier to learn. As the training continues, +the performance of the student will match the teacher so +that the hard labels will help students more. Upon this, we +apply cosine weight decay to α to dynamically adjust the +information from hard labels and soft ones from the teacher. +The formulation of α is: +α = −0.99 ∗ ((1 − cos(π ∗ Ei/Emax))/2) + 1, +(3) +where Ei denotes the current training epoch and Emax rep- +resents the maximum training epochs. +Notably, the introduction of DFL [8] requires extra pa- +rameters for the regression branch, which affects the in- +ference speed of small models significantly. Therefore, we +specifically design the Decoupled Localization Distillation +(DLD) for our small models to boost performance with- +out speed degradation. +Specifically, we append a heavy +auxiliary enhanced regression branch to incorporate DFL. +During the self-distillation, the student is equipped with a +na¨ıve regression branch and the enhanced regression branch +while the teacher only uses the auxiliary branch. +Note +that the na¨ıve regression branch is only trained with hard +labels while the auxiliary is updated according to signals +from both the teacher and hard labels. After the distillation, +the na¨ıve regression branch is retained whilst the auxiliary +branch is removed. With this strategy, the advantages of the +heavy regression branch for DFL in distillation is consider- +ably maintained without impacting the inference efficiency. +3. Experiments +3.1. Comparisons +The evaluation is made consistent with the early ver- +sions of YOLOv6 [7], which focuses on the throughput +and the GPU latency at deployment. +We test the speed +performance of all official models with FP16-precision on +the same Tesla T4 GPU with TensorRT [11]. We compare +the upgraded YOLOv6 with YOLOv5 [5], YOLOX [3], +PPYOLOE [17], YOLOv7 [16] and YOLOv8 [6]. +Note +that the performance of YOLOv7-Tiny is re-evaluated ac- +cording to their open-sourced code and weights at the in- +put size of 416 and 640. +Results are shown in Table 2 +and Fig. 1. Compared with YOLOv5-N/YOLOv7-Tiny (in- +put size=416), our YOLOv6-N has significantly advanced +by 9.5%/4.2% respectively. +It also comes with the best +speed performance in terms of both throughput and latency. +Compared with YOLOX-S/PPYOLOE-S, YOLOv6-S can +improve AP by 3.5%/0.9% with higher speed. YOLOv6- +M outperforms YOLOv5-M by 4.6% higher AP with a +similar speed, and it achieves 3.1%/1.0% higher AP than +YOLOX-M/PPYOLOE-M at a higher speed. +Besides, it +is more accurate and faster than YOLOv5-L. YOLOv6-L +is 3.1%/1.4% more accurate than YOLOX-L/PPYOLOE- +L under the same latency constraint. Compared with the +YOLOv8 series, our YOLOv6 achieves a similar perfor- +mance in accuracy and in the latency for models at all sizes, +while giving significant better throughput performance. +3 + +Head +Classification Head +Linear +I +Conv +TAuxiliary +Anchor-free +LosSaf +Label Assignment +Linear +Clsab +Regression Head +Anchor-based +Lossab +Linear +Regaf +Label Assignment +Conv +TAuxiliary +Linear +Regab +IMethod +Input Size +APval +APval +50 +FPS +FPS +Latency +Params +FLOPs +(bs=1) +(bs=32) +(bs=1) +YOLOv5-N [5] +640 +28.0% +45.7% +602 +735 +1.7 ms +1.9 M +4.5 G +YOLOv5-S [5] +640 +37.4% +56.8% +376 +444 +2.7 ms +7.2 M +16.5 G +YOLOv5-M [5] +640 +45.4% +64.1% +182 +209 +5.5 ms +21.2 M +49.0 G +YOLOv5-L [5] +640 +49.0% +67.3% +113 +126 +8.8 ms +46.5 M +109.1 G +YOLOv5-N6 [5] +1280 +36.0% +54.4% +172 +175 +5.8 ms +3.2 M +18.4 G +YOLOv5-S6 [5] +1280 +44.8% +63.7% +103 +103 +9.7 ms +12.6 M +67.2 G +YOLOv5-M6 [5] +1280 +51.3% +69.3% +49 +48 +20.1 ms +35.7 M +200.0 G +YOLOv5-L6 [5] +1280 +53.7% +71.3% +32 +30 +31.3 ms +76.8 M +445.6 G +YOLOv5-X6 [5] +1280 +55.0% +72.7% +17 +17 +58.6 ms +140.7 M +839.2 G +YOLOX-Tiny [3] +416 +32.8% +50.3%∗ +717 +1143 +1.4 ms +5.1 M +6.5 G +YOLOX-S [3] +640 +40.5% +59.3%∗ +333 +396 +3.0 ms +9.0 M +26.8 G +YOLOX-M [3] +640 +46.9% +65.6%∗ +155 +179 +6.4 ms +25.3 M +73.8 G +YOLOX-L [3] +640 +49.7% +68.0%∗ +94 +103 +10.6 ms +54.2 M +155.6 G +PPYOLOE-S [17] +640 +43.1% +59.6% +327 +419 +3.1 ms +7.9 M +17.4 G +PPYOLOE-M [17] +640 +49.0% +65.9% +152 +189 +6.6 ms +23.4 M +49.9 G +PPYOLOE-L [17] +640 +51.4% +68.6% +101 +127 +10.1 ms +52.2 M +110.1 G +YOLOv7-Tiny [16] +416 +33.3%∗ +49.9%∗ +787 +1196 +1.3 ms +6.2 M +5.8 G +YOLOv7-Tiny [16] +640 +37.4%∗ +55.2%∗ +424 +519 +2.4 ms +6.2 M +13.7 G∗ +YOLOv7 [16] +640 +51.2% +69.7%∗ +110 +122 +9.0 ms +36.9 M +104.7 G +YOLOv7-E6E [16] +1280 +56.8% +74.4%∗ +16 +17 +59.6 ms +151.7 M +843.2 G +YOLOv8-N [6] +640 +37.3% +52.6%∗ +561 +734 +1.8 ms +3.2 M +8.7 G +YOLOv8-S [6] +640 +44.9% +61.8%∗ +311 +387 +3.2 ms +11.2 M +28.6 G +YOLOv8-M [6] +640 +50.2% +67.2%∗ +143 +176 +7.0 ms +25.9 M +78.9 G +YOLOv8-L [6] +640 +52.9% +69.8%∗ +91 +105 +11.0 ms +43.7 M +165.2 G +YOLOv6-N +640 +37.0% / 37.5%‡ +52.7% / 53.1%‡ +779 +1187 +1.3 ms +4.7 M +11.4 G +YOLOv6-S +640 +44.3% / 45.0%‡ +61.2% / 61.8%‡ +339 +484 +2.9 ms +18.5 M +45.3 G +YOLOv6-M +640 +49.1% / 50.0%‡ +66.1% / 66.9%‡ +175 +226 +5.7 ms +34.9 M +85.8 G +YOLOv6-L +640 +51.8% / 52.8%‡ +69.2% / 70.3%‡ +98 +116 +10.3 ms +59.6 M +150.7 G +YOLOv6-N6 +1280 +44.9% +61.5% +228 +281 +4.4 ms +10.4 M +49.8 G +YOLOv6-S6 +1280 +50.3% +67.7% +98 +108 +10.2 ms +41.4 M +198.0 G +YOLOv6-M6 +1280 +55.2%‡ +72.4%‡ +47 +55 +21.0 ms +79.6 M +379.5 G +YOLOv6-L6 +1280 +57.2%‡ +74.5%‡ +26 +29 +38.5 ms +140.4 M +673.4 G +Table 2: Comparisons with other YOLO-series detectors on COCO 2017 val. FPS and latency are measured in FP16-precision +on a Tesla T4 in the same environment with TensorRT. All our models are trained for 300 epochs without pre-training or any +external data. Both the accuracy and the speed performance of our models are evaluated with the input resolution of 640×640. +‘‡’ represents that the proposed self-distillation method is utilized. ‘∗’ represents the re-evaluated result of the released model +through the official code. +To compare with the state-of-the-art methods, we fol- +low [5] to add an extra stage on the top of the backbone +to have a feature (C6) at a higher level for detecting extra- +large objects. The neck is also expanded accordingly. The +YOLOv6 of all sizes with C6 features are named YOLOv6- +N6/S6/M6/L6 respectively. Further, the image resolution is +adapted from 640 to 1280. The feature strides range from 8 +to 64, which benefits the accurate detection for rather small +and extra-large objects in high-resolution images. The ex- +perimental results listed in Table 2 show that the expanded +YOLOv6 obtain significant gains in accuracy. Compared +with expanded YOLOv5 (i.e., YOLOv5-N6/S6/M6/L6/X6), +ours have a much higher AP at a similar inference speed. +When compared with the state-of-the-art YOLOv7-E6E, +YOLOv6-L6 improves AP by 0.4% and runs 63% faster +with a batch size of 1. +BiC+SimCSPSPPF +AAT +DLD +APval +� +� +� +43.5% +� +� +� +44.1% +� +� +� +44.4% +� +� +� +45.1% +Table 3: Ablation study for all designs on YOLOv6-S. +4 + +Model +BiC +APval +APs +APm +APl +FPS +Bottom-up +Top-down +(bs=32) +YOLOv6-S +� +� +43.1% +23.4% +48.0% +59.9% +513 +� +� +43.7% +25.2% +48.7% +60.4% +492 +� +� +43.7% +25.0% +48.7% +59.7% +485 +YOLOv6-L +� +� +50.9% +32.4% +56.0% +68.0% +125 +� +� +51.3% +34.2% +56.5% +67.6% +120 +� +� +51.1% +33.6% +56.7% +67.9% +119 +Table 4: Effectiveness of the BiC module on YOLOv6. +3.2. Ablation Study +Experimental results in Table 3 exhibit the effectiveness +of all contributions in this work. The renovated network +with BiC and SimCSPSPPF has an enhanced AP by 0.6%. +With AAT and DLD, the accuracy is further improved by +0.3% and 0.7% incrementally. +3.2.1 +Network Design +We conduct a series of experiments to verify the effective- +ness of the proposed BiC module. As can be seen in Table 4, +applying the BiC module only on the top-down pathway of +PAN brings 0.6%/0.4% AP improvements on YOLOv6-S/L +respectively with negligible loss of efficiency. In contrast, +when we try to import the BiC module into the bottom- +up pathway, no positive gain in accuracy is obtained. The +probable reason is that the BiC module on the bottom-up +pathway would lead to confusion for detection heads about +features at different scales. +Therefore, we merely adopt +the BiC module on the top-down pathway. Besides, the +results indicate that the BiC module gives an impressive +boost to the performance of small object detection. For both +YOLOv6-S and YOLOv6-L, the detection performance on +small objects is improved by 1.8%. +Further, we explore the influence of different types of +SPP Blocks, including the simplified variants of SPPF [5] +and SPPCSPC [16] (denoted as SimSPPF and SimSPPC- +SPC respectively) and our SimCSPSPPF blocks. Addition- +ally, we apply SimSPPF blocks on the top three feature +maps (P3, P4 and P5) of our backbone to verify its ef- +fectiveness, which is denoted as SimSPPF*3. Experimen- +tal results are shown in Table 5. We observe that heavily +adopting SimSPPF brings little gain in accuracy with the +increased computational complexity. SimSPPCSPC outper- +forms SimSPPF by 1.6%/0.3% AP on YOLOv6-N/S re- +spectively while significantly decreasing inference speed. +Compared with SimSPPF, our SimCSPSPPF version can +obtain 1.1%/0.4%/0.1% performance gain for YOLOv6- +N/S/M respectively. In terms of inference efficiency, our +SimCSPSPPF block runs nearly 10% faster than SimSP- +PCSPC and is slightly slower than SimSPPF. For a better +accuracy-efficiency trade-off, the SimCSPSPPF blocks are +introduced in YOLOv6-N/S. For YOLOv6-M/L, SimSPPF +blocks are adopted. +Model +SPP Blocks +APval +FPS +(bs=32) +YOLOv6-N +SimSPPF [5] +35.8% +1190 +SimSPPF [5]*3 +35.9% +1072 +SimSPPCSPC [16] +37.4% +1078 +SimCSPSPPF +36.9% +1176 +YOLOv6-S +SimSPPF [5] +43.7% +492 +SimSPPF [5]*3 +43.6% +447 +SimSPPCSPC [16] +44.0% +432 +SimCSPSPPF +44.1% +477 +YOLOv6-M +SimSPPF [5] +48.6% +227 +SimCSPSPPF +48.7% +218 +YOLOv6-L +SimSPPF [5] +51.3% +120 +SimCSPSPPF +51.1% +117 +Table 5: Ablation study on different types of SPP Blocks. +All models are equipped with BiC modules. +3.2.2 +Anchor-Aided Training +The advantages of the AAT is verified in YOLOv6. +As +shown in Table 6, it brings about 0.3%/0.5%/0.5% AP gain +for YOLOv6-S/M/L respectively. +Notably, the accuracy +performance on small objects (AP s) is significantly en- +hanced for YOLOv6-N/S/M. For YOLOv6-L, the perfor- +mance on large objects (AP l) is improved even further. +3.2.3 +Self-distillation +We +verify +the +proposed +self-distillation +method +on +YOLOv6-L. For a fair comparison, we also verified the +model performance by doubling the training epochs besides +the baseline since the self-distillation needs an extra entire +training cycle to obtain the teacher model. As seen in Ta- +ble 7, no performance improvement is attained without the +weight decay strategy compared with the baseline. Dou- +bling the training epochs is even worse due to overfitting. +Method +AAT +APval +APs +APm +APl +YOLOv6-N +� +36.9% +17.2% +41.1% +52.9% +� +36.9% +18.7% +41.2% +53.0% +YOLOv6-S +� +44.1% +24.7% +48.7% +61.1% +� +44.4% +25.4% +49.6% +60.2% +YOLOv6-M +� +48.6% +29.7% +53.7% +65.5% +� +49.1% +31.1% +54.0% +65.4% +YOLOv6-L +� +51.3% +34.2% +56.5% +67.6% +� +51.8% +33.4% +56.8% +68.8% +Table 6: Ablation study about AAT. +5 + +Model +Weight Decay +APval +Baseline +- +51.8% +Double epochs +- +51.7% +Self-distillation +� +51.8% +� +52.4% +Table 7: Ablation on the self-distillation on YOLOv6-L. +DLD +Double epochs +APval +� +� +44.4% +� +� +44.6% +� +� +45.1% +Table 8: Ablation study of DLD on YOLOv6-S. +After the introduction of weight decay, the model is boosted +by 0.6% AP. +In addition, the DLD specifically designed for small +models is ablated on YOLOv6-S. As per self-distillation +for large models, we also compare the results with the +model trained with doubled epochs. 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In Proceedings of +the IEEE/CVF conference on computer vision and pattern +recognition, pages 10781–10790, 2020. 2 +[16] Chien-Yao +Wang, +Alexey +Bochkovskiy, +and +Hong- +Yuan Mark Liao. +Yolov7: Trainable bag-of-freebies sets +new state-of-the-art for real-time object detectors. +arXiv +preprint arXiv:2207.02696, 2022. 1, 2, 3, 4, 5, 7 +[17] Shangliang Xu, Xinxin Wang, Wenyu Lv, Qinyao Chang, +Cheng Cui, Kaipeng Deng, Guanzhong Wang, Qingqing +Dang, Shengyu Wei, Yuning Du, et al. Pp-yoloe: An evolved +version of yolo. arXiv preprint arXiv:2203.16250, 2022. 1, +3, 4, 7 +[18] Shifeng Zhang, Cheng Chi, Yongqiang Yao, Zhen Lei, and +Stan Z. Li. +Bridging the gap between anchor-based and +anchor-free detection via adaptive training sample selection. +In CVPR, 2020. 2 +[19] Zhaohui Zheng, Rongguang Ye, Qibin Hou, Dongwei Ren, +Ping Wang, Wangmeng Zuo, and Ming-Ming Cheng. Lo- +calization distillation for object detection. +arXiv preprint +arXiv:2204.05957, 2022. 3 +6 + +A. Detailed Latency and Throughput Bench- +mark +A.1. Setup +Unless otherwise stated, all the reported latency is mea- +sured on an NVIDIA Tesla T4 GPU with TensorRT ver- +sion 7.2.1.6. Due to the large variance of the hardware and +software settings, we re-measure latency and throughput of +all the models under the same configuration (both hardware +and software). For a handy reference, we also switch Ten- +sorRT versions (Table 9) for consistency check. Latency on +a V100 GPU (Table 10) is included for a convenient com- +parison. This gives us a full spectrum view of state-of-the- +art detectors. +A.2. T4 GPU Latency Table with TensorRT 8 +See Table 9. The throughput of YOLOv6 models still +emulates their peers. +Method +FPS +FPS +Latency +(bs=1) +(bs=32) +(bs=1) +YOLOv5-N [5] +702 +843 +1.4 ms +YOLOv5-S [5] +433 +515 +2.3 ms +YOLOv5-M [5] +202 +235 +4.9 ms +YOLOv5-L [5] +126 +137 +7.9 ms +YOLOX-Tiny [3] +766 +1393 +1.3 ms +YOLOX-S [3] +313 +489 +2.6 ms +YOLOX-M [3] +159 +204 +5.3 ms +YOLOX-L [3] +104 +117 +9.0 ms +PPYOLOE-S [17] +357 +493 +2.8 ms +PPYOLOE-M [17] +163 +210 +6.1 ms +PPYOLOE-L [17] +110 +145 +9.1 ms +YOLOv7-Tiny [16] +464 +568 +2.1 ms +YOLOv7 [16] +128 +135 +7.6 ms +YOLOv6-N +785 +1215 +1.3 ms +YOLOv6-S +345 +498 +2.9 ms +YOLOv6-M +178 +238 +5.6 ms +YOLOv6-L +105 +125 +9.5 ms +Table 9: YOLO-series comparison of latency and through- +put on a T4 GPU with a higher version of TensorRT (8.2). +A.3. V100 GPU Latency Table +See Table 10. +The speed advantage of YOLOv6 is +largely maintained. +A.4. CPU Latency +We evaluate the performance of our models and other +competitors on a 2.6 GHz Intel Core i7 CPU using OpenCV +Deep Neural Network (DNN), as shown in Table 11. +Method +FPS +FPS +Latency +(bs=1) +(bs=32) +(bs=1) +YOLOv5-N [5] +577 +1727 +1.4 ms +YOLOv5-S [5] +449 +1249 +1.7 ms +YOLOv5-M [5] +271 +698 +3.0 ms +YOLOv5-L [5] +178 +440 +4.7 ms +YOLOX-Tiny [3] +569 +2883 +1.4 ms +YOLOX-S [3] +386 +1206 +2.0 ms +YOLOX-M [3] +245 +600 +3.4 ms +YOLOX-L [3] +149 +361 +5.6 ms +PPYOLOE-S [17] +322 +1050 +2.4 ms +PPYOLOE-M [17] +222 +566 +4.0 ms +PPYOLOE-L [17] +153 +406 +5.5 ms +YOLOv7-Tiny [16] +453 +1565 +1.7 ms +YOLOv7 [16] +182 +412 +4.6 ms +YOLOv6-N +646 +2660 +1.2 ms +YOLOv6-S +399 +1330 +2.0 ms +YOLOv6-M +203 +676 +4.4 ms +YOLOv6-L +125 +385 +6.8 ms +Table 10: YOLO-series comparison of latency and through- +put on a V100 GPU. We measure all models at FP16- +precision with the input size 640×640 in the exact same +environment. +Method +Input +Latency +(bs=1) +YOLOv5-N [5] +640 +118.9 ms +YOLOv5-S [5] +640 +202.2 ms +YOLOX-Tiny [3] +416 +144.2 ms +YOLOX-S [3] +640 +164.6 ms +YOLOX-M [3] +640 +357.9 ms +YOLOv7-Tiny [16] +640 +137.5 ms +YOLOv6-N +640 +60.3 ms +YOLOv6-S +640 +148.0 ms +YOLOv6-M +640 +269.3 ms +Table 11: YOLO-series comparison of latency on a typical +CPU. We measure all models at FP32-precision with the +input size 640×640 in the exact same environment. +7 + diff --git a/SNE5T4oBgHgl3EQfaQ8G/content/tmp_files/load_file.txt b/SNE5T4oBgHgl3EQfaQ8G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..271845026e700c04bd92111b8f760d127f2ad338 --- /dev/null +++ b/SNE5T4oBgHgl3EQfaQ8G/content/tmp_files/load_file.txt @@ -0,0 +1,656 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf,len=655 +page_content='YOLOv6 v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0: A Full-Scale Reloading Chuyi Li∗ Lulu Li∗ Yifei Geng∗ Hongliang Jiang∗ MengCheng Bo Zhang Zaidan Ke Xiaoming Xu† Xiangxiang Chu Meituan Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' {lichuyi, lilulu05, gengyifei, jianghongliang02, chengmeng05, zhangbo97, kezaidan, xuxiaoming04, chuxiangxiang}@meituan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='com 0 200 400 600 800 1000 1200 Tesla T4 TensorRT FP16 Throughput (FPS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' BS=32 25 30 35 40 45 50 55 60 COCO AP (%) YOLOv6-N YOLOv6-S YOLOv6-M YOLOv6-L YOLOv6-M6 YOLOv6-L6 YOLOv6 YOLOv5 YOLOv7 YOLOv8 YOLOX PP-YOLOE 0 10 20 30 40 50 60 Tesla T4 TensorRT FP16 Latency (ms),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' BS=1 25 30 35 40 45 50 55 60 COCO AP (%) YOLOv6-N YOLOv6-S YOLOv6-M YOLOv6-L YOLOv6-M6 YOLOv6-L6 YOLOv6 YOLOv5 YOLOv7 YOLOv8 YOLOX PP-YOLOE Figure 1: Comparison of state-of-the-art efficient object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Both latency and throughput (at a batch size of 32) are given for a handy reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' All models are test with TensorRT 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Abstract The YOLO community has been in high spirits since our first two releases!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' By the advent of Chinese New Year 2023, which sees the Year of the Rabbit, we refurnish YOLOv6 with numerous novel enhancements on the network archi- tecture and the training scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' This release is identified as YOLOv6 v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' For a glimpse of performance, our YOLOv6- N hits 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5% AP on the COCO dataset at a throughput of 1187 FPS tested with an NVIDIA Tesla T4 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' YOLOv6- S strikes 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% AP at 484 FPS, outperforming other main- stream detectors at the same scale (YOLOv5-S, YOLOv8- S, YOLOX-S and PPYOLOE-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Whereas, YOLOv6-M/L also achieve better accuracy performance (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0%/52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% respectively) than other detectors at a similar inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Additionally, with an extended backbone and neck design, our YOLOv6-L6 achieves the state-of-the-art ac- curacy in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Extensive experiments are carefully conducted to validate the effectiveness of each improving component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Our code is made available at https:// Equal contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' † Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='com/meituan/YOLOv6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Introduction The YOLO series has been the most popular detection frameworks in industrial applications, for its excellent bal- ance between speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Pioneering works of YOLO series are YOLOv1-3 [12–14], which blaze a new trail of one-stage detectors along with the later substan- tial improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' YOLOv4 [1] reorganized the detection framework into several separate parts (backbone, neck and head), and verified bag-of-freebies and bag-of-specials at the time to design a framework suitable for training on a single GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' At present, YOLOv5 [5], YOLOX [3], PPY- OLOE [17], YOLOv7 [16] and most recently YOLOv8 [6] are all the competing candidates for efficient detectors to deploy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In this release, we strenuously renovate the network de- sign and the training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' We show the comparison of YOLOv6 with other peers at a similar scale in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The new features of YOLOv6 are summarized as follows: We renew the neck of the detector with a Bi-directional 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='05586v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='CV] 13 Jan 2023 Figure 2: (a) The neck of YOLOv6 (N and S are shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Note for M/L, RepBlocks is replaced with CSPStackRep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' (b) The structure of a BiC module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' (c) A SimCSPSPPF block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Concatenation (BiC) module to provide more accurate localization signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' SPPF [5] is simplified to form the SimCSPSPPF Block, which brings performance gains with negligible speed degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' We propose an anchor-aided training (AAT) strat- egy to enjoy the advantages of both anchor-based and anchor-free paradigms without touching inference ef- ficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' We deepen YOLOv6 to have another stage in the back- bone and the neck, which reinforces it to hit a new state-of-the-art performance on the COCO dataset at a high-resolution input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' We involve a new self-distillation strategy to boost the performance of small models of YOLOv6, in which the heavier branch for DFL [8] is taken as an enhanced auxiliary regression branch during training and is re- moved at inference to avoid the marked speed decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Network Design In practice, feature integration at multiple scales has been proven to be a critical and effective component of object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Feature Pyramid Network (FPN) [9] is proposed to aggregate the high-level semantic features and low-level features via a top-down pathway, which provides more accurate localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Subsequently, there have been several works [2, 4, 10, 15] on Bi-directional FPN in or- der to enhance the ability of hierarchical feature represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' PANet [10] adds an extra bottom-up pathway on top of FPN to shorten the information path of low-level and top-level features, which facilitates the propagation of accurate signals from low-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' BiFPN [15] in- troduces learnable weights for different input features and simplifies PAN to achieve better performance with high ef- ficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' PRB-FPN [2] is proposed to retain high-quality features for accurate localization by a parallel FP structure with bi-directional fusion and associated improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Motivated by the above works, we design an enhanced- PAN as our detection neck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In order to augment localiza- tion signals without bringing in excessive computation bur- den, we propose a Bi-directional Concatenation(BiC) mod- ule to integrate feature maps of three adjacent layers, which fuses an extra low-level feature from backbone Ci−1 into Pi (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In this case, more accurate localization signals can be preserved, which is significant for the localization of small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Moreover, we simplify the SPPF block [5] to have a CSP-like version called SimCSPSPPF Block, which strengthens the representational ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Particularly, we re- vise the SimSPPCSPC Block in [16] by shrinking the chan- nels of hidden layers and retouching space pyramid pool- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In addition, we upgrade the CSPBlock with RepBlock (for small models) or CSPStackRep Block (for large mod- els) and accordingly adjust the width and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The neck of YOLOv6 is denoted as RepBi-PAN, the framework of which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Anchor-Aided Training YOLOv6 is an anchor-free detector to pursue a higher inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' However, we experimentally find that the anchor-based paradigm brings additional performance gains on YOLOv6-N under the same settings when compared with the anchor-free one, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Moreover, anchor-based ATSS [18] is adopted as the warm-up label as- signment strategy in the early versions of YOLOv6, which stabilizes the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Paradigm APval APs APm APl Anchor-free 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% Anchor-based 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% Table 1: Comparisons of the anchor-free and the anchor- based paradigms on YOLOv6-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In light of this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' we propose anchor-aided training (AAT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='in which the anchor-based auxiliary branches are intro- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='Pi+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='RepBi-PAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=': Concatenation over channel dimension ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='--- ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1x1 Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='(a) RepBi-PAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='(b) BiC Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='(c) SimCSPSPPF Blockduced to combine the advantages of anchor-based and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='anchor-free paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' And they are applied both in the classification and the regression head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 3 shows the de- tection head with the auxiliaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Figure 3: The detection head with anchor-based auxiliary branches during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The auxiliary branches are re- moved at inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' ‘af’ and ‘ab’ are short for ‘anchor-free’ and ‘anchor-based’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' During the training stage, the auxiliary branches and the anchor-free branches learn from independent losses while signals are propagated altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Therefore, additional em- bedded guidance information from auxiliary branches is in- tegrated into the main anchor-free heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Worth mentioning that the auxiliary branches is removed at inference, which boosts the accuracy performance without decreasing speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Self-distillation In early versions of YOLOv6, the self-distillation is only introduced in large models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=', YOLOv6-M/L), which ap- plies the vanilla knowledge distillation technique by mini- mizing the KL-divergence between the class prediction of the teacher and the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Meanwhile DFL [8] is adopted as regression loss to perform self-distillation on box regres- sion similar to [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The knowledge distillation loss is formulated as: LKD = KL(pcls t ||pcls s ) + KL(preg t ||preg s ), (1) where pcls t and pcls s are class prediction of the teacher model and the student model respectively, and accordingly preg t and preg s are box regression predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The overall loss function is now formulated as: Ltotal = Ldet + αLKD, (2) where Ldet is the detection loss computed with predictions and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The hyperparameter α is introduced to balance two losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In the early stage of training, the soft labels from the teacher are easier to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' As the training continues, the performance of the student will match the teacher so that the hard labels will help students more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Upon this, we apply cosine weight decay to α to dynamically adjust the information from hard labels and soft ones from the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The formulation of α is: α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='99 ∗ ((1 − cos(π ∗ Ei/Emax))/2) + 1, (3) where Ei denotes the current training epoch and Emax rep- resents the maximum training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Notably, the introduction of DFL [8] requires extra pa- rameters for the regression branch, which affects the in- ference speed of small models significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Therefore, we specifically design the Decoupled Localization Distillation (DLD) for our small models to boost performance with- out speed degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Specifically, we append a heavy auxiliary enhanced regression branch to incorporate DFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' During the self-distillation, the student is equipped with a na¨ıve regression branch and the enhanced regression branch while the teacher only uses the auxiliary branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Note that the na¨ıve regression branch is only trained with hard labels while the auxiliary is updated according to signals from both the teacher and hard labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' After the distillation, the na¨ıve regression branch is retained whilst the auxiliary branch is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' With this strategy, the advantages of the heavy regression branch for DFL in distillation is consider- ably maintained without impacting the inference efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Experiments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Comparisons The evaluation is made consistent with the early ver- sions of YOLOv6 [7], which focuses on the throughput and the GPU latency at deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' We test the speed performance of all official models with FP16-precision on the same Tesla T4 GPU with TensorRT [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' We compare the upgraded YOLOv6 with YOLOv5 [5], YOLOX [3], PPYOLOE [17], YOLOv7 [16] and YOLOv8 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Note that the performance of YOLOv7-Tiny is re-evaluated ac- cording to their open-sourced code and weights at the in- put size of 416 and 640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Results are shown in Table 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Compared with YOLOv5-N/YOLOv7-Tiny (in- put size=416), our YOLOv6-N has significantly advanced by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5%/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' It also comes with the best speed performance in terms of both throughput and latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Compared with YOLOX-S/PPYOLOE-S, YOLOv6-S can improve AP by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5%/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9% with higher speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' YOLOv6- M outperforms YOLOv5-M by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% higher AP with a similar speed, and it achieves 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% higher AP than YOLOX-M/PPYOLOE-M at a higher speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Besides, it is more accurate and faster than YOLOv5-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' YOLOv6-L is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% more accurate than YOLOX-L/PPYOLOE- L under the same latency constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Compared with the YOLOv8 series, our YOLOv6 achieves a similar perfor- mance in accuracy and in the latency for models at all sizes, while giving significant better throughput performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 3 Head Classification Head Linear I Conv TAuxiliary Anchor-free LosSaf Label Assignment Linear Clsab Regression Head Anchor-based Lossab Linear Regaf Label Assignment Conv TAuxiliary Linear Regab IMethod Input Size APval APval 50 FPS FPS Latency Params FLOPs (bs=1) (bs=32) (bs=1) YOLOv5-N [5] 640 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 602 735 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7 ms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9 M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5 G YOLOv5-S [5] 640 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% 376 444 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7 ms 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2 M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5 G YOLOv5-M [5] 640 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% 182 209 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5 ms 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2 M 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0 G YOLOv5-L [5] 640 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3% 113 126 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8 ms 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5 M 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1 G YOLOv5-N6 [5] 1280 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 172 175 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8 ms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2 M 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4 G YOLOv5-S6 [5] 1280 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 103 103 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7 ms 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3% 32 30 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3 ms 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8 M 445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6 G YOLOv5-X6 [5] 1280 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 17 17 58.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5 G YOLOv6-L6 1280 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2%‡ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5%‡ 26 29 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5 ms 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4 M 673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4 G Table 2: Comparisons with other YOLO-series detectors on COCO 2017 val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' FPS and latency are measured in FP16-precision on a Tesla T4 in the same environment with TensorRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' All our models are trained for 300 epochs without pre-training or any external data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Both the accuracy and the speed performance of our models are evaluated with the input resolution of 640×640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' ‘‡’ represents that the proposed self-distillation method is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' ‘∗’ represents the re-evaluated result of the released model through the official code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' To compare with the state-of-the-art methods, we fol- low [5] to add an extra stage on the top of the backbone to have a feature (C6) at a higher level for detecting extra- large objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The neck is also expanded accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The YOLOv6 of all sizes with C6 features are named YOLOv6- N6/S6/M6/L6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Further, the image resolution is adapted from 640 to 1280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The feature strides range from 8 to 64, which benefits the accurate detection for rather small and extra-large objects in high-resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The ex- perimental results listed in Table 2 show that the expanded YOLOv6 obtain significant gains in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Compared with expanded YOLOv5 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=', YOLOv5-N6/S6/M6/L6/X6), ours have a much higher AP at a similar inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' When compared with the state-of-the-art YOLOv7-E6E, YOLOv6-L6 improves AP by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% and runs 63% faster with a batch size of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' BiC+SimCSPSPPF AAT DLD APval � � � 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5% � � � 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% � � � 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% � � � 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% Table 3: Ablation study for all designs on YOLOv6-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 4 Model BiC APval APs APm APl FPS Bottom-up Top-down (bs=32) YOLOv6-S � � 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9% 513 � � 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 492 � � 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 485 YOLOv6-L � � 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 125 � � 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% 120 � � 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9% 119 Table 4: Effectiveness of the BiC module on YOLOv6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Ablation Study Experimental results in Table 3 exhibit the effectiveness of all contributions in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The renovated network with BiC and SimCSPSPPF has an enhanced AP by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' With AAT and DLD, the accuracy is further improved by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% incrementally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1 Network Design We conduct a series of experiments to verify the effective- ness of the proposed BiC module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' As can be seen in Table 4, applying the BiC module only on the top-down pathway of PAN brings 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6%/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% AP improvements on YOLOv6-S/L respectively with negligible loss of efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In contrast, when we try to import the BiC module into the bottom- up pathway, no positive gain in accuracy is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The probable reason is that the BiC module on the bottom-up pathway would lead to confusion for detection heads about features at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Therefore, we merely adopt the BiC module on the top-down pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Besides, the results indicate that the BiC module gives an impressive boost to the performance of small object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' For both YOLOv6-S and YOLOv6-L, the detection performance on small objects is improved by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Further, we explore the influence of different types of SPP Blocks, including the simplified variants of SPPF [5] and SPPCSPC [16] (denoted as SimSPPF and SimSPPC- SPC respectively) and our SimCSPSPPF blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Addition- ally, we apply SimSPPF blocks on the top three feature maps (P3, P4 and P5) of our backbone to verify its ef- fectiveness, which is denoted as SimSPPF*3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Experimen- tal results are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' We observe that heavily adopting SimSPPF brings little gain in accuracy with the increased computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' SimSPPCSPC outper- forms SimSPPF by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6%/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3% AP on YOLOv6-N/S re- spectively while significantly decreasing inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Compared with SimSPPF, our SimCSPSPPF version can obtain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1%/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4%/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% performance gain for YOLOv6- N/S/M respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In terms of inference efficiency, our SimCSPSPPF block runs nearly 10% faster than SimSP- PCSPC and is slightly slower than SimSPPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' For a better accuracy-efficiency trade-off, the SimCSPSPPF blocks are introduced in YOLOv6-N/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' For YOLOv6-M/L, SimSPPF blocks are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Model SPP Blocks APval FPS (bs=32) YOLOv6-N SimSPPF [5] 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% 1190 SimSPPF [5]*3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9% 1072 SimSPPCSPC [16] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 1078 SimCSPSPPF 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9% 1176 YOLOv6-S SimSPPF [5] 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 492 SimSPPF [5]*3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% 447 SimSPPCSPC [16] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 432 SimCSPSPPF 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% 477 YOLOv6-M SimSPPF [5] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% 227 SimCSPSPPF 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 218 YOLOv6-L SimSPPF [5] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3% 120 SimCSPSPPF 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% 117 Table 5: Ablation study on different types of SPP Blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' All models are equipped with BiC modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2 Anchor-Aided Training The advantages of the AAT is verified in YOLOv6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' As shown in Table 6, it brings about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3%/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5%/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5% AP gain for YOLOv6-S/M/L respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Notably, the accuracy performance on small objects (AP s) is significantly en- hanced for YOLOv6-N/S/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' For YOLOv6-L, the perfor- mance on large objects (AP l) is improved even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3 Self-distillation We verify the proposed self-distillation method on YOLOv6-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' For a fair comparison, we also verified the model performance by doubling the training epochs besides the baseline since the self-distillation needs an extra entire training cycle to obtain the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' As seen in Ta- ble 7, no performance improvement is attained without the weight decay strategy compared with the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Dou- bling the training epochs is even worse due to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Method AAT APval APs APm APl YOLOv6-N � 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9% � 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% YOLOv6-S � 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% � 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2% YOLOv6-M � 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5% � 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% YOLOv6-L � 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% � 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% Table 6: Ablation study about AAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 5 Model Weight Decay APval Baseline 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% Double epochs 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% Self-distillation � 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8% � 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% Table 7: Ablation on the self-distillation on YOLOv6-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' DLD Double epochs APval � � 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4% � � 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% � � 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1% Table 8: Ablation study of DLD on YOLOv6-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' After the introduction of weight decay, the model is boosted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6% AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In addition, the DLD specifically designed for small models is ablated on YOLOv6-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' As per self-distillation for large models, we also compare the results with the model trained with doubled epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' As shown in Table 8, YOLOv6-S with DLD gives 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7% AP boost and performs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5% better than that of training with doubled epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Conclusion In this report, YOLOv6 is upgraded on aspects of the net- work design and training strategy, which boost YOLOv6 to achieve the state-of-the-art accuracy for real-time object de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In the future, we persistently work on the optimiza- tion of YOLOv6 to render an application-friendly detection framework as our research continues and the techniques in object detection advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' References [1] Alexey Bochkovskiy, Chien-Yao Wang, and Hong- Yuan Mark Liao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Yolov4: 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Bridging the gap between anchor-based and anchor-free detection via adaptive training sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 2 [19] Zhaohui Zheng, Rongguang Ye, Qibin Hou, Dongwei Ren, Ping Wang, Wangmeng Zuo, and Ming-Ming Cheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Lo- calization distillation for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='05957, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' 3 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Detailed Latency and Throughput Bench- mark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Setup Unless otherwise stated, all the reported latency is mea- sured on an NVIDIA Tesla T4 GPU with TensorRT ver- sion 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Due to the large variance of the hardware and software settings, we re-measure latency and throughput of all the models under the same configuration (both hardware and software).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' For a handy reference, we also switch Ten- sorRT versions (Table 9) for consistency check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Latency on a V100 GPU (Table 10) is included for a convenient com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' This gives us a full spectrum view of state-of-the- art detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' T4 GPU Latency Table with TensorRT 8 See Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The throughput of YOLOv6 models still emulates their peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Method FPS FPS Latency (bs=1) (bs=32) (bs=1) YOLOv5-N [5] 702 843 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4 ms YOLOv5-S [5] 433 515 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3 ms YOLOv5-M [5] 202 235 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9 ms YOLOv5-L [5] 126 137 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9 ms YOLOX-Tiny [3] 766 1393 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3 ms YOLOX-S [3] 313 489 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6 ms YOLOX-M [3] 159 204 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3 ms YOLOX-L [3] 104 117 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0 ms PPYOLOE-S [17] 357 493 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8 ms PPYOLOE-M [17] 163 210 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1 ms PPYOLOE-L [17] 110 145 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1 ms YOLOv7-Tiny [16] 464 568 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='1 ms YOLOv7 [16] 128 135 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6 ms YOLOv6-N 785 1215 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3 ms YOLOv6-S 345 498 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9 ms YOLOv6-M 178 238 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6 ms YOLOv6-L 105 125 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5 ms Table 9: YOLO-series comparison of latency and through- put on a T4 GPU with a higher version of TensorRT (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' V100 GPU Latency Table See Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' The speed advantage of YOLOv6 is largely maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' CPU Latency We evaluate the performance of our models and other competitors on a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6 GHz Intel Core i7 CPU using OpenCV Deep Neural Network (DNN), as shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Method FPS FPS Latency (bs=1) (bs=32) (bs=1) YOLOv5-N [5] 577 1727 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4 ms YOLOv5-S [5] 449 1249 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7 ms YOLOv5-M [5] 271 698 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0 ms YOLOv5-L [5] 178 440 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7 ms YOLOX-Tiny [3] 569 2883 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4 ms YOLOX-S [3] 386 1206 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0 ms YOLOX-M [3] 245 600 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4 ms YOLOX-L [3] 149 361 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6 ms PPYOLOE-S [17] 322 1050 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4 ms PPYOLOE-M [17] 222 566 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0 ms PPYOLOE-L [17] 153 406 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5 ms YOLOv7-Tiny [16] 453 1565 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='7 ms YOLOv7 [16] 182 412 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6 ms YOLOv6-N 646 2660 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2 ms YOLOv6-S 399 1330 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0 ms YOLOv6-M 203 676 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='4 ms YOLOv6-L 125 385 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='8 ms Table 10: YOLO-series comparison of latency and through- put on a V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' We measure all models at FP16- precision with the input size 640×640 in the exact same environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' Method Input Latency (bs=1) YOLOv5-N [5] 640 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9 ms YOLOv5-S [5] 640 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2 ms YOLOX-Tiny [3] 416 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='2 ms YOLOX-S [3] 640 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='6 ms YOLOX-M [3] 640 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='9 ms YOLOv7-Tiny [16] 640 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='5 ms YOLOv6-N 640 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3 ms YOLOv6-S 640 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='0 ms YOLOv6-M 640 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content='3 ms Table 11: YOLO-series comparison of latency on a typical CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} +page_content=' We measure all models at FP32-precision with the input size 640×640 in the exact same environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE5T4oBgHgl3EQfaQ8G/content/2301.05586v1.pdf'} 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file mode 100644 index 0000000000000000000000000000000000000000..9107c1e19838c74f7a5f10b1cafaf509e837ba83 --- /dev/null +++ b/WNE1T4oBgHgl3EQfJANe/content/tmp_files/2301.02946v1.pdf.txt @@ -0,0 +1,881 @@ +Presented at the Workshop on Visual Analytics in Healthcare (VAHC), jointly held with the Annual Symposium of the +American Medical Informatics Association (AMIA), Washington, DC, November, 2022 + + +Patterns of Social Vulnerability – An Interactive Dashboard to Explore +Risks to Public Health on the US County Level + +Darius Coelho* +Akai Kaeru LLC + +Nikita Gupta† +Stony Brook University + +Eric Papenhausen‡ +Akai Kaeru LLC + +Klaus Mueller§ +Akai Kaeru LLC +Stony Brook University + + + +Figure 1: The COVID-19 Risk Explorer dashboard for exploring risk patterns that drive COVID-19 death rates in the US. On the the +top left is an interactive geomap of the US counties shaded by their COVID-19 death rate. A county can be selected by clicking +the mouse on its map location. Here, the user selected Yuma, AZ, marked by an orange outline. The bottom panel has the risk +pattern browser which represents all the risk patterns as tiles, shaded by their COVID-19 death rate. Only the patterns relevant to +the selected county (Yuma, AZ) are shaded. Users can select a risk pattern by clicking the mouse on a pattern tile. Here the user +selected the first pattern, marked by an orange outline. All counties that share this pattern are then shaded on the map. The top +right panel shows information about the selected county while the panel below it offers information about the selected pattern. + +ABSTRACT +Social vulnerability is the susceptibility of a community to be +adversely impacted by natural hazards and public health emer- +gencies, such as drought, earthquakes, flooding, virus outbreaks, +and the like. Climate change is at the root of many recent natural +hazards while the COVID-19 pandemic is still an active threat. +Social vulnerability also refers to resilience, or the ability to +recover from such adverse events. To gauge the many aspects +of social vulnerability the US Center of Disease Control (CDC) +has subdivided social vulnerabilities into distinct themes, such as +socioeconomic status, household composition, and others. Knowing +a community’s social vulnerabilities can help policymakers and +responders to recognize risks to community health, prepare for +possible hazards, or recover from disasters. In this paper we study +social vulnerabilities on the US county level and present research +that suggests that there are certain combinations, or patterns, of +social vulnerability indicators into which US counties can be +grouped. We then present an interactive dashboard that allows +analysts to explore these patterns in various ways. We demonstrate +our methodology using COVID-19 death rate as the hazard and +show that the patterns we identified have high predictive capabilities +of the pandemic’s local impact. + +* e-mail: dcoelho@akaikaeru.com +† e-mail: nikigupta@cs.stonybrook.edu +‡ e-mail: epapenha@akaikaeru.com +§ e-mail: mueller@cs.stonybrook.edu + + +1 INTRODUCTION +Social vulnerability gauges the socioeconomic and demographic fac- +tors that affect the resilience of a community to external disastrous +events affecting human health, called stresses. These stresses can +range from natural or human-caused disasters to disease outbreaks. +A socially vulnerable community is less likely to recover and more +likely to perish as a result of these stresses, and by reducing social +vulnerability, one can lower both human suffering and economic +losses. Knowing the specific social vulnerabilities for a given com- +munity can help emergency response planners and public health +officials to quickly respond when a specific disaster strikes and to +build long-term residence for it to weaken its potential impact. +The concept of social vulnerability has been studied world-wide +and various measures have been established. In the US, the Agency +for Toxic Substances and Disease Registry (ATSDR) and the Center +of Disease Control (CDC) have used US Census data to determine a +Social Vulnerability Index (SVI) for every US census tract. Census +tracts are subdivisions of US counties and there are 3,006 counties in +the US. The CDC/ATSDR SVI ranks each tract on 15 social factors +which can be grouped into into four related themes: + +• Socioeconomic status: below poverty, unemployed, income, +no high school diploma +• Household composition & disability: aged 65 or older, aged +17 or younger, older than age 5 with a disability, single-parent +households +• Minority & language: minority, speak English “less than well” +• Housing type & transportation: multi-unit structures, mobile +homes, crowding, no vehicle, group quarters + +COVID-19RISKEXPLORER0 +U.S.MAP +COUNTY INFORMATION +Yuma,Arizona +Cont Res +Ma) +Oet +Top.3.RiskFeatutes. +I U.S.Range +e.State Range U.S:Average +1Count) +% poputation uninsured +20.0 +IVg.GPA +%gopuation.wohighscho +PATTERNINFORMATION +This pattern appears in 301 counties which have an avg. death rate of 119.4 per 100K +llU.S.RangePattemRange +PMz5perticle matter polution +%adufiswithfrequentphys. +122 +sminontypepulation +17.8 +ResetSeletlend +RISK PATTERN BROWSER +店Using a dataset from Kaggle [11] we have expanded this list to +a carefully curated set of 241 factors that refines these measures to +demographic features, such as race and gender. and adds further +health and social risks, such as smoking and drug use habits, teenage +pregnancies, sleep deprivedness, housing debt, vaccination rate, and +many others. The Kaggle dataset is composed of data collected from +200 publicly available COVID-19 related datasets, using sources like +Johns Hopkins, the WHO, the World Bank, the New York Times, +and many others. We found several redundancies in this dataset and +carefully trimmed it to the set of 241 factors which are the basis of +the methodology and study reported in this paper. +Visualizing the social vulnerabilities as a choropleth map can be +helpful to planners, responders, and policymakers to compare the +social vulnerability index across locations and so better understand +which communities are most susceptible to natural disasters and +disease outbreaks. Most prominent is the CDC’s own SVI Inter- +active Map [10] which colors counties by their overall SVI score - +the score is a value between 0 (lowest vulnerability) to 1 (highest +vulnerability); clicking the mouse on a specific county pops up a +scorecard that lists the score for each theme as a bar chart. The US +Federal Emergency Management Agency (FEMA) [6] provides a se- +ries of choropleth maps that aside from maps for social vulnerability +and community resilience also allows users to produce maps for the +risks and expected annualized loss for a variety of natural hazards, +such as flooding, lightning, tornadoes, and many others. A service +called County Health Rankings [3] allows users to produce detailed +factor-based comparative reports with visualizations for US states at +the granularity of counties. There are also several other institutions, +such as NASA, that produce factor-based SVI choropleth maps. +To the best of our knowledge there is no work thus far that allows +users to explore the multivariate nature of social vulnerabilities more +directly, in the form of patterns of multiple interacting factors. The +currently available maps all require users to switch from map to +map or perform side by side comparison to assess these correlations. +Our pattern analysis groups counties, which are not necessarily geo- +graphically related, in terms of vulnerability factors that cooperate +in making these counties more (or less) susceptible to a certain +target hazard. Enabling analysts to explore these patterns and the +locations of the affected counties on a single choropleth map allows +for a deeper and more efficient analysis. This paper describes the +outcomes of our research geared toward achieving such a map. +Our paper is organized as follows. Section 2 presents related +work. Section 3 describes our pattern analysis. Section 4 explains +our dashboard that allows users to explore these patterns in the +context of the US counties. Section 5 offers a conclusion. The +appendix presents a case study. + +2 RELATED WORK +Unlike previous large-scale global health crises the COVID-19 pan- +demic arrived in an era with an ubiquity of machine learning tools +and a widely developed information technology infrastructure. The +urgency of COVID-19 did not only boost efforts in biotech to de- +velop vaccines and medical treatments at rapid speeds, but it also +invigorated research in the predictive modeling of the spread of the +virus and in the development of visual information portals to inform +policy makers and the general public on death tolls, infections, and +the like. As such the COVID-19 health emergency brought about +much of the most recent related work on visual tools for public +health monitoring and risk assessment. + +2.1 Modeling and Prediction +Many predictive modeling approaches (such as [13], [24]) are based +on the mechanistic Susceptible–Exposed–Infectious–Recovered +(SEIR) compartment simulation model that, at the process level, +mimics the way COVID-19 spreads. Mechanistic models are at- +tractive since they allow one to simulate the effects of different +mitigation measures, such as quarantining, social distancing, school +closings, and so on. However, the model’s many parameters require +accurate estimates of the population in each compartment and their +transition rates which can be challenging due to the uncertainties +involved. +Popular are also statistical forecasting models such as that by U +Washington’s Institute for Health Metric and Education (IHME) [1]. +It uses a mixed effects nonlinear regression model to fit a curve to +data from world-wide geographical locations to create projections of +infections, death rates, and health resource demands at the local level. +Other approaches use more conventional statistical models, such +as correlation and linear regression to understand the influence of +certain socio-economic factors, such as county-level health variables, +urban density, poverty, commuting, and so on, while controlling for +other effects, such as race [18]. Typically these results are obtained +via standard step-wise modeling approaches that are not overly +scalable in the number of factors and local regions, making the +discovery of significant statistical relationships rather tedious. +Recognizing that models often produce a wide range of predictive +forecasts, ensemble methods have recently become popular (see for +e.g. [17]). Ensemble methods combine different individual models +together and weigh their outcomes into a unified forecast. This can +make predictions more robust and add stability to the process. + +2.2 Visualization +A primary source of information has been the Coronavirus Resource +Center at Johns Hopkins University [4]. They constructed dash- +boards for the US and for the entire world that each showed the +respective geographic maps overlaid with visual representations of +the numbers of people tested positive alongside various test and +death statics, leader boards, and temporal growth curve ensembles +that compare regions at various scales in terms of the increase of test +cases and deaths. Other dashboards and browser-based interactive +visualization of COVID-19 related data have been made available +by the companies Tableau [7], TIBCO [2], the open source project +Nextstrain [9], newspapers like the New York Times, and others. +These dashboards and visualizations illuminate specific aspects re- +lated to the outbreak, such as race, hospital overcrowding, test statis- +tics by state, mask compliance by county, unemployment rates and +claims, economic inequality, pathogen evolution, and more. +The IHME COVID-19 forcasting model has also become quite +popular due to its simple yet effective interactive visual dashboard +tools [5]. However, IHME is also known for its interactive ‘US +Health Map — Viz Hub’ [8]. This visual interface provides a menu +that allows users to select among four outcome or risk variables, +i.e. life expectancy, mortality rate, mortality risk, and others. Users +can then choose one of many diseases and health determinants and +display the outcome or risk on a choropleth map for states or counties. +However, similar to other maps, their map also can only display one +quantity at a time and so enables comparisons only on a factor- +wise basis. While juxtaposition [14] of several maps can facilitate +comparisons, it remains difficult to recognize general correlations +or groupings among the variables. Our geo-display enables it by +pairing pattern analysis with a set of information displays. +A recent visual interface is EnsembleVis [21] which is a web- +based geomap interface to view and compare model forecasts with +the ground truth on the US county level. It allows users to navigate +the ensemble models and so gain a better understanding of the ranges +and uncertainties. We also display our data on the county level but +our focus is mainly to explain why certain health risks occur, that is, +what are the social vulnerabilities that expose certain communities to +greater risk. While our method learns these risks from communities +that have already been exposed, there are others that fit this patterns +as well but have not suffered the same fate as yet. As such the +patterns we identify can also be used to predict or at least alert these +communities and associated responders. + + + +Figure 2: Scatterplot of a 3D pattern found in May 2020. The y-axis (target attribute) is the observed COVID-19 death rate and each point is a US +county. Each image shows the 2D projection into the plot’s pattern attribute and the target attribute after culling the points into the subspace +defined by the pattern attribute of the plot on its immediate left. The yellow points are inside the plot’s pattern subspace and the purple points are +outside of it. It can be seen that adding the third attribute is sufficient to eliminate all purple points. The culling of points is the reason why there +are progressively fewer points in the plots from left to right. The green bars on top show how much each of the subspace attributes (top to bottom, +and left to right in the plots: county poverty rate, % population aged over 65, population density) contributes the definition of the pattern. + +3 OUR APPROACH: PATTERN MINING AND DASHBOARD +In the following we summarize our pattern mining approach and then +focus on the dashboard we designed for exploring these patterns. + +3.1 Our Pattern Mining Approach +Our approach is rooted in pattern analysis, a well-studied area of +research in data science and AI [19]. A pattern is a subgroup of +data points that share similar characteristics, or features [12]. For +example, the data could be a set of counties that have a similar socio- +economic make-up. In our example each county has 241 features, +e.g., % adults w/excessive drinking habits, % adults in frequent +mental distress, unemployment rate, etc. These 241 features then +result in a 241-D feature space which is typically fairly sparse. +We have devised a pattern mining engine that automatically +searches this sparse feature space for regions occupied with similar +counties which all respond in a similar way to a given target variable +of interest. Some results of our pattern mining approach are reported +in [20]. In that work we specifically focused on COVID-19 and the +visualizations consisted of a simple choropleth map which did not +offer any capabilities to explore the patterns in terms of their features. +We now report on a dedicated dashboard that puts the human in the +loop and allows for detailed interrogations. +In our prior work we sought to identify the socio-economic con- +ditions that underlie higher than average COVID-19 death rates, and +so our target variable was a county’s COVID-19 deaths rate. We +note that counties that are considered similar do not need to be geo- +graphically connected; they just need to have similar characteristics +in terms of their feature values. A unique property of a pattern is +that it fits inside a hypercube with well-defined value ranges of the +features that describe it. This property and its inherent low dimen- +sionality [22], even when the overall feature space is not, makes +them easy to understand and explain. While deep neural networks, +random forests, etc. also learn low-D representations, these are not +easily described in terms of the native features. Hence, these types +of architectures are commonly referred to as black-box AI while +pattern analysis is an explainable AI approach. +Concretely, given a dataset with attributes {A1 A2 ....Am P} with +P being an attribute of interest, such as COVID-19 death rate, the +goal of pattern mining is to find a hypercube (or pattern) consisting +of constraints of the form Ai ∈ [vl , vr ] for i ∈ [1...m] (for example, +age > 45, race = Asian), where the points within the pattern are +“interesting.” For our purposes, a pattern of counties will be consid- +ered interesting if it is associated with a COVID-19 death rate that +is higher on average than the US county average. The definition of +what constitutes a consistently interesting pattern is based primarily + + + + +Figure 3: The locations of the counties in the pattern of Fig. 2. colored +by COVID-19 death rate. The newly disease-stricken counties in +June 2020 (inside the dotted ellipses but also elsewhere) are counties +located in the bottom of May 2020’s scatterplot in Fig. 2 or not yet +visualized there, but correctly predicted to get hit soon. + +on statistical hypothesis testing. For numerical attributes, we use +the Mann-Whitney test [23] to account for the often non-parametric +nature of the data, while for a binary target attribute, we use the +χ 2 test for independence. Extracting the patterns requires extensive +search; we use the FP-growth algorithm [16] which is fairly efficient +as it only requires scanning the full dataset twice during the mining. + +3.2 Our Pattern Mining: A Closer Look +For our prior COVID-19 study we used the 241-D dataset mentioned +in the introduction and used COVID-19 death rate in each US county +as the target variable. We found 297 2D and 3D patterns in May +2020. Fig. 2 visualizes one of them, a 3D pattern defined by high +poverty rate, high percentage of senior citizens, and low population +density. These three variables were sufficient to confirm the sta- +tistical significance for the elevated death rate average. The figure +caption explains the formation of the pattern in greater detail. +Let us have a closer look at the plot on the very right of Fig. 2 +which shows the projection of US counties into the pattern’s 3rd +attribute (yellow points). We notice that most US counties in the +pattern have a death rate above the US average (and the pattern’s +average itself is also above the US average which makes the pattern +”interesting”). But we also notice that there are a few US counties +that are below the US average bar; they have a death rate below the +US average. This can mean that there are other latent (unmeasured) +factors that protected the counties from contracting the virus. But it +can also mean that these counties were not yet hit by the COVID-19 +wave – recall that May 2020 was very early in the pandemic. +Fig. 3 gives more insight into this. It shows two maps where the +counties with significant death rates are shaded in blue – deeper blue +shades map to higher death rates. Note that we only colored counties +that matched the pattern shown in Fig. 2 (they may also match other +patterns but we did not consider these for this plot). On the left is the + +previouslyunaffectedcountiesnow +exhibitingsignificantdeathrates +May +June +10 +10Pattern Detail +two US counties with a +Pattern Detail +Pattern Detai +0.306 +very high death rate +0.306 +0.306 +eg to ex口gpeg tor spex口 +0.64 +0.64 ++ +It death rate (# of COVID-19 +C +death rate for the +D X-axis Log Scal +deaths per#of county +. +grey box's set of +Jt death rate (# of COVID-19 +two US counties with a +xatn Log.tcale +Yata Log Scan +residents,prob.log scale) +UScounties +deathsper#ofcounty +very high death rate +itdeathrate (#ofCOVID-19 +75 +residents,prob.logscale) +45- +deaths per#of county +4.0. +residents, prob. log scale) +40 +45- +45- +twoUS countieswitha +9.5- +1.0 +very high death rate +4.0- +10.0 +45- +10.5- +death rate averaged +greybox:thesetofuS countiesfor +110 +overall UScounties +whichtheCOVID-19deathrate was +. +45- +significantly higher than average +greybox: the set ofUS counties for +11.0. +-20- +grey box: the setofUS counties for +which the COVID-19death rate was +115 +-125- +significantly higher than average +-120- +whichtheCOVID-19deathrate was +13.0d, +significanty higher than average +10 +15 +20 +35 +4 +25 +county poverty rate +% population aged over 65 in counties with high poverty rate ++ +population density of aging counties with high poverty ratemap for May 2020, the month we used to learn our patterns. On the +right we see the corresponding map for the next month, June 2020. +We observe that there are quite a few counties now colored that were +not affected yet in May; see the areas encircled by ellipses, but there +are also new counties appearing in already affected regions. +All these newly affected counties are counties below the average +bar in Fig. 2 and so the pattern was able to predict their destiny. +In fact we observed that for 98% of all our patterns the death rate +growth was 2-3 times higher than the US average; the other 2% grew +at the average pace, none slowed in growth below the US-average. +These trends continued in July. This shows that our patterns are +highly predictive, and at the same time can also explain the socio- +economic conditions for higher-than-average COVID-19 death rate +in an easy to understand manner, in the language of the features. + +3.3 Our Interactive Visual Dashboard +Our dashboard is designed for people with varying levels of visu- +alization literacy to help them navigate and examine the patterns +mined with our approach. The dashboard consists of four main +panels - geomap, pattern browser, pattern information, and county +information - that are linked to each other. Each of these panels are +explained in the sections below and also in the caption of Fig. 1. +The data input is a standard CSV file with the data matrix. + +Risk Pattern Browser: As discussed in Section 3.1 patterns are +low-D hypercubes, however a collection of patterns can still span a +large number of dimensions. This makes it difficult if not impossible +to devise an easy to understand visual representation to explain a +pattern in its entirety. Thus we choose to represent the collection +of patterns as a list of tiles with each tile representing a pattern as +shown in the bottom panel in Fig. 1. The patterns are ordered from +left to right and top to bottom in descending order of the death rate. +This is re-enforced by coloring the tiles from dark to light blue based +on the COVID-19 death rate. Only the tiles that pertain to a county +selected in the geomap are shaded (more on this below). +Each of these tiles can be clicked which will then trigger updates +to the geomap view to indicate counties to which this pattern belongs +and to the pattern information panel to communicate the pattern +details (discussed below). Additionally, we change the shape of a +selected tile to a circle and give it a yellow border to make it easy for +users to locate the selected tile while their focus switches between +different elements of the dashboard. + + + +Figure 4: The representation of the ranges of features that define a +pattern where the gray bar represents the range of the feature across +the US and the blue bar indicates the range for that pattern. For +example the third feature ‘% minority population’ has a range of 0 to +99.2 across the US but the range of 37.6 to 99.2 is one of the features +of this pattern that drives a higher than average death rate. + +Pattern Information: This panel communicates the pattern infor- +mation to the user. A pattern is essentially a set of attributes with +specific ranges. Thus the pattern information panel reports these +ranges to the user while placing them in the context of the global +range of the dimension across all data points, in this case all counties. +To visualize these ranges, we make use of a bullet chart style visu- +alization that has been shown to be easy-to-understand by a wide +audience [15]. An example is shown in Fig. 4. Here each dimen- +sion’s range is represented as a horizontal bar. The gray portion +of the bar indicates the range across all counties in the US of the +dimension while the blue portion of the bar represents the range +of the dimension that defines this pattern. User can quickly scroll +through these bars and study the various ranges that define patterns. + +Geomap View: This panel consists of an interactive county-level +map of the United States (shown in Fig. 1). Each county in the map +is assigned a color based on its COVID-19 death rate. We use a +continuous color scale ranging from dark blue for high death rates +to white for a death rate of zero. Users can click on a county to learn +more about the factors leading to its COVID-19 death rate. Clicking +on a county will trigger an update to the risk pattern browser which +highlights the patterns that the county belongs to and grays out the +rest. Additionally, the county information panel is updated with the +top risk features for that county. We also allow the user to zoom and +pan the map in order to select smaller counties. + +County Information: This panel communicates the information of +a county selected by the user via the geomap. As shown in the top +right corner of Fig. 1 the panel reports the current COVID-19 county +death rate as well as the death rate over time. More importantly the +panel communicates the top 3 risk factors for the county. Here the +feature ranking is computed based on the frequency at which those +features appear across all patterns that contain the selected county. +We use the same bullet chart-like visualization used for the pattern +information to visually represent these features. Here again the gray +bars indicate the range of the dimension across all counties in the +US while the blue bars are ranges of the features across all counties +in the selected county’s state. In addition to the ranges shown in the +chart we also add markers for the value of the factor in the county +and the value for the US average. An example is shown in Fig. 5. + + + +Figure 5: A visual representation of the top 3 feature values of a +county in the context of state and US ranges. The gray bar represents +the range of the feature across the US and the blue indicates the +range of that feature across all counties in the county’s state. For +example, the county shown here has an ‘avg. GPA’ of 2.9 (solid black +marker) which is slightly lower than the US average (dotted black +marker). Additionally the US range for the ‘avg. GPA’ is 0 to 4 while +the range of this feature across all counties in the state is 2.4 to 3.7. + +4 CONCLUSIONS +We have outlined a methodology that can group socio-economic +indicators of public health into 1-3 factor patterns learnt from ob- +servational data. The patterns can be used by policy makers and +health officials to explain and predict the underlying risk a certain +community has with respect to some natural hazard or public health +emergency. To give easy access to these patterns we devised an +interactive visual dashboard by which the patterns can be explored +in the context of the communities’ geographical locations. While +we have used the early stages of the COVID-19 pandemic to show +an application of our methodology, we believe that its application is +far broader, which is being explored in ongoing work. +While we provide temporal context to our data – the COVID-19 +death rate over time - users currently cannot ”roll back” time to +examine the patterns at the selected time frame. This is a fairly +easy implementation and we plan to add this feature in the future. In +addition, while we have used a small cohort of users to gain feedback +during system development, we plan a broader task-based study in +the near future to gain further insight into utility and usability, + +U.s. Range +Pattern Range +8.6 +12.8 +PM25 particle matter pollution : +2. +8.9 +15.8 +12.2 +24.1 +% adults with frequent phys... : +15.6 +24.1 +37.6 +99.2 +% minority population :U.S. Range +State Range +U.S.Average +County +2.9 +avg. GPA : +0.0 +2.0 +2.4 +3.7 4.0 +24.5 +% population w/o high schoo... +0.0 +8.9 +30.7 +50.0 +100.0 +71.3 +% minority population : +00.2ACKNOWLEDGMENTS + +This research was funded by NSF SBIR contract 1926949. + + +REFERENCES +[1] 2020. university of washington institute for health metric and education +(ihme) covid-19 resources. retrieved from. http://www.healthdata. +org/covid. +[2] Coronavirus visual analysis hub. 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Mining frequent patterns without candidate +generation. ACM SIGMOD Record, 29(2):1–12, 2000. +[17] J.-S. Kim, H. Kavak, A. Zu¨ fle, and T. Anderson. Covid-19 ensemble +models using representative clustering. SIGSPATIAL Special, 12(2):33– +41, 2020. +[18] C. R. Knittel and B. Ozaltun. What does and does not correlate with +covid-19 death rates. Technical report, National Bureau of Economic +Research, 2020. +[19] H.-P. Kriegel, P. Kro¨ ger, and A. Zimek. Clustering high-dimensional +data: A survey on subspace clustering, pattern-based clustering, and +correlation clustering. ACM Transactions on Knowledge Discovery +from Data (TKKD), 3(1):1–58, 2009. +[20] K. Mueller and E. Papenhausen. Using demographic pattern analysis to +predict covid-19 fatalities on the us county level. Digital Government: +Research and Practice, 2(1):1–11, 2020. +[21] S. Srabanti, G. E. Marai, and F. Miranda. Covid-19 ensemblevis: +Visual analysis of county-level ensemble forecast models. In 2021 +IEEE Workshop on Visual Analytics in Healthcare (VAHC), pp. 1–5, +2021. +[22] B. Wang and K. Mueller. Does 3d really make sense for visual cluster +analysis? yes! In 2014 IEEE VIS International Workshop on 3DVis +(3DVis), pp. 37–44. IEEE, 2014. +[23] I. B. Weiner and W. E. Craighead. The corsini encyclopedia of psychol- +ogy, volume 4, vol. 4. John Wiley & Sons, 2010. +[24] Z. Yang, Z. Zeng, K. Wang, S.-S. Wong, W. Liang, M. Zanin, P. Liu, +X. Cao, Z. Gao, Z. Mai, et al. Modified seir and ai prediction of the +epidemics trend of covid-19 in china under public health interventions. +Journal of Thoracic Disease, 12(3):165, 2020. + +1 + +APPENDIX + +In this case study we follow Bob, a public health analyst, who uses the COVID-19 Risk Explorer to learn more about the +susceptibility of local communities to the spread of the COVID-19 virus. It’s December 2020 and a lot has happened. He starts +up the program and sees the screen below. + + + + +Bob observes that the areas that have seen the highest death rates overall are in Texas, Arizona, Montana, North Dakota, Idaho, +the South, Florida, and the Northern East Coast. Now Bob wonders about the timelines. He selects a darkly colored county (a +country with high death rates) in Connecticut – Hartford County. + + + + +When the screen transitions, Bob observes from the line chart showing the death rate over time in the “County Information” +panel that this county exceeded the US average death rate very early in 2020 and quite rapidly so, but then remained nearly flat +starting April. Apparently this county responded well and took good precautions to stem the spread. + +Bob looks at the “Pattern Information” panel to examine the risk factors of the most dominant pattern Hartford, CT participates +in. The “Risk Pattern Browser” indicates that this is the 3rd most dominant pattern in the database. It is a 2-feature pattern that +indicates that in Hartford, CT the percentage of non-Hispanics/Whites is at the low end of the US overall range and the ratio of +median household debt/income is on the high end of the US overall range. Looking at the “Top 3 Risk Factors” of Hartford in the +“County Information Panel” Bob learns that these two risk factors are actually the top 3 for Hartford in addition to PM25 particle + +COVID-19RISKEXPLORER0 +AkaiKaeru +U.S.MAP +COUNTYINFORMATION +Select a county.on.the.ms, +PATTERN INFORMATION +RISKPATTERNBROWSERCOVID-19RISKEXPLORER0 +Akai Kaeru +U.S.MAP +COUNTYINFORMATION +Hartford,Connecticut +161 DEATHS +PER100K +Top3RiskFeatures +U.S.Range +median household debt +%.populationNon-Hisp/Whites +81.4 +PM25D8 +PATTERNINFORMATION +ThispattemappearsIn315countieswhichhaveanavgdeathratf115per100K +U.s.RangePatemnRange +%population Non-Hisp/Whites +medianhouseholddebt/incom +Reset Seiections +RISKPATTERNBROWSER2 + +matter pollution and all its values compare unfavorably to the US average, and while they are not at the extreme ends for the +State of Connecticut they tend to be in the unfavorable value ranges. + +Next Bob clicks on an equally dark colored county in Southern Texas, Cameron County, TX and the screen transitions to what +is shown below. + + + + +Bob observes that for that county the death rate started to climb much later, in July 2020, and that it is still climbing now in +December, albeit at a shallower slope. While Cameron TX shares the 3rd risk pattern with Hartford CT, it has many more risk +patterns than Hartford CT, as can be seen by the many filled squares in the “Risk Pattern Browser”. It means that its conditions +for high death rates are more urgent than for Hartford CT. In fact, its top 3 risk factors are not those of the 3rd risk pattern. For all +of these Cameron TX fares unfavorably both within the value range found in Texas and with respect to the US average. + +Next, Bob clicks on the first risk pattern in the “Risk Pattern Browser” and sees the screen below. + + + + +This pattern is a 3-feature pattern with high and unfavorable value ranges in all three of these features. Only the feature “% +minority population” appears in the top 3 list for Cameron County, TX and upon further investigation Bob finds that the other +two risk features, “% population without high school degree” and “% uninsured”, are in risk pattern #4 (see picture next page). + +COVID-19RISK EXPLORER0 +Akai Kaer +U.S.MAP +COUNTYINFORMATION +Cameron,Texas +CountyRat +Top3RiskFeatures. +U.s.Range +%population +71:3 +90.2 +PATTERNINFORMATION +This patter +315counties which bave an avg.death rate of 115.1per 100K + U.S.Range Pattem Range +61.0 +12.0 +Reset Selecfons +RISKPATTERNBROWSERCOVID-19RISKEXPLORER0 +AkaiKaeru +U.S.MAP +7 +COUNTY INFORMATION +CeuntyRal +Cameron,Texas +Top3RiskFeatures +u.S. Range: +%populationuninsu +713 +90.2 +PATTERNINFORMATION +This pattern appears in 301 counties which have an avg.death rate of 119.4 per 100K +U.S.Range.PatternRange +PM25 particle matter pollution +%adultswithfrequentphys. +122 +37.0 +Reset Selections +RISK PATTERN BROWSER +二3 + + + +Bob now wants to investigate whether Cameron County, TX could have learned its fate from other counties with similar risk +factors but which had experienced high COVID-19 death rates earlier. He looks for counties that share some or all of its top 3 +risk features. So he examines counties with risk pattern #1 and risk pattern #4. + +He starts with risk pattern #1, clicks a few a counties on the map and eventually learns about Passaic County, NJ. + + + + +From the timeline he sees that Passaic County, NJ started its death rate at the earliest time and has a similar “% minority +population”. But this is only one out of the three top 3 risk factors of Cameron County, TX so Bob needs to search more to +complete the case. He turns to risk pattern #4 and after some search finds McKinley, NM (see next page). + +COVID-19RISKEXPLORER0 +AkaiKaeru +U.S.MAP +COUNTYINFORMATION +Cameron,Texas +May +Top3RiskFeatures +U.S.Range.StateRangeU.S.Average +Count +% popuiation w/o high schoo. +24.5 +14.7 +30 +%popuiationuninsured +20.0 +% minonty population +71.3 +PATTERNINFORMATION +Thisattepparin268countieswhichhandeathrate114er100 +%population vaccinated Blacks +3-0 +43.4 +tat +% population wio nigh schoo.. +Reset Selectlions +RISKPATTERNBROWSERCOVID-19RISKEXPLORERe +AkaiKaeru +U.S.MAP +COUNTY INFORMATION +-CountyRate +Passaic,New Jersey +Top3RiskFeatures +US.Rang +% adunts with frequent phys +10.2 +% minority populatio +PATTERNINFORMATION +This patternappearsin301counties whichhaveanavgdeathrateof119.4per100K +U.S.RangePatternRange +PM25particlen +12.8 +122 +4 +ResetSelecti +> +RISKPATTERN BROWSER4 + + + +McKinley, NM started its death rate climb later than Passaic, NJ but still 4 months earlier than Cameron, TX. Its top 3 risk +factors contain the two other risk factors of Cameron County, TX. So had Cameron County, TX looked at the fate of McKinley +County, NM and Passaic County, NJ it could have learned from them and adopt the precautionary measures they took. + +There are many more case studies like this one, where late risers could have learnt from early risers about their fate and prepared +better. As seen from the early risers’ timelines, all of them were able to get the spread under control. Late risers could have taken +similar measures and potentially save lives. + +As a conclusion, our results show that pattern analysis is a powerful tool for public health risk management and that a dashboard +like our Risk Explorer makes it easy to recognize risks and see how outbreaks and disaster in some communities can quickly +inform other communities that fit a similar vulnerability pattern to prevent further loss. + +COVID-19RISKEXPLORER0 +Akai Kaeru +U.S.MAP +COUNTYINFORMATION +McKiniey,NewMexico +Desthspe:100) +Top3RiskFeatures +U.S:Range +State RangeU.S.Average +14.7 +%population +20.0 +uoeindod.% +24.5 +81.4 +0 +PATTERNINFORMATION +%populationvaccinatedBlacks +38:0 +%populationuninsurec +13.4 +18-1 +%populationw/ohighschoo. +Reset Selections +RISKPATTERN BROWSER \ No newline at end of file diff --git a/WNE1T4oBgHgl3EQfJANe/content/tmp_files/load_file.txt b/WNE1T4oBgHgl3EQfJANe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df926c525c4c2dfcb95fc02ed07a6c2f9a3a8de7 --- /dev/null +++ b/WNE1T4oBgHgl3EQfJANe/content/tmp_files/load_file.txt @@ -0,0 +1,569 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf,len=568 +page_content='Presented at the Workshop on Visual Analytics in Healthcare (VAHC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' jointly held with the Annual Symposium of the American Medical Informatics Association (AMIA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' November,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2022 Patterns of Social Vulnerability – An Interactive Dashboard to Explore Risks to Public Health on the US County Level Darius Coelho* Akai Kaeru LLC Nikita Gupta† Stony Brook University Eric Papenhausen‡ Akai Kaeru LLC Klaus Mueller§ Akai Kaeru LLC Stony Brook University Figure 1: The COVID-19 Risk Explorer dashboard for exploring risk patterns that drive COVID-19 death rates in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' On the the top left is an interactive geomap of the US counties shaded by their COVID-19 death rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' A county can be selected by clicking the mouse on its map location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Here, the user selected Yuma, AZ, marked by an orange outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The bottom panel has the risk pattern browser which represents all the risk patterns as tiles, shaded by their COVID-19 death rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Only the patterns relevant to the selected county (Yuma, AZ) are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Users can select a risk pattern by clicking the mouse on a pattern tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Here the user selected the first pattern, marked by an orange outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' All counties that share this pattern are then shaded on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The top right panel shows information about the selected county while the panel below it offers information about the selected pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' ABSTRACT Social vulnerability is the susceptibility of a community to be adversely impacted by natural hazards and public health emer- gencies, such as drought, earthquakes, flooding, virus outbreaks, and the like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Climate change is at the root of many recent natural hazards while the COVID-19 pandemic is still an active threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Social vulnerability also refers to resilience, or the ability to recover from such adverse events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' To gauge the many aspects of social vulnerability the US Center of Disease Control (CDC) has subdivided social vulnerabilities into distinct themes, such as socioeconomic status, household composition, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Knowing a community’s social vulnerabilities can help policymakers and responders to recognize risks to community health, prepare for possible hazards, or recover from disasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' In this paper we study social vulnerabilities on the US county level and present research that suggests that there are certain combinations, or patterns, of social vulnerability indicators into which US counties can be grouped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We then present an interactive dashboard that allows analysts to explore these patterns in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We demonstrate our methodology using COVID-19 death rate as the hazard and show that the patterns we identified have high predictive capabilities of the pandemic’s local impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' e-mail: dcoelho@akaikaeru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='com † e-mail: nikigupta@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='edu ‡ e-mail: epapenha@akaikaeru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='com § e-mail: mueller@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='edu 1 INTRODUCTION Social vulnerability gauges the socioeconomic and demographic fac- tors that affect the resilience of a community to external disastrous events affecting human health, called stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' These stresses can range from natural or human-caused disasters to disease outbreaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' A socially vulnerable community is less likely to recover and more likely to perish as a result of these stresses, and by reducing social vulnerability, one can lower both human suffering and economic losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Knowing the specific social vulnerabilities for a given com- munity can help emergency response planners and public health officials to quickly respond when a specific disaster strikes and to build long-term residence for it to weaken its potential impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The concept of social vulnerability has been studied world-wide and various measures have been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' In the US, the Agency for Toxic Substances and Disease Registry (ATSDR) and the Center of Disease Control (CDC) have used US Census data to determine a Social Vulnerability Index (SVI) for every US census tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Census tracts are subdivisions of US counties and there are 3,006 counties in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The CDC/ATSDR SVI ranks each tract on 15 social factors which can be grouped into into four related themes: Socioeconomic status: below poverty,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' unemployed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' income,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' no high school diploma Household composition & disability: aged 65 or older,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' aged 17 or younger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' older than age 5 with a disability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' single-parent households Minority & language: minority,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' speak English “less than well” Housing type & transportation: multi-unit structures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' mobile homes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' crowding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' no vehicle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' group quarters COVID-19RISKEXPLORER0 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='MAP COUNTY INFORMATION Yuma,Arizona Cont Res Ma) Oet Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='RiskFeatutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' I U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Range e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='State Range U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S:Average 1Count) % poputation uninsured 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 IVg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='GPA %gopuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='wohighscho PATTERNINFORMATION This pattern appears in 301 counties which have an avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' death rate of 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='4 per 100K llU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='RangePattemRange PMz5perticle matter polution %adufiswithfrequentphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 122 sminontypepulation 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='8 ResetSeletlend RISK PATTERN BROWSER 店Using a dataset from Kaggle [11] we have expanded this list to a carefully curated set of 241 factors that refines these measures to demographic features, such as race and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' and adds further health and social risks, such as smoking and drug use habits, teenage pregnancies, sleep deprivedness, housing debt, vaccination rate, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The Kaggle dataset is composed of data collected from 200 publicly available COVID-19 related datasets, using sources like Johns Hopkins, the WHO, the World Bank, the New York Times, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We found several redundancies in this dataset and carefully trimmed it to the set of 241 factors which are the basis of the methodology and study reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Visualizing the social vulnerabilities as a choropleth map can be helpful to planners, responders, and policymakers to compare the social vulnerability index across locations and so better understand which communities are most susceptible to natural disasters and disease outbreaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Most prominent is the CDC’s own SVI Inter- active Map [10] which colors counties by their overall SVI score - the score is a value between 0 (lowest vulnerability) to 1 (highest vulnerability);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' clicking the mouse on a specific county pops up a scorecard that lists the score for each theme as a bar chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The US Federal Emergency Management Agency (FEMA) [6] provides a se- ries of choropleth maps that aside from maps for social vulnerability and community resilience also allows users to produce maps for the risks and expected annualized loss for a variety of natural hazards, such as flooding, lightning, tornadoes, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' A service called County Health Rankings [3] allows users to produce detailed factor-based comparative reports with visualizations for US states at the granularity of counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' There are also several other institutions, such as NASA, that produce factor-based SVI choropleth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' To the best of our knowledge there is no work thus far that allows users to explore the multivariate nature of social vulnerabilities more directly, in the form of patterns of multiple interacting factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The currently available maps all require users to switch from map to map or perform side by side comparison to assess these correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Our pattern analysis groups counties, which are not necessarily geo- graphically related, in terms of vulnerability factors that cooperate in making these counties more (or less) susceptible to a certain target hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Enabling analysts to explore these patterns and the locations of the affected counties on a single choropleth map allows for a deeper and more efficient analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This paper describes the outcomes of our research geared toward achieving such a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Section 2 presents related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Section 3 describes our pattern analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Section 4 explains our dashboard that allows users to explore these patterns in the context of the US counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Section 5 offers a conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The appendix presents a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2 RELATED WORK Unlike previous large-scale global health crises the COVID-19 pan- demic arrived in an era with an ubiquity of machine learning tools and a widely developed information technology infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The urgency of COVID-19 did not only boost efforts in biotech to de- velop vaccines and medical treatments at rapid speeds, but it also invigorated research in the predictive modeling of the spread of the virus and in the development of visual information portals to inform policy makers and the general public on death tolls, infections, and the like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' As such the COVID-19 health emergency brought about much of the most recent related work on visual tools for public health monitoring and risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='1 Modeling and Prediction Many predictive modeling approaches (such as [13], [24]) are based on the mechanistic Susceptible–Exposed–Infectious–Recovered (SEIR) compartment simulation model that, at the process level, mimics the way COVID-19 spreads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Mechanistic models are at- tractive since they allow one to simulate the effects of different mitigation measures, such as quarantining, social distancing, school closings, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' However, the model’s many parameters require accurate estimates of the population in each compartment and their transition rates which can be challenging due to the uncertainties involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Popular are also statistical forecasting models such as that by U Washington’s Institute for Health Metric and Education (IHME) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' It uses a mixed effects nonlinear regression model to fit a curve to data from world-wide geographical locations to create projections of infections, death rates, and health resource demands at the local level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Other approaches use more conventional statistical models, such as correlation and linear regression to understand the influence of certain socio-economic factors, such as county-level health variables, urban density, poverty, commuting, and so on, while controlling for other effects, such as race [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Typically these results are obtained via standard step-wise modeling approaches that are not overly scalable in the number of factors and local regions, making the discovery of significant statistical relationships rather tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Recognizing that models often produce a wide range of predictive forecasts, ensemble methods have recently become popular (see for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Ensemble methods combine different individual models together and weigh their outcomes into a unified forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This can make predictions more robust and add stability to the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2 Visualization A primary source of information has been the Coronavirus Resource Center at Johns Hopkins University [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' They constructed dash- boards for the US and for the entire world that each showed the respective geographic maps overlaid with visual representations of the numbers of people tested positive alongside various test and death statics, leader boards, and temporal growth curve ensembles that compare regions at various scales in terms of the increase of test cases and deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Other dashboards and browser-based interactive visualization of COVID-19 related data have been made available by the companies Tableau [7], TIBCO [2], the open source project Nextstrain [9], newspapers like the New York Times, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' These dashboards and visualizations illuminate specific aspects re- lated to the outbreak, such as race, hospital overcrowding, test statis- tics by state, mask compliance by county, unemployment rates and claims, economic inequality, pathogen evolution, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The IHME COVID-19 forcasting model has also become quite popular due to its simple yet effective interactive visual dashboard tools [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' However, IHME is also known for its interactive ‘US Health Map — Viz Hub’ [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This visual interface provides a menu that allows users to select among four outcome or risk variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' life expectancy, mortality rate, mortality risk, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Users can then choose one of many diseases and health determinants and display the outcome or risk on a choropleth map for states or counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' However, similar to other maps, their map also can only display one quantity at a time and so enables comparisons only on a factor- wise basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' While juxtaposition [14] of several maps can facilitate comparisons, it remains difficult to recognize general correlations or groupings among the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Our geo-display enables it by pairing pattern analysis with a set of information displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' A recent visual interface is EnsembleVis [21] which is a web- based geomap interface to view and compare model forecasts with the ground truth on the US county level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' It allows users to navigate the ensemble models and so gain a better understanding of the ranges and uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We also display our data on the county level but our focus is mainly to explain why certain health risks occur, that is, what are the social vulnerabilities that expose certain communities to greater risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' While our method learns these risks from communities that have already been exposed, there are others that fit this patterns as well but have not suffered the same fate as yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' As such the patterns we identify can also be used to predict or at least alert these communities and associated responders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Figure 2: Scatterplot of a 3D pattern found in May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The y-axis (target attribute) is the observed COVID-19 death rate and each point is a US county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Each image shows the 2D projection into the plot’s pattern attribute and the target attribute after culling the points into the subspace defined by the pattern attribute of the plot on its immediate left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The yellow points are inside the plot’s pattern subspace and the purple points are outside of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' It can be seen that adding the third attribute is sufficient to eliminate all purple points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The culling of points is the reason why there are progressively fewer points in the plots from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The green bars on top show how much each of the subspace attributes (top to bottom, and left to right in the plots: county poverty rate, % population aged over 65, population density) contributes the definition of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 3 OUR APPROACH: PATTERN MINING AND DASHBOARD In the following we summarize our pattern mining approach and then focus on the dashboard we designed for exploring these patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='1 Our Pattern Mining Approach Our approach is rooted in pattern analysis, a well-studied area of research in data science and AI [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' A pattern is a subgroup of data points that share similar characteristics, or features [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' For example, the data could be a set of counties that have a similar socio- economic make-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' In our example each county has 241 features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=', % adults w/excessive drinking habits, % adults in frequent mental distress, unemployment rate, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' These 241 features then result in a 241-D feature space which is typically fairly sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We have devised a pattern mining engine that automatically searches this sparse feature space for regions occupied with similar counties which all respond in a similar way to a given target variable of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Some results of our pattern mining approach are reported in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' In that work we specifically focused on COVID-19 and the visualizations consisted of a simple choropleth map which did not offer any capabilities to explore the patterns in terms of their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We now report on a dedicated dashboard that puts the human in the loop and allows for detailed interrogations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' In our prior work we sought to identify the socio-economic con- ditions that underlie higher than average COVID-19 death rates, and so our target variable was a county’s COVID-19 deaths rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We note that counties that are considered similar do not need to be geo- graphically connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' they just need to have similar characteristics in terms of their feature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' A unique property of a pattern is that it fits inside a hypercube with well-defined value ranges of the features that describe it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This property and its inherent low dimen- sionality [22], even when the overall feature space is not, makes them easy to understand and explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' While deep neural networks, random forests, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' also learn low-D representations, these are not easily described in terms of the native features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Hence, these types of architectures are commonly referred to as black-box AI while pattern analysis is an explainable AI approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Concretely, given a dataset with attributes {A1 A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='.Am P} with P being an attribute of interest, such as COVID-19 death rate, the goal of pattern mining is to find a hypercube (or pattern) consisting of constraints of the form Ai ∈ [vl , vr ] for i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='m] (for example, age > 45, race = Asian), where the points within the pattern are “interesting.” For our purposes, a pattern of counties will be consid- ered interesting if it is associated with a COVID-19 death rate that is higher on average than the US county average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The definition of what constitutes a consistently interesting pattern is based primarily Figure 3: The locations of the counties in the pattern of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' colored by COVID-19 death rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The newly disease-stricken counties in June 2020 (inside the dotted ellipses but also elsewhere) are counties located in the bottom of May 2020’s scatterplot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2 or not yet visualized there, but correctly predicted to get hit soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' on statistical hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' For numerical attributes, we use the Mann-Whitney test [23] to account for the often non-parametric nature of the data, while for a binary target attribute, we use the χ 2 test for independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Extracting the patterns requires extensive search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' we use the FP-growth algorithm [16] which is fairly efficient as it only requires scanning the full dataset twice during the mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2 Our Pattern Mining: A Closer Look For our prior COVID-19 study we used the 241-D dataset mentioned in the introduction and used COVID-19 death rate in each US county as the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We found 297 2D and 3D patterns in May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2 visualizes one of them, a 3D pattern defined by high poverty rate, high percentage of senior citizens, and low population density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' These three variables were sufficient to confirm the sta- tistical significance for the elevated death rate average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The figure caption explains the formation of the pattern in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Let us have a closer look at the plot on the very right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2 which shows the projection of US counties into the pattern’s 3rd attribute (yellow points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We notice that most US counties in the pattern have a death rate above the US average (and the pattern’s average itself is also above the US average which makes the pattern ”interesting”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' But we also notice that there are a few US counties that are below the US average bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' they have a death rate below the US average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This can mean that there are other latent (unmeasured) factors that protected the counties from contracting the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' But it can also mean that these counties were not yet hit by the COVID-19 wave – recall that May 2020 was very early in the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 3 gives more insight into this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' It shows two maps where the counties with significant death rates are shaded in blue – deeper blue shades map to higher death rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Note that we only colored counties that matched the pattern shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2 (they may also match other patterns but we did not consider these for this plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' On the left is the previouslyunaffectedcountiesnow exhibitingsignificantdeathrates May June 10 10Pattern Detail two US counties with a Pattern Detail Pattern Detai 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='306 very high death rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='306 eg to ex口gpeg tor spex口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='64 + It death rate (# of COVID-19 C death rate for the D X-axis Log Scal deaths per#of county .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=" grey box's set of Jt death rate (# of COVID-19 two US counties with a xatn Log." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='tcale Yata Log Scan residents,prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='log scale) UScounties deathsper#ofcounty very high death rate itdeathrate (#ofCOVID-19 75 residents,prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='logscale) 45- deaths per#of county 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' residents, prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' log scale) 40 45- 45- twoUS countieswitha 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='5- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 very high death rate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0- 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 45- 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='5- death rate averaged greybox:thesetofuS countiesfor 110 overall UScounties whichtheCOVID-19deathrate was .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 45- significantly higher than average greybox: the set ofUS counties for 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 20- grey box: the setofUS counties for which the COVID-19death rate was 115 125- significantly higher than average 120- whichtheCOVID-19deathrate was 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0d, significanty higher than average 10 15 20 35 4 25 county poverty rate % population aged over 65 in counties with high poverty rate + population density of aging counties with high poverty ratemap for May 2020, the month we used to learn our patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' On the right we see the corresponding map for the next month, June 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We observe that there are quite a few counties now colored that were not affected yet in May;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' see the areas encircled by ellipses, but there are also new counties appearing in already affected regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' All these newly affected counties are counties below the average bar in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 2 and so the pattern was able to predict their destiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' In fact we observed that for 98% of all our patterns the death rate growth was 2-3 times higher than the US average;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' the other 2% grew at the average pace, none slowed in growth below the US-average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' These trends continued in July.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This shows that our patterns are highly predictive, and at the same time can also explain the socio- economic conditions for higher-than-average COVID-19 death rate in an easy to understand manner, in the language of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='3 Our Interactive Visual Dashboard Our dashboard is designed for people with varying levels of visu- alization literacy to help them navigate and examine the patterns mined with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The dashboard consists of four main panels - geomap, pattern browser, pattern information, and county information - that are linked to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Each of these panels are explained in the sections below and also in the caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The data input is a standard CSV file with the data matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Risk Pattern Browser: As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='1 patterns are low-D hypercubes, however a collection of patterns can still span a large number of dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This makes it difficult if not impossible to devise an easy to understand visual representation to explain a pattern in its entirety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Thus we choose to represent the collection of patterns as a list of tiles with each tile representing a pattern as shown in the bottom panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The patterns are ordered from left to right and top to bottom in descending order of the death rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This is re-enforced by coloring the tiles from dark to light blue based on the COVID-19 death rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Only the tiles that pertain to a county selected in the geomap are shaded (more on this below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Each of these tiles can be clicked which will then trigger updates to the geomap view to indicate counties to which this pattern belongs and to the pattern information panel to communicate the pattern details (discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Additionally, we change the shape of a selected tile to a circle and give it a yellow border to make it easy for users to locate the selected tile while their focus switches between different elements of the dashboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Figure 4: The representation of the ranges of features that define a pattern where the gray bar represents the range of the feature across the US and the blue bar indicates the range for that pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' For example the third feature ‘% minority population’ has a range of 0 to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2 across the US but the range of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='6 to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2 is one of the features of this pattern that drives a higher than average death rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Pattern Information: This panel communicates the pattern infor- mation to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' A pattern is essentially a set of attributes with specific ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Thus the pattern information panel reports these ranges to the user while placing them in the context of the global range of the dimension across all data points, in this case all counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' To visualize these ranges, we make use of a bullet chart style visu- alization that has been shown to be easy-to-understand by a wide audience [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' An example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Here each dimen- sion’s range is represented as a horizontal bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The gray portion of the bar indicates the range across all counties in the US of the dimension while the blue portion of the bar represents the range of the dimension that defines this pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' User can quickly scroll through these bars and study the various ranges that define patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Geomap View: This panel consists of an interactive county-level map of the United States (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Each county in the map is assigned a color based on its COVID-19 death rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We use a continuous color scale ranging from dark blue for high death rates to white for a death rate of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Users can click on a county to learn more about the factors leading to its COVID-19 death rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Clicking on a county will trigger an update to the risk pattern browser which highlights the patterns that the county belongs to and grays out the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Additionally, the county information panel is updated with the top risk features for that county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We also allow the user to zoom and pan the map in order to select smaller counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' County Information: This panel communicates the information of a county selected by the user via the geomap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' As shown in the top right corner of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 1 the panel reports the current COVID-19 county death rate as well as the death rate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' More importantly the panel communicates the top 3 risk factors for the county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Here the feature ranking is computed based on the frequency at which those features appear across all patterns that contain the selected county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' We use the same bullet chart-like visualization used for the pattern information to visually represent these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Here again the gray bars indicate the range of the dimension across all counties in the US while the blue bars are ranges of the features across all counties in the selected county’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' In addition to the ranges shown in the chart we also add markers for the value of the factor in the county and the value for the US average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' An example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Figure 5: A visual representation of the top 3 feature values of a county in the context of state and US ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The gray bar represents the range of the feature across the US and the blue indicates the range of that feature across all counties in the county’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' For example, the county shown here has an ‘avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' GPA’ of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='9 (solid black marker) which is slightly lower than the US average (dotted black marker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Additionally the US range for the ‘avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' GPA’ is 0 to 4 while the range of this feature across all counties in the state is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='4 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 4 CONCLUSIONS We have outlined a methodology that can group socio-economic indicators of public health into 1-3 factor patterns learnt from ob- servational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The patterns can be used by policy makers and health officials to explain and predict the underlying risk a certain community has with respect to some natural hazard or public health emergency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' To give easy access to these patterns we devised an interactive visual dashboard by which the patterns can be explored in the context of the communities’ geographical locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' While we have used the early stages of the COVID-19 pandemic to show an application of our methodology, we believe that its application is far broader, which is being explored in ongoing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' While we provide temporal context to our data – the COVID-19 death rate over time - users currently cannot ”roll back” time to examine the patterns at the selected time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This is a fairly easy implementation and we plan to add this feature in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' In addition, while we have used a small cohort of users to gain feedback during system development, we plan a broader task-based study in the near future to gain further insight into utility and usability, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Range Pattern Range 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='8 PM25 particle matter pollution : 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='1 % adults with frequent phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' : 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2 % minority population :U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Range State Range U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Average County 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='9 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' GPA : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='5 % population w/o high schoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='9 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='3 % minority population : 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2ACKNOWLEDGMENTS This research was funded by NSF SBIR contract 1926949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' REFERENCES [1] 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' university of washington institute for health metric and education (ihme) covid-19 resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' retrieved from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' [6] Fema maps (accessed 8/14/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' https://hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='fema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='gov/ nri/social-vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' [7] Global covid-19 tracker (accessed 8/14/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' https://www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='org/ subnational/usa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' [9] Nextstrain real-time tracking of pathogen evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' (accessed 8/14/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' https://nextstrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' [10] Svi interactive map (accessed 8/14/2022).' metadata={'source': 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prediction of the epidemics trend of covid-19 in china under public health interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Journal of Thoracic Disease, 12(3):165, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 1 APPENDIX In this case study we follow Bob, a public health analyst, who uses the COVID-19 Risk Explorer to learn more about the susceptibility of local communities to the spread of the COVID-19 virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' It’s December 2020 and a lot has happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' He starts up the program and sees the screen below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Bob observes that the areas that have seen the highest death rates overall are in Texas, Arizona, Montana, North Dakota, Idaho, the South, Florida, and the Northern East Coast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Now Bob wonders about the timelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' He selects a darkly colored county (a country with high death rates) in Connecticut – Hartford County.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' When the screen transitions, Bob observes from the line chart showing the death rate over time in the “County Information” panel that this county exceeded the US average death rate very early in 2020 and quite rapidly so, but then remained nearly flat starting April.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Apparently this county responded well and took good precautions to stem the spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Bob looks at the “Pattern Information” panel to examine the risk factors of the most dominant pattern Hartford, CT participates in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' The “Risk Pattern Browser” indicates that this is the 3rd most dominant pattern in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' It is a 2-feature pattern that indicates that in Hartford, CT the percentage of non-Hispanics/Whites is at the low end of the US overall range and the ratio of median household debt/income is on the high end of the US overall range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Looking at the “Top 3 Risk Factors” of Hartford in the “County Information Panel” Bob learns that these two risk factors are actually the top 3 for Hartford in addition to PM25 particle COVID-19RISKEXPLORER0 AkaiKaeru U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='MAP COUNTYINFORMATION Select a county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='the.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='ms, PATTERN INFORMATION RISKPATTERNBROWSERCOVID-19RISKEXPLORER0 Akai Kaeru U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='MAP COUNTYINFORMATION Hartford,Connecticut 161 DEATHS PER100K Top3RiskFeatures U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Range median household debt %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='populationNon-Hisp/Whites 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='4 PM25D8 PATTERNINFORMATION ThispattemappearsIn315countieswhichhaveanavgdeathratf115per100K U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='RangePatemnRange %population Non-Hisp/Whites medianhouseholddebt/incom Reset Seiections RISKPATTERNBROWSER2 matter pollution and all its values compare unfavorably to the US average, and while they are not at the extreme ends for the State of Connecticut they tend to be in the unfavorable value ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Next Bob clicks on an equally dark colored county in Southern Texas, Cameron County, TX and the screen transitions to what is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Bob observes that for that county the death rate started to climb much later, in July 2020, and that it is still climbing now in December, albeit at a shallower slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' While Cameron TX shares the 3rd risk pattern with Hartford CT, it has many more risk patterns than Hartford CT, as can be seen by the many filled squares in the “Risk Pattern Browser”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' It means that its conditions for high death rates are more urgent than for Hartford CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' In fact, its top 3 risk factors are not those of the 3rd risk pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' For all of these Cameron TX fares unfavorably both within the value range found in Texas and with respect to the US average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Next, Bob clicks on the first risk pattern in the “Risk Pattern Browser” and sees the screen below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' This pattern is a 3-feature pattern with high and unfavorable value ranges in all three of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Only the feature “% minority population” appears in the top 3 list for Cameron County, TX and upon further investigation Bob finds that the other two risk features, “% population without high school degree” and “% uninsured”, are in risk pattern #4 (see picture next page).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' COVID-19RISK EXPLORER0 Akai Kaer U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='MAP COUNTYINFORMATION Cameron,Texas CountyRat Top3RiskFeatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Range %population 71:3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2 PATTERNINFORMATION This patter 315counties which bave an avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='death rate of 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='1per 100K U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Range Pattem Range 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 Reset Selecfons RISKPATTERNBROWSERCOVID-19RISKEXPLORER0 AkaiKaeru U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='MAP 7 COUNTY INFORMATION CeuntyRal Cameron,Texas Top3RiskFeatures u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Range: %populationuninsu 713 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2 PATTERNINFORMATION This pattern appears in 301 counties which have an avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='death rate of 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='4 per 100K U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='PatternRange PM25 particle matter pollution %adultswithfrequentphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 122 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 Reset Selections RISK PATTERN BROWSER 二3 Bob now wants to investigate whether Cameron County, TX could have learned its fate from other counties with similar risk factors but which had experienced high COVID-19 death rates earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' He looks for counties that share some or all of its top 3 risk features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' So he examines counties with risk pattern #1 and risk pattern #4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' He starts with risk pattern #1, clicks a few a counties on the map and eventually learns about Passaic County, NJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' From the timeline he sees that Passaic County, NJ started its death rate at the earliest time and has a similar “% minority population”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' But this is only one out of the three top 3 risk factors of Cameron County, TX so Bob needs to search more to complete the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' He turns to risk pattern #4 and after some search finds McKinley, NM (see next page).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' COVID-19RISKEXPLORER0 AkaiKaeru U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='MAP COUNTYINFORMATION Cameron,Texas May Top3RiskFeatures U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='StateRangeU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Average Count % popuiation w/o high schoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='7 30 %popuiationuninsured 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 % minonty population 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='3 PATTERNINFORMATION Thisattepparin268countieswhichhandeathrate114er100 %population vaccinated Blacks 3-0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='4 tat % population wio nigh schoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='. Reset Selectlions RISKPATTERNBROWSERCOVID-19RISKEXPLORERe AkaiKaeru U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='MAP COUNTY INFORMATION CountyRate Passaic,New Jersey Top3RiskFeatures US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Rang % adunts with frequent phys 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='2 % minority populatio PATTERNINFORMATION This patternappearsin301counties whichhaveanavgdeathrateof119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='4per100K U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='RangePatternRange PM25particlen 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='8 122 4 ResetSelecti > RISKPATTERN BROWSER4 McKinley, NM started its death rate climb later than Passaic, NJ but still 4 months earlier than Cameron, TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Its top 3 risk factors contain the two other risk factors of Cameron County, TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' So had Cameron County, TX looked at the fate of McKinley County, NM and Passaic County, NJ it could have learned from them and adopt the precautionary measures they took.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' There are many more case studies like this one, where late risers could have learnt from early risers about their fate and prepared better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' As seen from the early risers’ timelines, all of them were able to get the spread under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Late risers could have taken similar measures and potentially save lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' As a conclusion, our results show that pattern analysis is a powerful tool for public health risk management and that a dashboard like our Risk Explorer makes it easy to recognize risks and see how outbreaks and disaster in some communities can quickly inform other communities that fit a similar vulnerability pattern to prevent further loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' COVID-19RISKEXPLORER0 Akai Kaeru U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='MAP COUNTYINFORMATION McKiniey,NewMexico Desthspe:100) Top3RiskFeatures U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S:Range State RangeU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='Average 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='7 %population 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='0 uoeindod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='4 0 PATTERNINFORMATION %populationvaccinatedBlacks 38:0 %populationuninsurec 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content='4 18-1 %populationw/ohighschoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} +page_content=' Reset Selections RISKPATTERN BROWSER' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfJANe/content/2301.02946v1.pdf'} diff --git a/XdFJT4oBgHgl3EQf5i00/vector_store/index.pkl b/XdFJT4oBgHgl3EQf5i00/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..28d79c7294f14b05a80854eda0d5b680ef53c5a3 --- /dev/null +++ b/XdFJT4oBgHgl3EQf5i00/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8f462a2ea52420950193d32ed2c27e2b6b961ad78c4a293d0371579fb84e0f01 +size 326865 diff --git a/YtE1T4oBgHgl3EQfJwOa/content/tmp_files/2301.02956v1.pdf.txt b/YtE1T4oBgHgl3EQfJwOa/content/tmp_files/2301.02956v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..239fbd5e0bab1f8345b21fc7d418cae20bc4b575 --- /dev/null +++ b/YtE1T4oBgHgl3EQfJwOa/content/tmp_files/2301.02956v1.pdf.txt @@ -0,0 +1,1752 @@ +Vibronic excitations in resonant inelastic x-ray scattering spectra of K2RuCl6 +Naoya Iwahara1, ∗ and Shouta Shikano2 +1Graduate School of Engineering, Chiba University, +1-33 Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan +2Department of Materials Science, Faculty of Engineering, Chiba University, +1-33 Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan +(Dated: January 10, 2023) +We present the fingerprints of dynamic Jahn-Teller effect in resonant inelastic x-ray scattering +(RIXS) spectra of K2RuCl6. We determined the dynamic Jahn-Teller model Hamiltonian of an +embedded Ru4+ ion using post Hartree-Fock methods, and derived the vibronic states by numerically +diagonalizing the Hamiltonian. With the obtained vibronic states, we reproduced the RIXS spectra. +The shape and the temperature dependence of the RIXS spectrum agree well with the experimental +data. We found that some peaks emerge due to the dynamic Jahn-Teller effect rather than the +crystal field splitting. Our study indicates the significance of the Jahn-Teller coupling to adequately +interpret RIXS spectra. +I. +INTRODUCTION +Spin-orbit Mott insulators with heavy transition metal +ions exhibit diverse quantum phenomena [1–4]. A coun- +terintuitive excitonic magnetic phase could emerge in +compounds with nonmagnetic t4 +2g ions [5]. Heavy t4 +2g ion +embedded in an octahedral environment has a nonmag- +netic J = 0 ground state induced by strong spin-orbit +coupling, whereas sufficiently strong exchange interac- +tion between neighboring ions mixes the J = 0 and ex- +cited magnetic J = 1 multiplet states, and the admixed +magnetic quantum states may condensate. The excitonic +magnetism was attributed to the origin of the antifer- +romagnetism in Ca2RuO4 with a corner-shared struc- +ture. In the excitonic magnetic phase close to the quan- +tum critical point, amplitude fluctuation of magnetic mo- +ments (Higgs mode) develops, which was indeed observed +in the spin-wave excitation of the compound [6, 7]. This +theory predicts that different types of magnetism emerge +in other lattices with edge-shared octahedra; zigzag one- +dimensional magnetic order and bosonic Kitaev spin liq- +uid phase in honeycomb lattice [5, 8]. +Experimental exploration of the excitonic magnetism +in materials with edge-shared octahedra is underway. +Early attempts towards the realization of the excitonic +magnetism in Ir5+ double perovskites were prevented +by too strong spin-orbit coupling compared with the in- +tersite exchange interaction [9–11]. This situation lead +researchers to investigate 4d4 compounds with weaker +spin-orbit coupling than 5d4 compounds: +the investi- +gated compounds contain a honeycomb layered ruthen- +ate, Ag3LiRu2O6 [? ], and a cubic antifluorite, K2RuCl6 +[Fig. +1(a)] [12]. +In the former compound, three non- +magnetic phases arise under ambient and high-pressure +conditions, while the excitonic magnetism does not de- +velop [12]. The latter is a Van Vleck-type diamagnetic +material [13, 14], whereas the exchange interaction be- +∗ naoya.iwahara@gmail.com +tween Ru sites is tiny in ambient pressure according to +the dispersionless Ru L3 resonant inelastic x-ray scatter- +ing (RIXS) spectra [Fig. 1(d)] [15]. +An important factor controlling the magnetism in the +4d4 systems is the electron-phonon (vibronic) coupling. +In Ca2RuO4, the vibronic coupling between the J = 0 +and the excited states causes the development of the +pseudo JT deformation [16, 17], which triggers the de- +velopment of the spin-nematic phase above the magnetic +transition [18]. In Ag3LiRu2O6, the pseudo JT effect sta- +bilizes a singlet dimer phase, preventing the emergence +of excitonic magnetism [12]. +The vibronic coupling can give a significant influence +on the energy spectrum of Ru ion in K2RuCl6 [15]. In +the compound, the spin-orbit coupling (λ = 103 meV) +extracted from the RIXS spectra is largely reduced com- +pared with λ = 167 meV from the magnetic susceptibility +data [14] and λ = 150 meV in α-RuCl3 [19]. Takahashi et +al. attributed the large reduction of the spin-orbit cou- +pling to the dynamic JT stabilization of the J = 1 states +[15], while the magnitude of the vibronic coupling and +the impact of the dynamic JT effect on RIXS spectra +remain unclear. +In this work, we prove the existence of the dynamic +JT effect on the Ru sites in K2RuCl6 and elucidate its +fingerprints in the RIXS spectra based on ab initio cal- +culations. We derive a microscopic vibronic model of a +Ru site with post Hartree-Fock calculations, and numer- +ically diagonalize the vibronic Hamiltonian. +With the +obtained vibronic states, we simulate the RIXS spectra +of K2RuCl6. +II. +THEORY +A. +Model Hamiltonian for t2g orbitals +Let us set up our model for the RIXS in K2RuCl6. +This compound is a face-centered cubic crystal consist- +ing of RuCl2− +6 +octahedra [Fig. 1(a)]. In each octahedron, +arXiv:2301.02956v1 [cond-mat.str-el] 8 Jan 2023 + +2 +(a) +(b) +(c) +(d) +FIG. 1. +Crystal structure of K2RuCl6 and the experimental +RIXS spectra. (a) Conventional cell of K2RuCl6. The blue, +red, and green spheres are Ru, Cl, and K, respectively. The +JT active (b) Egu (3z2 − r2) and (c) Egv (x2 − y2) modes. +(d) Experimental RIXS spectra at 25 K (red circles) and 300 +K (green open triangles) taken from Ref. [15]. We took the +data with the momentum transfer of the light being (hkl) = +(0.19, 0.19, 0.19). +ligand field splits the 4d orbitals into a doublet (eg) and +a triplet (t2g), and the four 4d electrons populate the t2g +orbitals [20]. On each site, the electrons feel Coulomb, +spin-orbit, and electron-phonon (vibronic) couplings and +the interplay of these interactions determines the local +quantum states. The low-lying RIXS spectra display no +variation with respect to the crystal momentum, suggest- +ing that the intersite interactions between the neighbor- +ing octahedra are negligible [15]. +The model Hamiltonian for the embedded t4 +2g ion con- +sists of Coulomb ˆHC, spin-orbit ˆHSO [20], vibronic ˆHJT +[16, 21] interactions, and harmonic oscillator Hamiltonian +for the JT active modes ˆHvib: +ˆH = ˆHC + ˆHSO + ˆHJT + ˆHvib, +(1) +ˆHC = +� +γ +U ˆnγ↑ˆnγ↓ + +� +γ<γ′ +� +σσ′ +(U − 2JH)ˆnγσˆnγ′σ′ ++ +� +γ̸=γ′ +JH +� +ˆd† +γ↑ ˆd† +γ′↓ ˆdγ↓ ˆdγ′↑ + ˆd† +γ↑ ˆd† +γ↓ ˆdγ′↓ ˆdγ′↓ +� +− +� +γ<γ′ +� +σ +JH ˆnγσˆnγ′σ, +(2) +ˆHSO = +� +γσ +� +γ′σ′ +λ⟨γσ|ˆl · ˆs|γ′σ′⟩ ˆd† +γσ ˆdγ′σ′, +(3) +ˆHJT = +� +σ +V +� +ˆnyz,σ +� +−1 +2 +ˆQu + +√ +3 +2 +ˆQv +� ++ ˆnzx,σ +� +−1 +2 +ˆQu − +√ +3 +2 +ˆQv +� ++ ˆnxy,σ ˆQu +� +, +(4) +ˆHvib = +� +γ=u,v +1 +2 +� +ˆP 2 +γ + ω2 ˆQ2 +γ +� +. +(5) +Here ˆd† +γσ and ˆdγσ are, respectively, electron creation and +annihilation operators in orbital γ (γ = yz, zx, xy) with +spin projection σ, ˆnγσ = ˆd† +γσ ˆdγσ the electron number op- +erator, ˆl the t2g orbital angular momenta [20], ˆs the spin +angular momenta, ˆQγ the mass-weighted normal coor- +dinates (see e.g. §10.1 in Ref. [22]), ˆPγ the conjugate +momenta, and U, JH, λ, V , and ω are, respectively, the +Coulomb, Hund’s rule, spin-orbit, vibronic coupling pa- +rameters, and frequency. For the JT active modes, see +Fig. 1(b), (c). +Since the t2g orbitals are more than half-filled, we in- +troduce hole operators. The hole creation ˜d† and annihi- +lation ˜d operators are, respectively, +ˆd† +γσ = (−1)s+σ ˜dγ,−σ, +ˆdγσ = (−1)s+σ ˜d† +γ,−σ. +(6) +The vacuum state corresponds to the t6 +2g electron config- +uration. The Coulomb interaction for the holes remains +the same as Eq. +(2) except for a constant term. +We +obtain it by replacing the ˆd ( ˆd†) with ˜d ( ˜d†) and us- +ing the constraint on the number of the holes per site, +� +γσ ˜d† +γσ ˜dγσ = � +γσ ˜nγσ = 2. +Similarly, by using Eq. +(6), ˆHSO and ˆHJT remain the same form with opposite +sign (Appendix A). +We also introduce dimensionless coordinates ˆq and mo- +menta ˆp: +ˆQγ = +� +ℏ +2ω ˆqγ, +ˆPγ = +� +ℏω +2 ˆpγ. +(7) +With ˆq and ˆp and the hole operators (6), the vibronic +coupling and harmonic oscillator Hamiltonian become, +respectively, +˜HJT = +� +σ +−ℏωg +� +˜nyz,σ +� +−1 +2 ˆqu + +√ +3 +2 ˆqv +� ++ ˜nzx,σ +� +−1 +2 ˆqu − +√ +3 +2 ˆqv +� ++ ˜nxy,σˆqu +� +, +(8) +ˆHvib = +� +γ=u,v +ℏω +2 +� +ˆp2 +γ + ˆq2 +γ +� +. +(9) +Here g stands for the dimensionless vibronic coupling pa- +rameter defined by +g = +V +√ +ℏω3 . +(10) + +(z) +C +b +() +(x) +a25 K +Intensity (arb. units) +300 K +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +hw (eV)3 +Now we diagonalize the interactions in the descending +order of the energy scale. The Coulomb interaction splits +the t2 +2g hole configurations into four terms, 3T1 ⊕ 1E ⊕ +1T2 ⊕ 1A1 [20]. Since each representation appears once, +we can uniquely determine the term states as +|[S]ΓγMS⟩ = 1 +√ +2! +� +γ1σ1 +� +γ2σ2 +˜d† +γ1σ1 ˜d† +γ2σ2|0⟩ +× (t2γ1, t2γ2|Γγ)(sσ1, sσ2|SMS), +(11) +where +(t2γ1, t2γ2|Γγ) +and +(sσ1, sσ2|SMS) +are +the +Clebsch-Gordan coefficients [23, 24], and [S] = 2S + 1. +Using the term states (11) as the basis, the Coulomb +Hamiltonian is +ˆHC = 2JH +� +ˆP1E + ˆP1T2 +� ++ 5JH ˆP1A1, +(12) +with ˆP[S]Γ = � +γMS |[S]ΓγMS⟩⟨[S]ΓγMS|. In Eq. (12), +we set the 3T1 term energy to zero. +The spin-orbit coupling splits the [S]Γ terms into mul- +tiplet states. ˆHSO linearly couple to the 3T1 term states, +and the latter become J = 0 (A1), J = 1 (T1), and J = 2 +(E ⊕ T2) multiplet states: +|JMM⟩ = +� +γMS +|3T1γMS⟩(t1γ, t1MS|JMJ). +(13) +Using the |JMJ⟩ and the spin singlet 1Γ terms (Γ = +E, T2, A1) as the basis for ˆHSO, we obtain +ˆHSO = λ +� +− ˆPJ=0 − 1 +2 +ˆPJ=1 + 1 +2 +ˆPJ=2 +− i +√ +2 +� +|J = 0⟩⟨1A1| − |1A1⟩⟨J = 0| +� ++ +� +Γ=E,T2 +� +γ +i +√ +2 +� +|Γγ⟩⟨1Γγ| − |1Γγ⟩⟨Γγ| +� +� +. +(14) +Here ˆPJ = �J +MJ=−J |JMJ⟩⟨JMJ|. The second and third +lines in Eq. (14) are the interactions between the J = 0 +(A1) multiplet and 1A1 term and between the J = 2 +(E ⊕ T2) multiplet and 1E ⊕ 1T2 terms. +The spin-orbit multiplet energy levels [the energy +eigenstates of ˆHC + ˆHSO] are as follows: +EA1 = 1 +2 +� +5JH − λ − +� +25J2 +H + 10JHλ + 9λ2 +� +, +ET1 = −1 +2λ, +EE/T2 = 1 +4 +� +4JH + λ − +� +16J2 +H − 8JHλ + 9λ2 +� +, +E1E/1T2 = 1 +4 +� +4JH + λ + +� +16J2 +H − 8JHλ + 9λ2 +� +, +E1A1 = 1 +2 +� +5JH − λ + +� +25J2 +H + 10JHλ + 9λ2 +� +. +(15) +The 1E, 1T2, and 1A1 states are no longer pure term +states (11), while we continue using the same symbols. +The JT interaction is active in orbitally degenerate +terms. In the 3T1 term, the orbital part of the vibronic +interaction is +− ℏωg +� � +−1 +2 ˆqu + +√ +3 +2 ˆqv +� +|3T1x⟩⟨3T1x| ++ +� +−1 +2 ˆqu − +√ +3 +2 ˆqv +� +|3T1y⟩⟨3T1y| + ˆqu|3T1z⟩⟨3T1z| +� +. +(16) +Transforming the 3T1 term into the spin-orbit multiplet +states (13), Eq. (16) reduces to +−ℏωg +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +0 +0 +0 +0 +− 1 +√ +2 ˆqv − 1 +√ +2 ˆqu +0 +0 +0 +0 +− 1 +4 ˆqu + +√ +3 +4 ˆqv +0 +0 +0 +0 +− 3 +4 ˆqu − +√ +3 +4 ˆqv +0 +0 +0 +0 +− 1 +4 ˆqu − +√ +3 +4 ˆqv +0 +0 +0 +0 +3 +4 ˆqu − +√ +3 +4 ˆqv +0 +0 +0 +0 +1 +2 ˆqu +0 +0 +0 +0 +√ +3 +2 ˆqv +− 1 +√ +2 ˆqv +0 +0 +0 +1 +2 ˆqu +1 +2 ˆqv +0 +0 +0 +− 1 +√ +2 ˆqu +0 +0 +0 +1 +2 ˆqv +− 1 +2 ˆqu +0 +0 +0 +0 +− 3 +4 ˆqu − +√ +3 +4 ˆqv +0 +0 +0 +0 +− 1 +4 ˆqu + +√ +3 +4 ˆqv +0 +0 +0 +0 +3 +4 ˆqu − +√ +3 +4 ˆqv +0 +0 +0 +0 +− 1 +4 ˆqu − +√ +3 +4 ˆqv +0 +0 +0 +0 +√ +3 +2 ˆqv +0 +0 +0 +0 +1 +2 ˆqu +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(17) +in the increasing order of J [J = 0, J = 1 (x, y, z), E +(u, v) and T2 (yz, zx, xy) from J = 2]. Eq. (17) consists + +4 +of the (A ⊕ E) ⊗ E and the (T1 ⊕ T2) ⊗ E JT interaction +blocks. The diagonal blocks of Eq. (17) indicate that +the spin-orbit coupling quenches the vibronic coupling +by half in comparison with Eq. (8). +The vibronic coupling is active within the 1E ⊕ 1T2 +terms too. The JT Hamiltonian matrix for the terms is +−ℏωg +� +� +� +� +� +� +ˆqu +−ˆqv +0 +0 +0 +−ˆqv −ˆqu +0 +0 +0 +0 +0 +1 +2 ˆqu − +√ +3 +2 ˆqv +0 +0 +0 +0 +0 +1 +2 ˆqu + +√ +3 +2 ˆqv +0 +0 +0 +0 +0 +−ˆqu +� +� +� +� +� +� +, +(18) +in the order of 1E (u, v) and 1T2 (yz, zx, xy). Eq. (18) +is the direct sum of E ⊗ E type and T2 ⊗ E type JT +interactions. The vibronic coupling in the 1E ⊕ 1T2 term +states is unquenched. +We ignore the vibronic coupling (4) between different +[S]Γ terms. We show the validity of the assumption for +K2RuCl6 in Sec. IV B. +The vibronic coupling of the JT type can drive the +formation of the quantum entanglement of the spin-orbit +multiplet and the vibrational states (dynamic JT effect). +The energy eigenstates (vibronic states) of Eq. (1) gen- +erally have the form of +|ν⟩ = +� +Γγ +|Γγ⟩ ⊗ |χΓγ,ν⟩, +(19) +where |Γγ⟩ indicate the spin-orbit multiplets, and |χ⟩ are +the vibrational states of the JT modes. We determine |χ⟩ +by a numerical method (Sec. III C). With the vibronic +states (19) as the basis, the Hamiltonian is +ˆH = +� +ν +Eν|ν⟩⟨ν|, +(20) +where Eν are the energy eigenvalues. +B. +RIXS +Here we describe the cross section for the Ru-L3 RIXS +taking account of the dynamic JT effect. The process +consists of two steps: Excitation of an electron from the +2p3/2 orbitals to the empty 4d (t2g) orbitals absorbing a +photon followed by a transition of a 4d, t2g electron into +the empty 2p3/2 emitting a photon. We can derive the +cross section for the dynamic JT system by combining +the vibronic states and the second order time-dependent +perturbation theory (Kramers-Heisenberg formula). +The free Hamiltonian consists of the valence and core +electron Hamiltonians and the radiation field Hamilto- +nian. We have described the valence Hamiltonian (20) in +Sec. II A. The core level Hamiltonian is +ˆHc = +j +� +mj=−j +ϵjˆc† +jmj ˆcjmj, +(21) +where j = 3 +2, and ˆc† +jmj and ˆcjmj are the electron creation +and annihilation operators in 2p3/2 orbital with projec- +tion mj: +|jmj⟩ = +� +γpσ +|2pγp, sσ⟩(Γ4γp, Γ6σ|Γ8mj). +(22) +Here γp = x, y, z are the components of the 2p orbitals. +Single electron spin states and j = +3 +2 states belong to +the Γ6 and Γ8 representations in the octahedron, respec- +tively. +The radiation field Hamiltonian is +ˆHrad = +� +kλ +ℏωk +� +ˆa† +kλˆakλ + 1 +2 +� +. +(23) +Here k are the momenta, λ the polarization, ωk = ck is +the frequency of light, k = |k|, and c the speed of light. +ˆa† +kλ and ˆakλ are the creation and annihilation operators +of the photon with kλ, respectively. With ˆa† +kλ and ˆakλ, +vector potential ˆ +A at the Ru site (r = 0) is +ˆ +A = +� +kλ +� +ℏ +2V ε0ωk +� +ekλˆakλ + e∗ +−kλˆa† +−kλ +� +. +(24) +Here ekλ are the polarization vectors, V is the volume, +and ε0 is the permittivity of the vacuum. +We take +Coulomb gauge, and hence, k · ekλ = 0. +We assume that the bilinear interaction of the 4d elec- +tron’s momentum and the vector field be dominant and +the field around the Ru site be uniform (dipole approxi- +mation): +ˆH′ = e +m +ˆ +A · ˆp. +(25) +Here e is the elementary charge, m is the mass of electron, +and ˆp is the momentum operator. The electron momen- +tum operator between the core and valence orbitals is +ˆp = +� +γσ +� +mj +� +⟨t2gγ, sσ|ˆp|jmj⟩ ˆd† +γσˆcjmj ++ ⟨jmj|ˆp|t2gγ, sσ⟩ˆc† +jmj ˆdγσ +� +. +(26) +The matrix elements of ˆp are, by using Eq. +(22) and +Wigner-Eckart theorem [20, 22–24], +⟨t2gγ, sσ|ˆpα|jmj⟩ = (t2g∥ˆp∥2p) +� +dt2 +� +γp +(Γ4γp, Γ6σ|Γ8mj) +× (Γ5γ|Γ4γp, Γ4α). +(27) +Here α = x, y, z, dt2 = 3 is the dimension of the t2 (Γ5) +representation, and (t2g∥ˆp∥2p) is the reduced matrix el- +ement. +Applying the second-order time-dependent perturba- +tion theory to our model under resonant condition, we +obtain the cross section of the RIXS processes. The ini- +tial and final states are the products of the vibronic states + +5 +(19) and one-photon states, |kλ⟩ = ˆa† +kλ|0⟩, and the inter- +mediate states are those with one 2p3/2 core-hole. When +the initial and intermediate energies are close to each +other, the cross section is +d2σ +dΩdk′ = V 2ω2 +k′ +(2π)2ℏc3 +���⟨ν′; k′λ′| ˆH′ ˆG(zνk) ˆH′|ν; kλ⟩ +��� +2 +× δ (Eν + ℏωk − Eν′ − ℏωk′) . +(28) +Here ˆG is the propagator for the intermediate states, +ˆG(z) = � +n +|n⟩⟨n| +z−En , and zνk = Eν + ℏωk + iΓ. Substi- +tuting ˆ +H′ (25) into Eq. (28), we obtain an explicit form +for the vibronic RIXS spectrum: +d2σ +dΩdk′ =ℏca2 +0 +m2 +ωk′ +ωk +����� +� +αα′ +e∗ +k′λ′,α′ekλ,α⟨ν′|ˆpα′ ˆG(zνk)ˆpα|ν⟩ +����� +2 +× δ (Eν + ℏωk − Eν′ − ℏωk′) . +(29) +The vibronic cross-section indicates that the dynamic JT +effect modulates the RIXS spectrum in two ways: (1) the +vibronic reduction of the electronic operator ˆpα′ ˆG(zνk)ˆpα +and (2) the emergence of new peaks. +We continue simplifying the cross section for our nu- +merical calculations by applying the fast collision ap- +proximation [25, 26]. +This approximation ignores the +detailed energy structures and dynamics of the interme- +diate states by replacing En and ˆG by a typical value ¯E +and ¯G(z) = +1 +z− ¯ +E , respectively. With the approximation, +Eq. (29) reduces to +d2σ +dΩdk′ =ℏca2 +0 +m2 +ωk′ +ωk +�� ¯G(zνk) +��2 +× +����� +� +αα′ +e∗ +k′λ′,α′ekλ,α⟨ν′| ˆFα′α|ν⟩ +����� +2 +× δ (Eν + ℏωk − Eν′ − ℏωk′) , +(30) +where ˆFα′α = ˆpα ˆPchˆpα′, and ˆPch the projection operator +into the intermediate states. Using Eq. (26) in ˆF, +ˆFα′α = (t2g∥ˆp∥2p)2 +dt2 +� +γ′σ′ +� +γσ +(−1)σ−σ′ +× +� � +γ′ +pγp +� +mj +(t1γ′ +p, Γ6σ′|Γ8mj)∗(t1γ′ +p, t1α′|t2γ′) +× (t1γp, Γ6σ|Γ8mj)(t1γp, t1α|t2γ) +� +˜d† +γ′−σ′ ˜dγ,−σ. +(31) +Finally, we include the thermal effect. +The cross- +section at finite T is +d2σ +dΩdk′ = ℏca2 +0 +m2 +ωk′ +ωk +�� ¯G(zνk) +��2 +× +� +ν +ρν +����� +� +αα′ +e∗ +k′λ′,α′ekλ,α⟨ν′| ˆFα′α|ν⟩ +����� +2 +× δ (Eν + ℏωk − Eν′ − ℏωk′) , +(32) +with the canonical distribution of the dynamic JT sys- +tem, ρν = exp(−Eνβ)/Z. Here β is the inverse temper- +ature and Z = � +ν exp(−Eνβ). +III. +METHODS +A. +Ab initio method +We quantitatively determined the electronic structure +of a single Ru site by cluster calculations with post +Hartree-Fock methods. +We constructed the Ru clus- +ter from the x-ray structure at 300 K [14]. +The clus- +ter consists of three parts. The first part contains one +Ru atom, the nearest six Cl, and the nearest eight +K atoms. +We treated the electrons in this part fully +quantum mechanically with the atomic-natural-orbital +relativistic-correlation consistent-valence triple zeta po- +larization (ANO-RCC-VTZP) basis functions. The sec- +ond part consists of surrounding atoms (12 Zr atoms at +the Ru sites, 48 K, and 72 Cl). We treated them within +the ab initio embedding model potential method [27]. +The last part consists of 1554 point charges surround- +ing the first and the second parts. The total charge of +the cluster is neutral. +We calculated the electronic states of the cluster em- +ploying a series of post Hartree-Fock methods. First, we +derived the [S]Γ term states using the complete active +space self-consistent field (CASSCF) method [28]. In the +CASSCF calculations, we treated the five 4d orbitals as +the active space and calculated all the term states with +S = 0, 1, 2. +We expressed the atomic bielectronic in- +tegrals using Cholesky decomposition with a threshold +of 5 × 10−7 Eh and set the ionization potential electron +affinity (IPEA) shift to zero and the imaginary (IMAG) +shift to 0.1. After the CASSCF calculations, we included +the dynamic electron correction effect on the [S]Γ term +energies with the extended multistate complete active +space second-order perturbation theory (XMS-CASPT2) +[29, 30]. Then, we included the spin-orbit coupling us- +ing the spin-orbit restricted active space state interaction +(SO-RASSI) method. For all the calculations, we used +OpenMolcas [31, 32]. +B. +Vibronic coupling rameters +We derived the vibronic coupling parameters by fit- +ting the 3T1 energy levels for JT deformed structures to + +6 +the JT model as in Refs. [33, 34]. We constructed the +JT deformed structures of the Ru cluster by varying the +normal coordinate Qu from 0 to 10 by 2 (in atomic unit): +RA = R(0) +A + +Qu +√MA +(eu)A . +(33) +Here A indicates the nearest neighbor Cl atoms, RA the +Cartesian coordinates of atom A, R(0) +A the coordinates at +the perfect octahedral structure, MA the mass of atom +A, eu the eigenvector of the dynamical matrix, and (eu)A +the components for atom A in eu. We chose the phase +of eu to give the deformation in Fig. 1(b) with positive +Qu. At each JT deformed structure, we performed the +CASSCF/XMS-CASPT2 calculations. +We obtained ω and the vibronic coupling parameter g +by fitting the ab initio term energies to the potential en- +ergy surface of the JT model. The model potential con- +tains the harmonic potential and the vibronic coupling +(16): +U(Qu) = ω2 +2 Q2 +u − V +� +−1 +2Qu|3T1x⟩⟨3T1x| +− 1 +2Qu|3T1y⟩⟨3T1y| + Qu|3T1z⟩⟨3T1z| +� +. +(34) +C. +Vibronic states +We calculated the vibronic states by numerically diag- +onalizing the dynamic JT Hamiltonian (1). We expand +the nuclear part |χ⟩ of the vibronic states (19) with the +energy eigenstates of ˆHvib, |nu, nv⟩ (nu, nv = 0, 1, 2, ...), +and expansion coefficients, χΓγnunv,ν: +|χΓγ,ν⟩ = +� +nu,nv +|nu, nv⟩χΓγnunv,ν. +(35) +Thus, the vibronic basis for the dynamic JT Hamiltonian +is a set of the direct products of |Γγ⟩ ⊗ |nu, nv⟩. +To numerically diagonalize the vibronic Hamiltonian, +we introduced the following approximations. We treated +the vibronic states related to the 3T1 terms and the 1E ⊕ +1T2 terms separately. This is valid when the pseudo JT +couplings between the terms are negligible. We truncated +the vibronic basis by introducing the maximum number +of the vibrational quanta, 0 ≤ nu + nv ≤ 20. This basis +is sufficiently large [See Ref. [34]]. +With the vibronic basis, we constructed the vibronic +Hamiltonian matrix, and numerically diagonalized it. +For the diagonalization of the Hamiltonian matrix, we +used dsyevd in Lapack library [35]. +IV. +RESULTS +A. +Electronic states +We performed the ab initio electronic state calculations +of the cubic Ru cluster. +Table I shows the calculated +(a) +(b) +FIG. 2. +The spin-orbit energy levels with respect to λ/JH +for the Oh cluster. The spin-orbit multiplet energies originate +from (a) the t4 +2 states and (b) the 3T1 term. At the vertical +line (λ/JH = 0.357), the ratio of the excited energy levels +with respect to the ground one coincides with the ab initio +data. +TABLE I. Ab initio [S]Γ term and spin-orbit multiplet energies +of the cubic Ru cluster (eV). +[S]Γ term +Spin-orbit multiplet +3T1 +0 +J = 0 (A1) +−0.1604 +J = 1 (T1) +−0.0756 +J = 2 (T2) +0.0455 +J = 2 (E) +0.0461 +1T2 +0.9585 +0.9581 +1E +0.9629 +0.9619 +1A1 +2.1275 +2.1350 +electronic energy levels: the values of the left and right +columns correspond to the [S]Γ term and spin-orbit mul- +tiplet energies, respectively. The splittings of the J = 2 +and the excited E ⊗T2 multiplet energy levels amount to +only a few meV. +We determined the electronic interaction parameters +by fitting the ab initio data to the model Hamiltonians. +We obtained JH = 443.7 meV from the fitting of the +CASSCF/XMS-CASPT2 levels to Eq. (12). Since the +energy splitting of the 1T2 and 1E is only 4 meV and +much smaller than the other energy gaps, we ignored the +gap in the fitting. +The present Hund rule coupling is +close to the experimental JH = 420 meV extracted from +the RIXS spectra of K2RuCl6 [15]. +Similarly, we derived the spin-orbit coupling parame- +ter from the SO-RASSI levels and Eq. (15). The energy +levels of the first (J = 1) and the second (J = 2) ex- +cited states with respect to the ground (J = 0) level are, +respectively, ∆EJ=1 = 84.8 meV and ∆EJ=2 = 206.1 +meV ignoring the small splitting of the latter. We de- +termined λ/JH to be 0.357 by reproducing the ratio of +∆EJ=2/∆EJ=1 = 2.4 with Eq. (15) [Fig. 2]. Our spin- +orbit coupling λ is 158 meV. +The ab initio λ deviates from the experimental esti- +mate in Ref. [15]. The ratio of the excitation energies, +∆EJ=2/∆EJ=1 = 2.4, is smaller than the ratio of 2.7 +extracted from the RIXS data. The present λ is close to + +5 +1 A1 +4 +3 +1E④1T2 +E/JH +2 +1 +J=2 +0 +J=1 +-1 +J=0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +^/JH0.2 +J=2 +0.0 +J=1 +-0.2 +-0.4 +J=0 +-0.6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +^/JH7 +(a) +(c) +(b) +FIG. 3. +The adiabatic potential energies and the vibronic +levels. +(a) The 3T1 term energies with respect to the JT +deformation (Qu) in atomic units. The red points are the ab +initio energies and the solid lines are the eigenvalues of U (34). +(b) The spin-orbit multiplet energies (meV) with respect to +the JT deformation (Qu). (c) The spin-orbit and vibrational +energy levels (SO + vib) and vibronic energy levels (DJT) (in +meV). In the left column, the black and gray lines are the +spin-orbit energies without and with vibrational excitations. +In the right column, the solid and dashed lines are the energy +eigenstates of the (A ⊕ E) ⊗ E and the (T1 ⊕ T2) ⊗ e dynamic +JT models, respectively. +λ = 167 meV derived from the magnetic susceptibility +data of K2RuCl6 [14] and λ = 150 meV for α-RuCl3 [19], +while by about 50 % larger than λ = 103 meV derived +from the RIXS spectra [15]. Takahashi et al. ascribed +this reduction to the dynamic JT effect. +We examine +this idea below. +B. +Vibronic coupling parameters +We derived the vibronic coupling parameters from the +gradients of the 3T1 term energies with respect to the +JT deformation. Figure 3(a) indicates the ab initio 3T1 +term energies for several JT deformed structures (the red +points). By fitting the data to Eq. (34), we derived ℏω = +42.1 meV and the vibronic coupling parameter g = 1.07. +The solid curves in Fig. 3(a) are the best fit. +Our ab initio calculations show that the pseudo JT +couplings between the 3T1 term and the other terms are +weak. We transformed the 3T1 term states into the spin- +orbit multiplet states (13), and draw the adiabatic po- +tential energy surfaces in Fig. 3(b). The figure indicates +FIG. 4. +Temperature dependence of the effective magnetic +moment, Meff. The black solid line indicates Kotani’s model +[36, 37] with λ from Ref. +[14] and the red points are the +present theoretical data. +a good agreement between the ab initio (the red points) +and the model (the blue solid lines), meaning that the +pseudo JT coupling between the different multiplets is +negligible. +The 3T1 term states could vibronically couple to the +T2g modes, while it is negligible. We calculated the term +energies for the geometries with the T2g deformations, +and found that the JT coupling is only a few % of the +V for the Eg mode. Therefore, we ignored the vibronic +coupling to the T2g mode in this work. +C. +Vibronic states +With the derived parameters, we calculated the vi- +bronic states. Figure 3(c) shows that the vibronic cou- +pling modulates the distribution of energy levels (the +right column) with respect to the decoupled ones (the +left column). In the right column, the solid lines are the +vibronic states from the (A ⊕ E) ⊗ E JT part and the +others from the (T1 ⊕ T2) ⊗ e JT part. +Now we closely look at the vibronic states which turn +out to be important in the RIXS spectrum of K2RuCl6. +The arrows identify the pairs of the spin-orbit and vi- +bronic states that are close to each other. The vibronic +states have large contributions of |Γγ⟩ ⊗ |nu = nv = 0⟩ +type: the weights (χ2 +Γγnunv,ν) are 0.98 (J = 0), 0.83 +(J = 1), 0.82 (E), and 0.78 (T2). Although the ground +J = 0 spin-orbit multiplet state does not linearly couple +to the JT active vibrations, the pseudo JT coupling be- +tween the J = 0 and Eg levels (17) stabilizes the J = 0 +vibronic state by 4 meV. The dynamic JT effect stabi- +lizes the J = 1 multiplet state by 11 meV, while it does +not stabilize much the J = 2 states due to the pseudo JT +coupling between the J = 1 and the T2 part of the J = 2 +multiplet states. + +3.0 +2.5 +2.0 +Meff/μB +1.5 +1.0 +0.5 +0.0 +0 +50 +100 +150 +200 +250 +300 +T (K)0.8 +0.6 +0.4 +0.2 +0.0 +-0.2 +0 +2 +4 +6 +8 +10 +Qu (atomic unit)250 +J=2 +200 +150 +J=1 +100 +50 +J=0 +0 +SO+vib +DJT50. +0 +-50 +-100 +-150 +0 +2 +4 +6 +8 +10 +Qu (atomic unit)8 +D. +Effective magnetic moment +Before moving to the simulations of the RIXS spectra, +let us discuss the effective magnetic moments. The mag- +netic moment operators ˆµα (α = x, y, z) within the t2g +orbitals are +ˆµα = +� +γσ +� +γ′σ′ +−µB +� +kˆlα + geˆsα +� +γσ,γ′σ′ ˆd† +γσ ˆdγ′σ′, +(36) +where µB is the Bohr magneton, ge the g-factor of the +electron, and k is the reduction factor of the orbital an- +gular momentum due to the covalency between the Ru 4d +and Cl 3p orbitals. By fitting the ab initio magnetic mo- +ments at the CASSCF level to Eq. (36), we determined +the reduction factor k to be 0.920. Then, we projected +the magnetic moments (36) into the vibronic states (19). +With the magnetic moments, we simulated the tem- +perature dependence of the effective magnetic moment. +Our model consists of the vibronic Hamiltonian (20) and +the Zeeman Hamiltonian, ˆHZee = −ˆµ·H, where H is the +external magnetic field along the c axis. We calculated +Meff as +Meff = +� +3(kBT)2 ∂2lnZ(H) +∂H2 +���� +H→+0 +, +(37) +with Z(H) being the partition function for the model. +Finally, we compared the calculated Meff with the ex- +perimental one from Ref. [14] [Fig. 4]. The theoretical +and the experimental Meff are overall in good agreement +with each other. The deviation between them is only 5-6 +% of Meff at 300 K. The deviation might come from the +underestimations of the metal-ligand covalency (1 − k) +within the post Hartree-Fock method and Van Vleck’s +contribution due to the lack of the high-energy states +such as t3 +2ge1 +g within our calculations. The present result +suggests that our model is accurate enough to adequately +describe the dynamic JT effect in K2RuCl6. +E. +RIXS spectra +Using the numerical vibronic states in Sec. IV C, we +simulated the RIXS spectra. We derived the RIXS spec- +tra by substituting the calculated vibronic states (19) +into Eq. +(32), and then convoluting the latter with +Lorentzian function, L(x) = 1 +π +Γ/2 +x2+(Γ/2)2 , where Γ is the +line width. For all the simulations below, Γ = 0.075 eV. +The polarizations and the directions (θ = π/4 in the lit- +erature) of the incident and scattered lights are the same +as the experimental ones [15]. +To clarify the vibronic +effect, we also calculated the RIXS spectrum with the +electronic model at the same level of approximations. +Let us compare the electronic and vibronic RIXS spec- +tra at 25 K [Fig. 5(a)]. The strongest peak in the vibronic +RIXS spectrum is lower than that in the electronic one +(a) +(b) +(c) +(d) +(e) +FIG. 5. +RIXS spectra. (a) Comparison between the elec- +tronic (the blue dotted line) and vibronic (the red solid line) +RIXS spectra at 25 K. (b), (c) The vibronic RIXS spectra at +25 K (the red dotted line) and 300 K (the green solid line). +Γ = 0.075 eV in all cases. (d), (e) The electronic and vibronic +transitions. +due to the vibronic reduction of ˆF. The peaks in the vi- +bronic RIXS spectrum tend to be broader than those in +the electronic spectrum. In particular, the broadening is +significant at ≈ 1.1 eV. We will discuss the origin below. +The ratio of the first and second excitation energies +becomes close to the experimental one due to the vibronic +effect. The peak positions of the low-energy region are +at 0.078 eV and 0.212 eV, and the ratio is about 2.7, +which agrees well with the experimental value [15]. The +ratio becomes larger than our electronic one in Sec. IV A +because of the dynamic JT effect. +The broadening of the RIXS spectrum occurs due +to the transitions from the ground state to various ex- +cited vibronic states. +Since the ground vibronic state + +25 K +Intensity (arb. units) +Electronic +Vibronic +0.0 +0.5 +1.0 +1.5 +hw (eV)25 K +300 K +Intensity (arb. units) +-0.1 +0.0 +0.1 +0.2 +0.3 +hw (eV)25 K +300 K +Intensity (arb. units) +0.9 +1.0 +1.1 +1.2 +1.3 +hw (eV)250 +J=2 +200 +150 +J=1 +100 +50 +J=0 +0 +SO +DJT1250 +1200 +1150 +1100 +1050 +SO +DJT9 +≈ |J = 0⟩ ⊗ |nu = nv = 0⟩ in K2RuCl6, the main fea- +tures of the peaks in the electronic RIXS spectrum persist +in the vibronic RIXS spectrum, whereas more vibronic +transitions exist in the latter and they make the spec- +trum at ≈ 0.2 eV and at ≈ 1 eV broad. [Fig. 5(d), (e)]. +In particular, the peaks in the high-energy region emerge +due to the presence of the dynamic JT effect rather than +crystal-field splitting of the E and T2 multiplet levels. +As temperature rises to 300 K, the vibronic RIXS spec- +trum again becomes broader [Fig. 5(b), (c)]. With the +increase in temperature, the height of the peak in the low- +energy region (0.078 eV) becomes lower than the one at +25 K and the peak has a new shoulder at ≈ −0.08 eV +[Fig. 5(b)]. The peak in the high-energy region (about +1.1 eV) also becomes broader and has a new shoulder at +about 1 eV [Fig. 5(c)]. The patterns of the broadening +agree well with the changes in the experimental data [Fig. +1(d)]. The new shoulder peaks appear due to the transi- +tions from the third excited vibronic states to the J = 0 +ground state (the green dotted line) and E vibronic state +(the green dashed line), respectively [Fig. 5(d), (e)]. +The vibronic RIXS spectrum has all the important fea- +tures of the experimental spectrum, while quantitative +discrepancy exists. The theoretical peak positions (and +λ) are about 20 % larger than the experimental data. The +deviation comes from the underestimated covalency, and +consequently overestimated λ, within the post Hartree- +Fock method. +Since all the parameters are somewhat +enlarged, the qualitative features would not be affected +much by the quantitative difference. +V. +CONCLUSION +We developed the ab initio based theory of RIXS spec- +tra of a t4 +2g dynamic JT ion in cubic spin-orbit Mott +insulators. We derived the electronic and vibronic pa- +rameters of an embedded Ru center by using the post +Hartree-Fock calculations, and derived the low-lying vi- +bronic states. +Using the ab initio data and Kramers- +Heisenberg formula, we simulated the Ru-L3 RIXS spec- +tra. The shape and the temperature dependence of the +vibronic RIXS spectrum agree well with the experimental +data, confirming the presence of the dynamic Jahn-Teller +effect in K2RuCl6. Our simulation indicates that several +peaks emerge due to vibronic levels rather than ligand- +field split spin-orbit multiplet levels. +We also demon- +strated that the dynamic JT effect enlarges the line width +of the RIXS spectrum by increasing temperature. The +present results call for the reconsideration of the assign- +ments of the RIXS spectra of cubic spin-orbit Mott insu- +lators fully taking account of the vibronic effects. +ACKNOWLEDGMENTS +We are grateful to Veacheslav Vieru for providing +his ab initio data of the cubic cluster and reading this +manuscript. This work was partly supported by the Ike- +tani Science and Technology Foundation and Grant-in- +Aid for Scientific Research (Grant No. 22K03507) from +the Japan Society for the Promotion of Science. +Appendix A: Hole operators +In the hole picture, the one-electron operators ˆO acting +on the t2g electrons are transformed as follows: +ˆO = +� +γdσ +� +γ′ +dσ′ +⟨γdσ| ˆO|γ′ +dσ′⟩ ˆd† +γdσ ˆdγ′ +dσ′ +→ +� +γdσ +� +γ′ +dσ′ +[⟨γ′ +d, σ′|Θ] +� +Θ ˆO|γd, σ⟩ +� +× (−1)s+σ(−1)s+σ′ ˜dγd−σ ˜d† +γ′ +d−σ′ += −σTR( ˆO) +� +γdσ +� +γ′ +dσ′ +⟨γ′ +dσ′| ˆO|γdσ⟩ ˜d† +γ′ +dσ′ ˜dγdσ. +(A1) +The time-inversion Θ of operators and spin states are +[20, 38] +Θ|γdσ⟩ = (−1)s−σ|γd, −σ⟩, +(A2) +Θ ˆO = σTR( ˆO) ˆOΘ, +(A3) +respectively. Here s = 1/2, and σTR( ˆO) is the sign. We +assumed that ˆO is traceless. +[1] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Ba- +lents, Correlated quantum phenomena in the strong spin- +orbit regime, Annu. Rev. Condens. Matter Phys. 5, 57 +(2014). +[2] J. G. Rau, E. K.-H. 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Bleaney, Electron Paramagnetic Res- +onance of Transition Ions (Clarendon Press, Oxford, +1970). + diff --git a/YtE1T4oBgHgl3EQfJwOa/content/tmp_files/load_file.txt b/YtE1T4oBgHgl3EQfJwOa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..784d132c69559714a8b7c0b8070f4afd9a35c045 --- /dev/null +++ b/YtE1T4oBgHgl3EQfJwOa/content/tmp_files/load_file.txt @@ -0,0 +1,1163 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf,len=1162 +page_content='Vibronic excitations in resonant inelastic x-ray scattering spectra of K2RuCl6 Naoya Iwahara1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' ∗ and Shouta Shikano2 1Graduate School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Chiba University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 1-33 Yayoi-cho,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Inage-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Chiba-shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Chiba 263-8522,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Japan 2Department of Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Faculty of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Chiba University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 1-33 Yayoi-cho,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Inage-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Chiba-shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Chiba 263-8522,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Japan (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 2023) We present the fingerprints of dynamic Jahn-Teller effect in resonant inelastic x-ray scattering (RIXS) spectra of K2RuCl6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We determined the dynamic Jahn-Teller model Hamiltonian of an embedded Ru4+ ion using post Hartree-Fock methods, and derived the vibronic states by numerically diagonalizing the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' With the obtained vibronic states, we reproduced the RIXS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The shape and the temperature dependence of the RIXS spectrum agree well with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We found that some peaks emerge due to the dynamic Jahn-Teller effect rather than the crystal field splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Our study indicates the significance of the Jahn-Teller coupling to adequately interpret RIXS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' INTRODUCTION Spin-orbit Mott insulators with heavy transition metal ions exhibit diverse quantum phenomena [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' A coun- terintuitive excitonic magnetic phase could emerge in compounds with nonmagnetic t4 2g ions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Heavy t4 2g ion embedded in an octahedral environment has a nonmag- netic J = 0 ground state induced by strong spin-orbit coupling, whereas sufficiently strong exchange interac- tion between neighboring ions mixes the J = 0 and ex- cited magnetic J = 1 multiplet states, and the admixed magnetic quantum states may condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The excitonic magnetism was attributed to the origin of the antifer- romagnetism in Ca2RuO4 with a corner-shared struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In the excitonic magnetic phase close to the quan- tum critical point, amplitude fluctuation of magnetic mo- ments (Higgs mode) develops, which was indeed observed in the spin-wave excitation of the compound [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' This theory predicts that different types of magnetism emerge in other lattices with edge-shared octahedra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' zigzag one- dimensional magnetic order and bosonic Kitaev spin liq- uid phase in honeycomb lattice [5, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Experimental exploration of the excitonic magnetism in materials with edge-shared octahedra is underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Early attempts towards the realization of the excitonic magnetism in Ir5+ double perovskites were prevented by too strong spin-orbit coupling compared with the in- tersite exchange interaction [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' This situation lead researchers to investigate 4d4 compounds with weaker spin-orbit coupling than 5d4 compounds: the investi- gated compounds contain a honeycomb layered ruthen- ate, Ag3LiRu2O6 [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' ], and a cubic antifluorite, K2RuCl6 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 1(a)] [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In the former compound, three non- magnetic phases arise under ambient and high-pressure conditions, while the excitonic magnetism does not de- velop [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The latter is a Van Vleck-type diamagnetic material [13, 14], whereas the exchange interaction be- ∗ naoya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='iwahara@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='com tween Ru sites is tiny in ambient pressure according to the dispersionless Ru L3 resonant inelastic x-ray scatter- ing (RIXS) spectra [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 1(d)] [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' An important factor controlling the magnetism in the 4d4 systems is the electron-phonon (vibronic) coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In Ca2RuO4, the vibronic coupling between the J = 0 and the excited states causes the development of the pseudo JT deformation [16, 17], which triggers the de- velopment of the spin-nematic phase above the magnetic transition [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In Ag3LiRu2O6, the pseudo JT effect sta- bilizes a singlet dimer phase, preventing the emergence of excitonic magnetism [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The vibronic coupling can give a significant influence on the energy spectrum of Ru ion in K2RuCl6 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In the compound, the spin-orbit coupling (λ = 103 meV) extracted from the RIXS spectra is largely reduced com- pared with λ = 167 meV from the magnetic susceptibility data [14] and λ = 150 meV in α-RuCl3 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' attributed the large reduction of the spin-orbit cou- pling to the dynamic JT stabilization of the J = 1 states [15], while the magnitude of the vibronic coupling and the impact of the dynamic JT effect on RIXS spectra remain unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In this work, we prove the existence of the dynamic JT effect on the Ru sites in K2RuCl6 and elucidate its fingerprints in the RIXS spectra based on ab initio cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We derive a microscopic vibronic model of a Ru site with post Hartree-Fock calculations, and numer- ically diagonalize the vibronic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' With the obtained vibronic states, we simulate the RIXS spectra of K2RuCl6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Model Hamiltonian for t2g orbitals Let us set up our model for the RIXS in K2RuCl6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' This compound is a face-centered cubic crystal consist- ing of RuCl2− 6 octahedra [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In each octahedron, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='02956v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='str-el] 8 Jan 2023 2 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Crystal structure of K2RuCl6 and the experimental RIXS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (a) Conventional cell of K2RuCl6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The blue, red, and green spheres are Ru, Cl, and K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The JT active (b) Egu (3z2 − r2) and (c) Egv (x2 − y2) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (d) Experimental RIXS spectra at 25 K (red circles) and 300 K (green open triangles) taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We took the data with the momentum transfer of the light being (hkl) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='19, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='19, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' ligand field splits the 4d orbitals into a doublet (eg) and a triplet (t2g), and the four 4d electrons populate the t2g orbitals [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' On each site, the electrons feel Coulomb, spin-orbit, and electron-phonon (vibronic) couplings and the interplay of these interactions determines the local quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The low-lying RIXS spectra display no variation with respect to the crystal momentum, suggest- ing that the intersite interactions between the neighbor- ing octahedra are negligible [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The model Hamiltonian for the embedded t4 2g ion con- sists of Coulomb ˆHC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' spin-orbit ˆHSO [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' vibronic ˆHJT [16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 21] interactions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' and harmonic oscillator Hamiltonian for the JT active modes ˆHvib: ˆH = ˆHC + ˆHSO + ˆHJT + ˆHvib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (1) ˆHC = � γ U ˆnγ↑ˆnγ↓ + � γ<γ′ � σσ′ (U − 2JH)ˆnγσˆnγ′σ′ + � γ̸=γ′ JH � ˆd† γ↑ ˆd† γ′↓ ˆdγ↓ ˆdγ′↑ + ˆd† γ↑ ˆd† γ↓ ˆdγ′↓ ˆdγ′↓ � − � γ<γ′ � σ JH ˆnγσˆnγ′σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (2) ˆHSO = � γσ � γ′σ′ λ⟨γσ|ˆl · ˆs|γ′σ′⟩ ˆd† γσ ˆdγ′σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (3) ˆHJT = � σ V � ˆnyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='σ � −1 2 ˆQu + √ 3 2 ˆQv � + ˆnzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='σ � −1 2 ˆQu − √ 3 2 ˆQv � + ˆnxy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='σ ˆQu � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (4) ˆHvib = � γ=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='v 1 2 � ˆP 2 γ + ω2 ˆQ2 γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (5) Here ˆd† γσ and ˆdγσ are, respectively, electron creation and annihilation operators in orbital γ (γ = yz, zx, xy) with spin projection σ, ˆnγσ = ˆd† γσ ˆdγσ the electron number op- erator, ˆl the t2g orbital angular momenta [20], ˆs the spin angular momenta, ˆQγ the mass-weighted normal coor- dinates (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [22]), ˆPγ the conjugate momenta, and U, JH, λ, V , and ω are, respectively, the Coulomb, Hund’s rule, spin-orbit, vibronic coupling pa- rameters, and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' For the JT active modes, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 1(b), (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Since the t2g orbitals are more than half-filled, we in- troduce hole operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The hole creation ˜d† and annihi- lation ˜d operators are, respectively, ˆd† γσ = (−1)s+σ ˜dγ,−σ, ˆdγσ = (−1)s+σ ˜d† γ,−σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (6) The vacuum state corresponds to the t6 2g electron config- uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The Coulomb interaction for the holes remains the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (2) except for a constant term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We obtain it by replacing the ˆd ( ˆd†) with ˜d ( ˜d†) and us- ing the constraint on the number of the holes per site, � γσ ˜d† γσ ˜dγσ = � γσ ˜nγσ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Similarly, by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (6), ˆHSO and ˆHJT remain the same form with opposite sign (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We also introduce dimensionless coordinates ˆq and mo- menta ˆp: ˆQγ = � ℏ 2ω ˆqγ, ˆPγ = � ℏω 2 ˆpγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (7) With ˆq and ˆp and the hole operators (6), the vibronic coupling and harmonic oscillator Hamiltonian become, respectively, ˜HJT = � σ −ℏωg � ˜nyz,σ � −1 2 ˆqu + √ 3 2 ˆqv � + ˜nzx,σ � −1 2 ˆqu − √ 3 2 ˆqv � + ˜nxy,σˆqu � , (8) ˆHvib = � γ=u,v ℏω 2 � ˆp2 γ + ˆq2 γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (9) Here g stands for the dimensionless vibronic coupling pa- rameter defined by g = V √ ℏω3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (10) (z) C b () (x) a25 K Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' units) 300 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 hw (eV)3 Now we diagonalize the interactions in the descending order of the energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The Coulomb interaction splits the t2 2g hole configurations into four terms, 3T1 ⊕ 1E ⊕ 1T2 ⊕ 1A1 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Since each representation appears once, we can uniquely determine the term states as |[S]ΓγMS⟩ = 1 √ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' � γ1σ1 � γ2σ2 ˜d† γ1σ1 ˜d† γ2σ2|0⟩ × (t2γ1, t2γ2|Γγ)(sσ1, sσ2|SMS), (11) where (t2γ1, t2γ2|Γγ) and (sσ1, sσ2|SMS) are the Clebsch-Gordan coefficients [23, 24], and [S] = 2S + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Using the term states (11) as the basis, the Coulomb Hamiltonian is ˆHC = 2JH � ˆP1E + ˆP1T2 � + 5JH ˆP1A1, (12) with ˆP[S]Γ = � γMS |[S]ΓγMS⟩⟨[S]ΓγMS|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (12), we set the 3T1 term energy to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The spin-orbit coupling splits the [S]Γ terms into mul- tiplet states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' ˆHSO linearly couple to the 3T1 term states, and the latter become J = 0 (A1), J = 1 (T1), and J = 2 (E ⊕ T2) multiplet states: |JMM⟩ = � γMS |3T1γMS⟩(t1γ, t1MS|JMJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (13) Using the |JMJ⟩ and the spin singlet 1Γ terms (Γ = E, T2, A1) as the basis for ˆHSO, we obtain ˆHSO = λ � − ˆPJ=0 − 1 2 ˆPJ=1 + 1 2 ˆPJ=2 − i √ 2 � |J = 0⟩⟨1A1| − |1A1⟩⟨J = 0| � + � Γ=E,T2 � γ i √ 2 � |Γγ⟩⟨1Γγ| − |1Γγ⟩⟨Γγ| � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (14) Here ˆPJ = �J MJ=−J |JMJ⟩⟨JMJ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The second and third lines in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (14) are the interactions between the J = 0 (A1) multiplet and 1A1 term and between the J = 2 (E ⊕ T2) multiplet and 1E ⊕ 1T2 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The spin-orbit multiplet energy levels [the energy eigenstates of ˆHC + ˆHSO] are as follows: EA1 = 1 2 � 5JH − λ − � 25J2 H + 10JHλ + 9λ2 � , ET1 = −1 2λ, EE/T2 = 1 4 � 4JH + λ − � 16J2 H − 8JHλ + 9λ2 � , E1E/1T2 = 1 4 � 4JH + λ + � 16J2 H − 8JHλ + 9λ2 � , E1A1 = 1 2 � 5JH − λ + � 25J2 H + 10JHλ + 9λ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (15) The 1E, 1T2, and 1A1 states are no longer pure term states (11), while we continue using the same symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The JT interaction is active in orbitally degenerate terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In the 3T1 term, the orbital part of the vibronic interaction is − ℏωg � � −1 2 ˆqu + √ 3 2 ˆqv � |3T1x⟩⟨3T1x| + � −1 2 ˆqu − √ 3 2 ˆqv � |3T1y⟩⟨3T1y| + ˆqu|3T1z⟩⟨3T1z| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (16) Transforming the 3T1 term into the spin-orbit multiplet states (13), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (16) reduces to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='−ℏωg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (17) in the increasing order of J [J = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' J = 1 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' E (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' v) and T2 (yz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' zx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' xy) from J = 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (17) consists 4 of the (A ⊕ E) ⊗ E and the (T1 ⊕ T2) ⊗ E JT interaction blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The diagonal blocks of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (17) indicate that the spin-orbit coupling quenches the vibronic coupling by half in comparison with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The vibronic coupling is active within the 1E ⊕ 1T2 terms too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The JT Hamiltonian matrix for the terms is −ℏωg � � � � � � ˆqu −ˆqv 0 0 0 −ˆqv −ˆqu 0 0 0 0 0 1 2 ˆqu − √ 3 2 ˆqv 0 0 0 0 0 1 2 ˆqu + √ 3 2 ˆqv 0 0 0 0 0 −ˆqu � � � � � � , (18) in the order of 1E (u, v) and 1T2 (yz, zx, xy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (18) is the direct sum of E ⊗ E type and T2 ⊗ E type JT interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The vibronic coupling in the 1E ⊕ 1T2 term states is unquenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We ignore the vibronic coupling (4) between different [S]Γ terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We show the validity of the assumption for K2RuCl6 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The vibronic coupling of the JT type can drive the formation of the quantum entanglement of the spin-orbit multiplet and the vibrational states (dynamic JT effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The energy eigenstates (vibronic states) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (1) gen- erally have the form of |ν⟩ = � Γγ |Γγ⟩ ⊗ |χΓγ,ν⟩, (19) where |Γγ⟩ indicate the spin-orbit multiplets, and |χ⟩ are the vibrational states of the JT modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We determine |χ⟩ by a numerical method (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' With the vibronic states (19) as the basis, the Hamiltonian is ˆH = � ν Eν|ν⟩⟨ν|, (20) where Eν are the energy eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' RIXS Here we describe the cross section for the Ru-L3 RIXS taking account of the dynamic JT effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The process consists of two steps: Excitation of an electron from the 2p3/2 orbitals to the empty 4d (t2g) orbitals absorbing a photon followed by a transition of a 4d, t2g electron into the empty 2p3/2 emitting a photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We can derive the cross section for the dynamic JT system by combining the vibronic states and the second order time-dependent perturbation theory (Kramers-Heisenberg formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The free Hamiltonian consists of the valence and core electron Hamiltonians and the radiation field Hamilto- nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We have described the valence Hamiltonian (20) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The core level Hamiltonian is ˆHc = j � mj=−j ϵjˆc† jmj ˆcjmj, (21) where j = 3 2, and ˆc† jmj and ˆcjmj are the electron creation and annihilation operators in 2p3/2 orbital with projec- tion mj: |jmj⟩ = � γpσ |2pγp, sσ⟩(Γ4γp, Γ6σ|Γ8mj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (22) Here γp = x, y, z are the components of the 2p orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Single electron spin states and j = 3 2 states belong to the Γ6 and Γ8 representations in the octahedron, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The radiation field Hamiltonian is ˆHrad = � kλ ℏωk � ˆa† kλˆakλ + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (23) Here k are the momenta, λ the polarization, ωk = ck is the frequency of light, k = |k|, and c the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' ˆa† kλ and ˆakλ are the creation and annihilation operators of the photon with kλ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' With ˆa† kλ and ˆakλ, vector potential ˆ A at the Ru site (r = 0) is ˆ A = � kλ � ℏ 2V ε0ωk � ekλˆakλ + e∗ −kλˆa† −kλ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (24) Here ekλ are the polarization vectors, V is the volume, and ε0 is the permittivity of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We take Coulomb gauge, and hence, k · ekλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We assume that the bilinear interaction of the 4d elec- tron’s momentum and the vector field be dominant and the field around the Ru site be uniform (dipole approxi- mation): ˆH′ = e m ˆ A · ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (25) Here e is the elementary charge, m is the mass of electron, and ˆp is the momentum operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The electron momen- tum operator between the core and valence orbitals is ˆp = � γσ � mj � ⟨t2gγ, sσ|ˆp|jmj⟩ ˆd† γσˆcjmj + ⟨jmj|ˆp|t2gγ, sσ⟩ˆc† jmj ˆdγσ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (26) The matrix elements of ˆp are, by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (22) and Wigner-Eckart theorem [20, 22–24], ⟨t2gγ, sσ|ˆpα|jmj⟩ = (t2g∥ˆp∥2p) � dt2 � γp (Γ4γp, Γ6σ|Γ8mj) × (Γ5γ|Γ4γp, Γ4α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (27) Here α = x, y, z, dt2 = 3 is the dimension of the t2 (Γ5) representation, and (t2g∥ˆp∥2p) is the reduced matrix el- ement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Applying the second-order time-dependent perturba- tion theory to our model under resonant condition, we obtain the cross section of the RIXS processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The ini- tial and final states are the products of the vibronic states 5 (19) and one-photon states, |kλ⟩ = ˆa† kλ|0⟩, and the inter- mediate states are those with one 2p3/2 core-hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' When the initial and intermediate energies are close to each other, the cross section is d2σ dΩdk′ = V 2ω2 k′ (2π)2ℏc3 ���⟨ν′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' k′λ′| ˆH′ ˆG(zνk) ˆH′|ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' kλ⟩ ��� 2 × δ (Eν + ℏωk − Eν′ − ℏωk′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (28) Here ˆG is the propagator for the intermediate states, ˆG(z) = � n |n⟩⟨n| z−En , and zνk = Eν + ℏωk + iΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Substi- tuting ˆ H′ (25) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (28), we obtain an explicit form for the vibronic RIXS spectrum: d2σ dΩdk′ =ℏca2 0 m2 ωk′ ωk ����� � αα′ e∗ k′λ′,α′ekλ,α⟨ν′|ˆpα′ ˆG(zνk)ˆpα|ν⟩ ����� 2 × δ (Eν + ℏωk − Eν′ − ℏωk′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (29) The vibronic cross-section indicates that the dynamic JT effect modulates the RIXS spectrum in two ways: (1) the vibronic reduction of the electronic operator ˆpα′ ˆG(zνk)ˆpα and (2) the emergence of new peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We continue simplifying the cross section for our nu- merical calculations by applying the fast collision ap- proximation [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' This approximation ignores the detailed energy structures and dynamics of the interme- diate states by replacing En and ˆG by a typical value ¯E and ¯G(z) = 1 z− ¯ E , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' With the approximation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (29) reduces to d2σ dΩdk′ =ℏca2 0 m2 ωk′ ωk �� ¯G(zνk) ��2 × ����� � αα′ e∗ k′λ′,α′ekλ,α⟨ν′| ˆFα′α|ν⟩ ����� 2 × δ (Eν + ℏωk − Eν′ − ℏωk′) , (30) where ˆFα′α = ˆpα ˆPchˆpα′, and ˆPch the projection operator into the intermediate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (26) in ˆF, ˆFα′α = (t2g∥ˆp∥2p)2 dt2 � γ′σ′ � γσ (−1)σ−σ′ × � � γ′ pγp � mj (t1γ′ p, Γ6σ′|Γ8mj)∗(t1γ′ p, t1α′|t2γ′) × (t1γp, Γ6σ|Γ8mj)(t1γp, t1α|t2γ) � ˜d† γ′−σ′ ˜dγ,−σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (31) Finally, we include the thermal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The cross- section at finite T is d2σ dΩdk′ = ℏca2 0 m2 ωk′ ωk �� ¯G(zνk) ��2 × � ν ρν ����� � αα′ e∗ k′λ′,α′ekλ,α⟨ν′| ˆFα′α|ν⟩ ����� 2 × δ (Eν + ℏωk − Eν′ − ℏωk′) , (32) with the canonical distribution of the dynamic JT sys- tem, ρν = exp(−Eνβ)/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Here β is the inverse temper- ature and Z = � ν exp(−Eνβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Ab initio method We quantitatively determined the electronic structure of a single Ru site by cluster calculations with post Hartree-Fock methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We constructed the Ru clus- ter from the x-ray structure at 300 K [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The clus- ter consists of three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The first part contains one Ru atom, the nearest six Cl, and the nearest eight K atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We treated the electrons in this part fully quantum mechanically with the atomic-natural-orbital relativistic-correlation consistent-valence triple zeta po- larization (ANO-RCC-VTZP) basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The sec- ond part consists of surrounding atoms (12 Zr atoms at the Ru sites, 48 K, and 72 Cl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We treated them within the ab initio embedding model potential method [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The last part consists of 1554 point charges surround- ing the first and the second parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The total charge of the cluster is neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We calculated the electronic states of the cluster em- ploying a series of post Hartree-Fock methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' First, we derived the [S]Γ term states using the complete active space self-consistent field (CASSCF) method [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In the CASSCF calculations, we treated the five 4d orbitals as the active space and calculated all the term states with S = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We expressed the atomic bielectronic in- tegrals using Cholesky decomposition with a threshold of 5 × 10−7 Eh and set the ionization potential electron affinity (IPEA) shift to zero and the imaginary (IMAG) shift to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' After the CASSCF calculations, we included the dynamic electron correction effect on the [S]Γ term energies with the extended multistate complete active space second-order perturbation theory (XMS-CASPT2) [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Then, we included the spin-orbit coupling us- ing the spin-orbit restricted active space state interaction (SO-RASSI) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' For all the calculations, we used OpenMolcas [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Vibronic coupling rameters We derived the vibronic coupling parameters by fit- ting the 3T1 energy levels for JT deformed structures to 6 the JT model as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We constructed the JT deformed structures of the Ru cluster by varying the normal coordinate Qu from 0 to 10 by 2 (in atomic unit): RA = R(0) A + Qu √MA (eu)A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (33) Here A indicates the nearest neighbor Cl atoms, RA the Cartesian coordinates of atom A, R(0) A the coordinates at the perfect octahedral structure, MA the mass of atom A, eu the eigenvector of the dynamical matrix, and (eu)A the components for atom A in eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We chose the phase of eu to give the deformation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 1(b) with positive Qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' At each JT deformed structure, we performed the CASSCF/XMS-CASPT2 calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We obtained ω and the vibronic coupling parameter g by fitting the ab initio term energies to the potential en- ergy surface of the JT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The model potential con- tains the harmonic potential and the vibronic coupling (16): U(Qu) = ω2 2 Q2 u − V � −1 2Qu|3T1x⟩⟨3T1x| − 1 2Qu|3T1y⟩⟨3T1y| + Qu|3T1z⟩⟨3T1z| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (34) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Vibronic states We calculated the vibronic states by numerically diag- onalizing the dynamic JT Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We expand the nuclear part |χ⟩ of the vibronic states (19) with the energy eigenstates of ˆHvib, |nu, nv⟩ (nu, nv = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='), and expansion coefficients, χΓγnunv,ν: |χΓγ,ν⟩ = � nu,nv |nu, nv⟩χΓγnunv,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (35) Thus, the vibronic basis for the dynamic JT Hamiltonian is a set of the direct products of |Γγ⟩ ⊗ |nu, nv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' To numerically diagonalize the vibronic Hamiltonian, we introduced the following approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We treated the vibronic states related to the 3T1 terms and the 1E ⊕ 1T2 terms separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' This is valid when the pseudo JT couplings between the terms are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We truncated the vibronic basis by introducing the maximum number of the vibrational quanta, 0 ≤ nu + nv ≤ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' This basis is sufficiently large [See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [34]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' With the vibronic basis, we constructed the vibronic Hamiltonian matrix, and numerically diagonalized it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' For the diagonalization of the Hamiltonian matrix, we used dsyevd in Lapack library [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Electronic states We performed the ab initio electronic state calculations of the cubic Ru cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Table I shows the calculated (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The spin-orbit energy levels with respect to λ/JH for the Oh cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The spin-orbit multiplet energies originate from (a) the t4 2 states and (b) the 3T1 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' At the vertical line (λ/JH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='357), the ratio of the excited energy levels with respect to the ground one coincides with the ab initio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Ab initio [S]Γ term and spin-orbit multiplet energies of the cubic Ru cluster (eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [S]Γ term Spin-orbit multiplet 3T1 0 J = 0 (A1) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1604 J = 1 (T1) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0756 J = 2 (T2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0455 J = 2 (E) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0461 1T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='9585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='9581 1E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='9629 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='9619 1A1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1275 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1350 electronic energy levels: the values of the left and right columns correspond to the [S]Γ term and spin-orbit mul- tiplet energies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The splittings of the J = 2 and the excited E ⊗T2 multiplet energy levels amount to only a few meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We determined the electronic interaction parameters by fitting the ab initio data to the model Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We obtained JH = 443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='7 meV from the fitting of the CASSCF/XMS-CASPT2 levels to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Since the energy splitting of the 1T2 and 1E is only 4 meV and much smaller than the other energy gaps, we ignored the gap in the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The present Hund rule coupling is close to the experimental JH = 420 meV extracted from the RIXS spectra of K2RuCl6 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Similarly, we derived the spin-orbit coupling parame- ter from the SO-RASSI levels and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The energy levels of the first (J = 1) and the second (J = 2) ex- cited states with respect to the ground (J = 0) level are, respectively, ∆EJ=1 = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='8 meV and ∆EJ=2 = 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1 meV ignoring the small splitting of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We de- termined λ/JH to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='357 by reproducing the ratio of ∆EJ=2/∆EJ=1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='4 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (15) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Our spin- orbit coupling λ is 158 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The ab initio λ deviates from the experimental esti- mate in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The ratio of the excitation energies, ∆EJ=2/∆EJ=1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='4, is smaller than the ratio of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='7 extracted from the RIXS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The present λ is close to 5 1 A1 4 3 1E④1T2 E/JH 2 1 J=2 0 J=1 1 J=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 ^/JH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 J=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 J=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='4 J=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='5 ^/JH7 (a) (c) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The adiabatic potential energies and the vibronic levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (a) The 3T1 term energies with respect to the JT deformation (Qu) in atomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The red points are the ab initio energies and the solid lines are the eigenvalues of U (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (b) The spin-orbit multiplet energies (meV) with respect to the JT deformation (Qu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (c) The spin-orbit and vibrational energy levels (SO + vib) and vibronic energy levels (DJT) (in meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In the left column, the black and gray lines are the spin-orbit energies without and with vibrational excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In the right column, the solid and dashed lines are the energy eigenstates of the (A ⊕ E) ⊗ E and the (T1 ⊕ T2) ⊗ e dynamic JT models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' λ = 167 meV derived from the magnetic susceptibility data of K2RuCl6 [14] and λ = 150 meV for α-RuCl3 [19], while by about 50 % larger than λ = 103 meV derived from the RIXS spectra [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' ascribed this reduction to the dynamic JT effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We examine this idea below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Vibronic coupling parameters We derived the vibronic coupling parameters from the gradients of the 3T1 term energies with respect to the JT deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Figure 3(a) indicates the ab initio 3T1 term energies for several JT deformed structures (the red points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' By fitting the data to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (34), we derived ℏω = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1 meV and the vibronic coupling parameter g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The solid curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 3(a) are the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Our ab initio calculations show that the pseudo JT couplings between the 3T1 term and the other terms are weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We transformed the 3T1 term states into the spin- orbit multiplet states (13), and draw the adiabatic po- tential energy surfaces in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The figure indicates FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Temperature dependence of the effective magnetic moment, Meff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The black solid line indicates Kotani’s model [36, 37] with λ from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [14] and the red points are the present theoretical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' a good agreement between the ab initio (the red points) and the model (the blue solid lines), meaning that the pseudo JT coupling between the different multiplets is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The 3T1 term states could vibronically couple to the T2g modes, while it is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We calculated the term energies for the geometries with the T2g deformations, and found that the JT coupling is only a few % of the V for the Eg mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Therefore, we ignored the vibronic coupling to the T2g mode in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Vibronic states With the derived parameters, we calculated the vi- bronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Figure 3(c) shows that the vibronic cou- pling modulates the distribution of energy levels (the right column) with respect to the decoupled ones (the left column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In the right column, the solid lines are the vibronic states from the (A ⊕ E) ⊗ E JT part and the others from the (T1 ⊕ T2) ⊗ e JT part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Now we closely look at the vibronic states which turn out to be important in the RIXS spectrum of K2RuCl6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The arrows identify the pairs of the spin-orbit and vi- bronic states that are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The vibronic states have large contributions of |Γγ⟩ ⊗ |nu = nv = 0⟩ type: the weights (χ2 Γγnunv,ν) are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='98 (J = 0), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='83 (J = 1), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='82 (E), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='78 (T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Although the ground J = 0 spin-orbit multiplet state does not linearly couple to the JT active vibrations, the pseudo JT coupling be- tween the J = 0 and Eg levels (17) stabilizes the J = 0 vibronic state by 4 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The dynamic JT effect stabi- lizes the J = 1 multiplet state by 11 meV, while it does not stabilize much the J = 2 states due to the pseudo JT coupling between the J = 1 and the T2 part of the J = 2 multiplet states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 Meff/μB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 0 50 100 150 200 250 300 T (K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 0 2 4 6 8 10 Qu (atomic unit)250 J=2 200 150 J=1 100 50 J=0 0 SO+vib DJT50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 0 50 100 150 0 2 4 6 8 10 Qu (atomic unit)8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Effective magnetic moment Before moving to the simulations of the RIXS spectra, let us discuss the effective magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The mag- netic moment operators ˆµα (α = x, y, z) within the t2g orbitals are ˆµα = � γσ � γ′σ′ −µB � kˆlα + geˆsα � γσ,γ′σ′ ˆd† γσ ˆdγ′σ′, (36) where µB is the Bohr magneton, ge the g-factor of the electron, and k is the reduction factor of the orbital an- gular momentum due to the covalency between the Ru 4d and Cl 3p orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' By fitting the ab initio magnetic mo- ments at the CASSCF level to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (36), we determined the reduction factor k to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Then, we projected the magnetic moments (36) into the vibronic states (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' With the magnetic moments, we simulated the tem- perature dependence of the effective magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Our model consists of the vibronic Hamiltonian (20) and the Zeeman Hamiltonian, ˆHZee = −ˆµ·H, where H is the external magnetic field along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We calculated Meff as Meff = � 3(kBT)2 ∂2lnZ(H) ∂H2 ���� H→+0 , (37) with Z(H) being the partition function for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Finally, we compared the calculated Meff with the ex- perimental one from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [14] [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The theoretical and the experimental Meff are overall in good agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The deviation between them is only 5-6 % of Meff at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The deviation might come from the underestimations of the metal-ligand covalency (1 − k) within the post Hartree-Fock method and Van Vleck’s contribution due to the lack of the high-energy states such as t3 2ge1 g within our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The present result suggests that our model is accurate enough to adequately describe the dynamic JT effect in K2RuCl6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' RIXS spectra Using the numerical vibronic states in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' IV C, we simulated the RIXS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We derived the RIXS spec- tra by substituting the calculated vibronic states (19) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (32), and then convoluting the latter with Lorentzian function, L(x) = 1 π Γ/2 x2+(Γ/2)2 , where Γ is the line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' For all the simulations below, Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='075 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The polarizations and the directions (θ = π/4 in the lit- erature) of the incident and scattered lights are the same as the experimental ones [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' To clarify the vibronic effect, we also calculated the RIXS spectrum with the electronic model at the same level of approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Let us compare the electronic and vibronic RIXS spec- tra at 25 K [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 5(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The strongest peak in the vibronic RIXS spectrum is lower than that in the electronic one (a) (b) (c) (d) (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' RIXS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (a) Comparison between the elec- tronic (the blue dotted line) and vibronic (the red solid line) RIXS spectra at 25 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (b), (c) The vibronic RIXS spectra at 25 K (the red dotted line) and 300 K (the green solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='075 eV in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (d), (e) The electronic and vibronic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' due to the vibronic reduction of ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The peaks in the vi- bronic RIXS spectrum tend to be broader than those in the electronic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In particular, the broadening is significant at ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We will discuss the origin below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The ratio of the first and second excitation energies becomes close to the experimental one due to the vibronic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The peak positions of the low-energy region are at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='078 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='212 eV, and the ratio is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='7, which agrees well with the experimental value [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The ratio becomes larger than our electronic one in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' IV A because of the dynamic JT effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The broadening of the RIXS spectrum occurs due to the transitions from the ground state to various ex- cited vibronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Since the ground vibronic state 25 K Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' units) Electronic Vibronic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='5 hw (eV)25 K 300 K Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='3 hw (eV)25 K 300 K Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='3 hw (eV)250 J=2 200 150 J=1 100 50 J=0 0 SO DJT1250 1200 1150 1100 1050 SO DJT9 ≈ |J = 0⟩ ⊗ |nu = nv = 0⟩ in K2RuCl6, the main fea- tures of the peaks in the electronic RIXS spectrum persist in the vibronic RIXS spectrum, whereas more vibronic transitions exist in the latter and they make the spec- trum at ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='2 eV and at ≈ 1 eV broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 5(d), (e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' In particular, the peaks in the high-energy region emerge due to the presence of the dynamic JT effect rather than crystal-field splitting of the E and T2 multiplet levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' As temperature rises to 300 K, the vibronic RIXS spec- trum again becomes broader [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 5(b), (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' With the increase in temperature, the height of the peak in the low- energy region (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='078 eV) becomes lower than the one at 25 K and the peak has a new shoulder at ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='08 eV [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 5(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The peak in the high-energy region (about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content='1 eV) also becomes broader and has a new shoulder at about 1 eV [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 5(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The patterns of the broadening agree well with the changes in the experimental data [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 1(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The new shoulder peaks appear due to the transi- tions from the third excited vibronic states to the J = 0 ground state (the green dotted line) and E vibronic state (the green dashed line), respectively [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 5(d), (e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The vibronic RIXS spectrum has all the important fea- tures of the experimental spectrum, while quantitative discrepancy exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The theoretical peak positions (and λ) are about 20 % larger than the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The deviation comes from the underestimated covalency, and consequently overestimated λ, within the post Hartree- Fock method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Since all the parameters are somewhat enlarged, the qualitative features would not be affected much by the quantitative difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' CONCLUSION We developed the ab initio based theory of RIXS spec- tra of a t4 2g dynamic JT ion in cubic spin-orbit Mott insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We derived the electronic and vibronic pa- rameters of an embedded Ru center by using the post Hartree-Fock calculations, and derived the low-lying vi- bronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Using the ab initio data and Kramers- Heisenberg formula, we simulated the Ru-L3 RIXS spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The shape and the temperature dependence of the vibronic RIXS spectrum agree well with the experimental data, confirming the presence of the dynamic Jahn-Teller effect in K2RuCl6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Our simulation indicates that several peaks emerge due to vibronic levels rather than ligand- field split spin-orbit multiplet levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We also demon- strated that the dynamic JT effect enlarges the line width of the RIXS spectrum by increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' The present results call for the reconsideration of the assign- ments of the RIXS spectra of cubic spin-orbit Mott insu- lators fully taking account of the vibronic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' ACKNOWLEDGMENTS We are grateful to Veacheslav Vieru for providing his ab initio data of the cubic cluster and reading this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' This work was partly supported by the Ike- tani Science and Technology Foundation and Grant-in- Aid for Scientific Research (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' 22K03507) from the Japan Society for the Promotion of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Appendix A: Hole operators In the hole picture, the one-electron operators ˆO acting on the t2g electrons are transformed as follows: ˆO = � γdσ � γ′ dσ′ ⟨γdσ| ˆO|γ′ dσ′⟩ ˆd† γdσ ˆdγ′ dσ′ → � γdσ � γ′ dσ′ [⟨γ′ d, σ′|Θ] � Θ ˆO|γd, σ⟩ � × (−1)s+σ(−1)s+σ′ ˜dγd−σ ˜d† γ′ d−σ′ = −σTR( ˆO) � γdσ � γ′ dσ′ ⟨γ′ dσ′| ˆO|γdσ⟩ ˜d† γ′ dσ′ ˜dγdσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' (A1) The time-inversion Θ of operators and spin states are [20, 38] Θ|γdσ⟩ = (−1)s−σ|γd, −σ⟩, (A2) Θ ˆO = σTR( ˆO) ˆOΘ, (A3) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Here s = 1/2, and σTR( ˆO) is the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' We assumed that ˆO is traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfJwOa/content/2301.02956v1.pdf'} +page_content=' Witczak-Krempa, G.' metadata={'source': 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2023 +COMPANION WEAKLY PERIODIC MATRICES OVER +FINITE AND COUNTABLE FIELDS +PETER DANCHEV AND ANDRADA POJAR +Abstract. We explore the situation where all companion n × n matrices over +a field F are weakly periodic of index of nilpotence 2 and prove that this can be +happen uniquely when F is a countable field of positive characteristic, which is +an algebraic extension of its minimal simple (finite) subfield, with all subfields of +order greater than n. In particular, in the commuting case, we show even that F +is a finite field of order greater than n. +Our obtained results somewhat generalize those obtained by Breaz-Modoi in +Lin. Algebra & Appl. (2016). +1. Introduction and Principalities +Let F be a field and n an arbitrary non-negative integer, say n ≥ 1. We denote +by Mn(F) the matrix ring consisting of all squared n × n matrices over F. For a +non-negative integer m ≥ 2, we denote by τ the cycle τm = (1 2 . . . m) ∈ Sm, +and by Pτm the m × m permutation matrix with only 0s and 1s over F, that is, +Pτm = (aij)1≤i,j≤m, where (aij) = 1 if j = τm(i), and (aij) = 0 if j ̸= τm(i). +The matrix M ∈ Mn(F) is called potent if there exists an integer k ≥ 1 such +that Mk+1 = M. So, let STn the set of traces of all potent companion matrices +C ∈ Mn(F). By a root of unity, we denote a root of the polynomial over F which is +of the form g = Xi − 1, where i is an integer with i ≥ 2. Thus, let SRn be the set +of sums of at most n-th roots of unity which are not necessarily distinct. +Likewise, we denote by Lm,n the set of polynomials having degree at most m and +with non-negative integer multiples of unity coefficients such that their sum is not +exceeding n. We also denote Wm = ∪f∈Lm,nSpec(f(Pτm)), and the symbol d(m) +stands for the number of non-negative divisors of m. +Let R be a ring. An element x ∈ R is said to be potent if there exists a non- +negative integer q ≥ 1 with xq+1 = a, and x ∈ R is said to be weakly periodic with +nilpotence index 2, provided x is the sum of a potent and a square-zero nilpotent +of R. Furthermore, we shall say that the ring R is weakly periodic with nilpotence +index 2, provided every element of R is weakly periodic with nilpotence index 2. +Historically, the concept of weak periodicity arisen quite normally in the existing +on the subject literature. In fact, it was showed in [3] that an element x of a ring +R is periodic, i.e., xn = xm for some two different positive integers m, n, if and only +if x can be written as a sum of a potent element and a nilpotent element which +commute each other. Thus, by removing the ”commuting property”, it is rather +natural to consider the sum of such two elements. In addition, a ring R is called +weakly periodic if all its elements are weakly periodic. +On the other hand, Diesl defined in [5] the notion of a nil-clean ring R as the ring +for which, for every a ∈ R, there are an idempotent e and a nilpotent b such that +2010 Mathematics Subject Classification. Primary 15A23, 15B33; Secondary 16S50, 16U60. +Key words and phrases. companion matrices, fields, potents, square-zero nilpotents. +1 + +2 +P. DANCHEV AND A. POJAR +a = e+b. Moreover, Ye introduced in [7] the notion of semi-clean ring R as the ring +for which, for each a ∈ R, there are a potent c and a unit u such that a = c + u. +Henceforth, it is pretty obvious that the weakly periodic rings are situated between +nil-clean and semi-clean rings. So, they really need a detailed exploration to which +is devoted the present article being our basic motivation. Our major purpose is to +decide what is the power structure of the base field, provided that all companion +matrices over it are either commuting weakly periodic with index of nilpotence 2 or +just weakly periodic with index of nilpotence 2, respectively. In what follows, we +shall show that such a field is either finite or countably infinite, respectively. +2. Main Results +We divide our basic results into two subsections as follows: +2.1. The general case. We begin our work with the following plain but useful +technicality. +Lemma 2.1. Let n ≥ 1 be an integer. Then, the inclusion STn ⊆ SRn holds. +Proof. Let t ∈ STn be the trace of a potent companion matrix C such that Cm+1 = +C for some non-negative integer m ≥ 1. +Let λ be an eigenvalue of C. +Then, +λm+1 = λ. +It follows that either λ = 0 or λm = 1. +Suppose k is the number +of distinct non-zero distinct eigenvalues of C, say λ1, λ2, . . . , λk. Thus, it follows +that there exist non-negative integers α1, α2, . . . , αk, αk+1 with sum n, and non-zero +integers α1, α2, . . . , αk such that +t = α1λ1 + α2λ2 + . . . + αkλk + αk+1 · 0. +Therefore, +t = α1λ1 + α2λ2 + . . . + αkλk, +such that the inequality α1 + α2 + . . . + αk ≤ n is fulfilled. Hence, t is a sum of at +most n m-th roots of unity. Consequently, t ∈ SRn, as required. +□ +We continue with a few pivotal for our work statements. +Proposition 2.2. Let n ≥ 1 be a non-negative integer and C ∈ Mn(F) a companion +matrix over a field F. Then the following two claims are true: +(1) If C is weakly periodic with nilpotence index 2, then trace C ∈ SRn. +(2) If trace C ∈ STn, then C is weakly periodic with nilpotence index 2. +Proof. (1) Assume C = M + N with M potent and N a square-zero nilpotent such +that traceC /∈ SRn. So, it is obvious that traceC /∈ STn. Since trace N = 0, it +follows that trace C = trace M. Thus, one deduces that M is not similar to a +companion matrix. We shall now consider the next two cases: +Case 1: M is similar to a direct sum of two companion matrices C1 of size l +and C2 of size n − l, so that there exists an invertible matrix of size n over F +with M = U(C1 ⊕ C2)U −1. Hence, trace M = trace C = trace C1 + trace C2. +Since the direct sum C1 ⊕ C2 is obviously potent, one sees that trace C1 ∈ STl +and trace C2 ∈ STn−l. By virtue of Lemma 2.1, one gets that trace C1 ∈ SRl and +trace C2 ∈ SRn−l. Therefore, trace C1+trace C2 ∈ SRn, and hence trace M ∈ SRn. +But trace C = trace M, and so we obtain that trace C ∈ SRn. +Case 2: M is similar to a direct sum of u companion matrices, where 3 ≤ u ≤ n. +It can easily be seen that with induction on u we will have trace C ∈ SRn. + +COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS +3 +(2) Since trace C ∈ STn, there is a potent companion matrix C′ such that +trace C′ = trace C, whence (C −C′)2 = 0. In conclusion, C is weakly periodic with +nilpotence index 2, as claimed. +□ +Proposition 2.3. Let n ≥ 2 be an integer. If all n × n companion matrices are +weakly periodic with nilpotence index 2, then F ⊆ SRn. +Proof. This follows immediately with the aid of Proposition 2.2. +□ +Lemma 2.4. Let n ≥ 2 be an integer. Then the following containment is valid: +∪k≤n,d(m)≥kWm ⊆ ∪m∈N,m≥2Wm. +Proof. This follows directly from the easy fact that, for any non-negative integer +m ≥ 2, if m has at least k divisors, then m has at least one divisor. +□ +Lemma 2.5. Let n ≥ 1 be a non-negative integer. Then the following relation +holds: +SRn ⊆ ∪m∈N,m≥2Wm. +Proof. Letting t ∈ SRn, then there exist non-zero integers 1 ≤ k ≤ n and α1, α2, . . . αk +with sum not exceeding n as well as there are roots of unity λ1, λ2, ...λk and non- +negative integers m1, m2, . . . , mk such that λm1 +1 += λm2 +2 += . . . = λmk +k += 1 and such +that +t = α1λ1 + α2λ1 + . . . αkλk. +If we take m to be the common multiple of m1, m2, . . . , mk, then λm +i = 1 for every +i ∈ {1, 2, . . . , k}. Consequently, k ≤ m and +{λ1, λ2, . . . , λk} ⊆ {ǫa1 + ǫa2, . . . ǫak}, +where {a1, a2, . . . , ak} ⊆ {0, 1, . . . , m − 1} and ǫ is so that it generates all m-th +roots of unity. Without loss of generality, we can assume that a1 < a2 < . . . < ak. +Therefore, +t = α1ǫa1 + α2ǫa2 + . . . + αkǫak. +Suppose now that +f = α1Xa1 + α2Xa2 + . . . + αkXak. +Since α1, α2, . . . , αk are non-negative integers with sum not exceeding n, it will +follow that f ∈ Lm,n. We also will have the equality +t = f(ǫ). +Since ǫ is a primitive m-th root of unity, one extracts that ǫ is an eigenvalue for +Pτm, and hence f(ǫ) is an eigenvalue of f(Pτm). +Furthermore, since m is a common multiple k non-zero integers greater than +1, m1, m2, . . . , mk, it must be that m ≥ 2 is a non-negative integer. Now, as m is +a non-negative integer, m is a common multiple of number k of its divisors if, and +only if, m has at least k divisors and thus, in conjunction with Lemma 2.4, one +derives that +SRn ⊆ ∪k≤n,d(m)≥kWm ⊆ ∪m∈N,m≥2Wm, +as asserted. +□ +Lemma 2.6. Let m ≥ 2 and n ≥ 1 be integers. If ω ∈ Wm, then there exist two +integer multiples of unity u = u(ω) and α = α(ω) such that (ω − α)m = u. + +4 +P. DANCHEV AND A. POJAR +Proof. Given ω ∈ Wm. Then, for every f ∈ Lm,n, we have ω ∈ Spec(f(Pτm). It +follows now that ω is a root of the polynomial det(XIm − f(Pτm). But as f ∈ +Lm,n, there exist 1 ≤ k ≤ m and non-zero integers α1, α2, . . . , αk and non-negative +pairwise distinct integers a1, a2, . . . , ak with values at most n − 1 such that +f = α1Xa1 + α2Xa2 + . . . + αkXak. +Therefore, +f(Pτm) = α1(Pτm)a1 + α2(Pτm)a2 + . . . + αk(Pτm)ak. +Since τm is a cycle in Sm, and τm = (1 2 . . . m), it follows that τm is a product of +m − 1 transpositions, and hence τm and m obviously have different parities. +Furthermore, assuming m is even, it follows then that τm is odd, and thus τ ai +m +and ai have the same parity for every i ∈ {1, 2, . . . , k}. Now, let (bij)1≤i,j≤m = +XIm − f(Pτm) and αi = f(0). Therefore, +det(f(Pτm)−XIm) = (αi−XIm)m+ +� +σ̸=e,σeven +b1σ(1) · · · bmσ(m)− +� +σodd +b1σ(1) · · · bmσ(m) = += (αi − X)m + (−1)a1+1αm +1 + (−1)a2+1αm +2 + . . . + (−1)ak+1αm +k − (−1)ai+1αm +i . +Consequently, for even m, we obtain: +(αi − ω)m = (−1)a1αm +1 + (−1)a2αm +2 + . . . + (−1)akαm +k − (−1)aiαm +i . +Assuming that m is now odd, we then infer that τm is even, and so τ ai +m is even +for each i ∈ {1, 2, . . . , k}. Now, let (bij)1≤i,j≤m = f(Pτm) − XIm and αi = f(0) +Therefore, +det(f(Pτm)−XIm) = (αi −X)m + +� +σ̸=e,σeven +b1σ(1) · · · bmσ(m) − +� +σodd +b1σ(1) · · · bmσ(m) = += (αi − X)m + αm +1 + αm +2 + . . . + αm +k − 0 − αm +i . +Consequently, for odd m, we conclude: +(ω − αi)m = αm +1 + αm +2 + . . . + αm +k − αm +i . +Finally, there exist multiple integers of unity, say α = αi and u, such that (ω − +α)m = u, as required. +□ +The next claim is closely related to assertions from [4]. +Lemma 2.7. Every diagonalizable matrix over the finite Galois field GF(pl) for +some l ≥ 1 with no multiple eigenvalues is a non-derogatory potent matrix. +Proof. Since every element x ∈ GF(pl) satisfies the equation xpl = x, it follows that +each diagonal over GF(pl) is potent. Hence, any diagonalizable matrix over GF(pl) +is necessarily potent. +Since the matrix has no multiple eigenvalues, the algebraic multiplicity of every +eigenvalue is exactly 1. And since the matrix is diagonalizable, the geometric mul- +tiplicity of any eigenvalue equals to the algebraic multiplicity, so will be the 1 too. +Therefore, the matrix is non-derogatory as well. +In conclusion, the matrix is a non-derogatory potent matrix, as expected. +□ +The following technical claim from field theory is our key to establish the chief +result stated below. + +COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS +5 +Proposition 2.8. Each infinite field, which is an algebraic extension of its minimal +simple (finite) subfield, is an infinite countable union of its finite subfields, and thus +it is countable. In addition, these finite subfields can be taken of the sort GF(pli) +such that GF(pli) is contained in GF(pli+1) and li divides li+1 for each i ≥ 1. +Proof. Given F is an infinite field of characteristic p > 0, which is an algebraic +extension of its simple subfield Fp of p elements, and u is an arbitrary element of +F. Hence, each Fp(u) which means the extension of Fp generated by u, is a finite +extension of Fp and so it is a finite subfield of F. Precisely, all finite subfields of F +are of this type, because for the finite extensions of Fp is valid the classical theorem +for the ”primitive element”. We also know that Fp(u) ∼= Fp[X]/(gu(X)), where X +is a transcendental element over Fp, and gu(X) is the minimal polynomial of u over +Fp. To ensure an uniqueness of gu(X), as usual we will assume that it is normed, +that is, its chief coefficient is exactly 1. Also, the degree [Fp(u) : Fp] of the extension +Fp(u)/Fp coincides with the degree of the polynomial gu(X). +Furthermore, the extensions Fp(u)/Fp are normal with cyclic Galois group. More- +over, for every positive integer n, at most (exactly) one field of the form Fp(u) is an +extension of the field Fp of degree n. In other words, there is a bijection between the +set of all subfields of F of the kind Fp(u) onto some subset of the set N consisting of +all natural numbers. Concretely, this subset consists of those elements of N which +are equal to the degree [Fp(u) : Fp] for some element u ∈ F. Since any subset of +N is either finite or infinitely countable, our arguments so far allow us to conclude +that F can be presented as a finite or countably infinite union of finite (sub)fields. +This union is properly countable uniquely when F is an infinitely countable field, +as pursued. +Next, by what we have already shown, the base field F is countable, and hence +the elements of F can be linearly ordered as f1, . . . , fn, . . . . Set Fn = Fp(f1, . . . , fn) +for every n ∈ N. It is now easily inspected that F is equal to the countable union +of all its subfields Fn, where n ≥ 1. Besides, it easily follows that Fn/Fp is a finite +extension, because it is simultaneously an algebraic extension and finitely generated. +Likewise, Fn is obviously a subfield of Fn+1 for each natural n ≥ 1. +Thus, we +arrive at the conclusion that Fn is a finite field of order pln, where ln = [Fn : Fp]. +Since Fn ≤ Fn+1, one deduces that ln+1 is divided by ln and that the equality +ln+1/ln = [Fn+1 : Fn] holds for all n ∈ N, as desired. +As for the second part that such a field is necessarily is now an immediate con- +sequence of the first one. +□ +The next comments are worthwhile to explain that the conditions in the previous +statement cannot be weakened. +Remark 2.9. Knowing that Fp is the simple (finite) field of prime characteristic +p, routine arguments show that the field Fp(X) of rational functions of the variable +X with coefficients from Fp, which is actually a transcendental extension of Fp, is +an example of a countable field of characteristic p which cannot be constructed as +a countable union of finite fields. Moreover, arguing in the same manner, one can +see that the field of rational numbers, Q, also cannnot be presented as a countable +union of finite fields. +We now arrive at our first main result in the present paper. Specifically, the +following is true: + +6 +P. DANCHEV AND A. POJAR +Theorem 2.10. Let n ≥ 1 be an integer and let F be a field. Then the following +two conditions are equivalent: +(1) Every n × n companion matrix over F is the sum of a potent and a square- +zero nilpotent over F. +(2) F is a countable field of positive characteristic, which is an algebraic exten- +sion of its minimal simple (finite) subfield, with all subfields of order greater +than n. +Proof. (1) +=⇒ +(2). +Assume that each n × n companion matrix C over F is +weakly periodic with nilpotence index 2. +Referring to Proposition 2.5, we have +F ⊆ ∩m∈N,m≥2Wm. But, for every ω ∈ Wm, there exist integer multiples of unity, +say α and u, such that (ω − α)m = u by owing to Lemma 2.6. +Assume in a way of contradiction that F has zero characteristic. It then follows +that each non-zero integer multiple of unity s is invertible. Set ω = r·s−1, and let r +and ω be integers multiple of unity. Hence, r · s−1 ∈ ∩m∈N,m≥2Wm. Fix an m ≥ 2. +Thus, there exist integer multiples of unity α and u such that (r · s−1 − α)m = u. +By the classical Newton’s binomial formula, we find that +(r · s−1)m + (−1)1 +�m +1 +� +(r · s−1)m−1α + (−1)2 +�m +2 +� +(r · s−1)m−2α2 + . . . ++(−1)m−1 +� +m +m − 1 +� +(r · s−1)1αm−1 + αm = u +and so +rm + (−1)1 +�m +1 +� +rm−1sα + (−1)2 +�m +2 +� +rm−2s2α2+ +. . . + (−1)m−1r1sm−1αm−1 + αm · sm = u · sm, +which is equivalent to +rm = ((−1)0 +�m +1 +� +rm−1 · α + (−1)1 +�m +2 +� +rm−2α2s+ +. . . + (−1)m−2rαm−1sm−2 + αm · sm−1 + usm−1) · s. +As by assumption the field is of characteristic zero, we may consider with no harm of +generality that the above equation is satisfied in Z, whence it follows that s divides +rm for any s ∈ Z∗, r ∈ Z, which is manifestly false; for example, by considering +s = 2 and r = 3. Hence, the characteristic of the field has to be some non-zero +prime, say p > 0. Besides, Proposition 2.3 ensures that the field is of necessity +countable, because SRn is countable and F ⊆ SRn. +Now, assume that the containment GF(pl) ⊆ F is fulfilled for some integer l +satisfying the inequalities 2t ̸= pl − 1 ≤ n − 1 for any non-negative integer t. Thus, +there exists an odd proper divisor d of pl − 1. +Further, take q a non-negative +odd integer such that q and pl − 1 are co-prime. +Then, m = d · q is odd and +d = gcd(pl − 1, m). Let the equation xm − 1 = 0 holds over F. It has d solutions +in GF(pl), namely 1, h, h2, . . . , hd−1, for h = g +pl−1 +d , where g is a generator of the +multiplicative cyclic group of the field GF(pl). Since hd = 1, we have +1 + h + h2 + . . . + hd−1 = 0. + +COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS +7 +Therefore, +−1 = h + h2 + . . . + hd−1. +So, −1 is the sum of d − 1 m-th roots of unity with each of them not equal to 1. +Since d − 1 < d ≤ pl − 1 ≤ n − 1, we detect that +−1 = α1h + α2h2 + . . . + αd−1hd−1 +with +α1 = α2 = . . . = αd−1 = 1, +and hence it follows from the proof of Lemma 2.6 that +(−1 − 0)m = αm +1 + αm +2 + . . . + αm +d−1 − 0. +Also, (−1)m = d − 1, and since m is odd, we deduce that p divides d. But d divides +pl − 1. So, by the transitivity law, p must divide pl − 1 and thus p = 1, which is a +contradiction. +Therefore, we have GF(pl) ⊆ F for some integer l with 2t = pl − 1 ≤ n − 1 for +some non-negative integer t. It follows now that p−1 and r = pl−1+pl−2+. . .+p+1 +are powers of 2 and p − 1 divides r. But since p − 1 divides r − l, we receive that +p − 1 divides l. So, since p ̸= 2, we deduce that l = 2u with u a positive integer. +That is why, (pu − 1)(pu + 1) = 2t and hence pu − 1 divides pu + 1. Finally, pu − 1 +divides 2 and since p ̸= 2, we obtain p = 3 and u = 1. +So, l = 2u = 2 and GF(pl) = GF(32). Let g be a generator of the multiplicative +group of GF(9). Since gcd(32 − 1, 12) = 4, there are four 12-th roots of unity in +the field GF(32). They are ǫ0, ǫ6, ǫa3 and ǫa4, where ǫ is a primitive 12-th root of +unity, whereas a3 and a4 are positive distinct integers less then 12 and not equal to +6. However, as 32−1 +4 += 2, we have +{1, −1, ǫa3, ǫa4} = {1, g2, (g2)2, (g2)3}. +Since (g2)4 = 1, it follows that +1 + g2 + (g2)2 + (g2)3 = 0. +Consequently, +−1 = −1 + ǫa3 + ǫa4. +We thus derive two things: ǫa3 = −ǫa4, and −1 is the sum of at most n 12−th roots +of unity, because 3 < 32 ≤ n. Therefore, imitating the proof of Lemma 2.6, taking +into account that 12 is even, it follows that +(0 − (−1))12 = (−1)6 · 112 + (−1)a3 · 112 + (−1)a4 · 112. +We thus inspect that the numbers a3 and a4 cannot be simultaneously odd or +simultaneously even. In this way, without loss of generality, we can assume that a3 is +even and a4 is odd, and since ǫa3 = −ǫa4, we obtain that (−ǫ)a3 = (−ǫ)a4. However, +because ǫ is not lying in the set {−1, 0, 1}, while a3 and a4 are positive integers less +then 12, it follows automatically that a3 = a4, which is a new contradiction. +Now, in order to demonstrate that F is an algebraic extension of its minimal +simple (finite) subfield, we will prove that every non-zero element of F is a root +of unity. +In fact, owing to Proposition 2.3, we have that F ∈ SRn, the set of +all sums of at most n roots of unity. Also, according to Lemma 2.5, we get that +SRn ⊆ ∪m∈N,m≥2Wm, whereas Lemma 2.6 enables us that if x ∈ Wm, then there +exist u ∈ Fp and α ∈ Fp such that (x − α)m = u. Likewise, the proofs of the +mentioned lemmas tell us that if x is the sum of at most n m-th roots of unity +within there are exactly α values of 1, then (x − α)m ∈ Fp. So, we may assume + +8 +P. DANCHEV AND A. POJAR +that x = α. If x ̸= 0, then xp−1 = 1 and thus x is a root of unity. If, however, +x is the sum of at most n roots of unity within there are no values of 1, then +(x − α)m ∈ Fp − {0}. Hence, x = α + y with αp−1 = 1 and ym(p−1) = 1. If α ̸= 1, +then x is the sum of two roots of unity not equal to 1. So, xm(p−1) ∈ Fp − {0} and, +therefore, xm(p−1)2 = 1 Thus, x is a root of unity. Now take α = 1. But as +x = 1 · 1 + y = (p − 1) · (−1) + y = ((n − 1) · (−1) + y) + (p − n) · (−1), +we have that (n − 1)(−1) + y is the sum of n − 1 roots of unity equal to (−1) and +one m(p − 1) − th root of unity not equal to 1. It follows now that ((n − 1) · (−1) + +y − 1 · 0)m(p−1) = 1. Moreover, if (p − n) · (−1) = 1, then p divides n − 1. But +n − 1 ≤ p − 1, so that p ≤ p − 1 which is false. By this contradiction we derive +that x is the sum of two m(p − 1)-th roots of unity not equal to 1. Consequently, +xm(p−1) ∈ Fp − {0}, whence xm(p−1)2 = 1. Finally, x is a root of unity, as promised. +In conclusion, F is a countable field of positive characteristic, which is an algebraic +extension of its minimal simple (finite) subfield, with all subfields of order greater +than n, as desired. +(2) +=⇒ +(1). Let GF(pl) be contained in F. Take C1 to be an n × n com- +panion matrix over GF(pl). Then, as pl ≥ n + 1, it follows by the application of +the main result from [4] that we may decompose C1 = D1 + N1, where D1 is a +diagonalizable matrix over GF(pl) with no multiple eigenvalues and N1 is a nilpo- +tent matrix of nilpotence index 2 over GF(pl). Now, Lemma 2.7 helps us to have +that D1 is a non-derogatory potent matrix, and since trace N = 0 we can get that +trace C1 = trace D1. Therefore, any element in GF(pl) can be the trace of a non- +derogatory potent matrix. Hence, each element in GF(pl) can be the trace of a +potent companion matrix. +Now, according to Proposition 2.8, every countable field of positive characteristic, +which is an algebraic extension of its minimal simple (finite) subfield, is a countable +union of its finite subfields, it follows at once that every element in F can be the +trace of an n × n potent companion matrix over F. Thus, letting C be an arbitrary +n × n companion matrix over F. Then, trace C ∈ STn. In conclusion, applying +Proposition 2.2(2), one infers that C can be written as a sum of a potent matrix +and a nilpotent matrix of order 2 over F, as wanted. +□ +We now come to the case when the potent and square-zero matrices commute +each other. +2.2. The commuting case. We will attack here the commuting case of weakly +periodic with nilpotence index 2 companion matrices, that is, the existing potent +and square-zero nilpotent matrices commute each other. To that aim, we first need +the following two technical conventions. +Lemma 2.11. Let n be a positive integer and F a field. If, for every n×n companion +matrix C over F, there exist an integer t > 1, a potent matrix P such that P = P t +and a square-zero nilpotent N such that C = P + N with PN = NP, then the +following two points are true: +(1) If C is invertible, then χC divides (Xt−1 − 1)2. +(2) If λ1, λ2, . . . , λn are non-zero elements of F, then there exists an integer +t > 1 such that λt−1 +i += 1 for every i ∈ {1, 2, . . . , n}. +Proof. (1) Since C = P + N and PN = NP, it follows that CP = PC and +CN = NC. But we have (C − N)t = C − N. Since CN = NC, we can apply + +COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS +9 +Newton’s binomial formula to argue that there exists an n × n matrix over F such +that +Ct − tCN + MN 2 = C − N. +But as N 2 = 0, we obtain that +C(Ct−1 − tN) = C − N. +Multiplying with Ct−1 + tN, we get that +C(C2t−2 − t2N 2) = Ct + tCN − Ct−1N + tN 2, +and using that N 2 = 0, we write the equality +C2t−1 = Ct + (tC − Ct−1)N. +But C is invertible, and so +C2t−2 − Ct−1 = (tIdn − Ct−2)N. +Now, bearing in mind that CN = NC and N 2 = 0, we infer +((Ct−1)2 − Ct−1)2 = 0, +and since C is invertible, we conclude +(Ct−1 − 1)2 = 0. +Therefore, the minimal polynomial of C divides (Xt−1 − 1)2. +But the minimal +polynomial of C is the characteristic polynomial, say χC, of C. Summarizing all +the information so far, χC divides (Xt−1 − 1)2, as promised. +(2) Just take C to be the companion matrix with eigenvalues λ1, λ2, . . . , λn and +apply the preceding point (1) along with the fact that λ1, λ2, . . . , λn are from F. +□ +Lemma 2.12. Let n be a positive integer and F a field. If, for every n×n companion +matrix C over F, there exist an integer t > 1, a potent matrix P such that P = P t +and a square zero nilpotent N such that C = P + N with PN = NP, then the +following two conditions are valid: +(1) If C is invertible, then there exists a polynomial q ∈ F[X] such that χC +divides (q(X) − X)2. +(2) If λ1, λ2, . . . , λn are non-zero elements of F, then there exists a polynomial +q ∈ F[X] such that q(λi) = λi for every i ∈ {1, 2, . . . , n}. +Proof. (1) Since C = P + N and PN = NP, it follows that CP = PC and CN = +NC. It is well known that all matrices that commute with a companion matrix C +can be interpreted just as polynomials in C over F. So, there exists q ∈ F[X] with +C = q(C) + N. Now, since N 2 = 0, it follows that (q(C) − C)2 = 0. Therefore, +the minimal polynomial of C obviously divides (q(X) − X)2. +But the minimal +polynomial of C is the characteristic polynomial, say χC, of C. Summarizing all +the information thus far, χC divides (q(X) − X)2, as asked for. +(2) Just take C to be the companion matrix with eigenvalues λ1, λ2, . . . , λn and +employ the preceding condition (1) together with the fact that λ1, λ2, . . . , λn are +from F. +□ +The following claim from number theory is a well-known folklore fact, but we +state it here only for the sake of completeness and the reader’s convenience. +Lemma 2.13. Let a, b, c be three positive integers. Then, gcd(bc, a) divides gcd(b, a)· +gcd(c, a). + +10 +P. DANCHEV AND A. POJAR +Proof. Let we set f = gcd(a, bc), f1 = gcd(a, b) and f2 = gcd(a, c). Then, there +exist integers s1, s2, t1, t2 such that +f1 = s1a + t1b +and +f2 = s2a + t2c. +Thus, +f1f2 = s1s2a2 + s1t2ac + t1s2ba + t1t2bc. +But f divides a and f divides bc, so f divides a2, f divides ac, f divides ba and +f divides bc. Hence, f divides f1f2 whence gcd(bc, a) divides gcd(b, a) · gcd(c, a), as +formulated. +□ +The next comments are needed to explain the complicated situation in the com- +muting case. +Remark 2.14. Let λ1, λ2, . . . , λn be distinct non-zero elements of F. By Lemma +2.12(2), there exists q ∈ F[X] such that q(λi) = λi for every i ∈ {1, 2, . . . , n}. +Take GF(pl1) such that λ1, λ2, . . . , λn are in GF(pl1), GF(pl2) and such that q ∈ +GF(pl2)[X] with l = max(l1, l2). Then, λ1, λ2, . . . , λn are in GF(pl), while q ∈ +GF(pl)[X]. +Furthermore, since λ1, λ2, . . . , λn are in F, it follows from Lemma 2.11(2) that +there exists a non-negative integer t > 1 such that λt−1 +i += 1 for every i ∈ {1, 2, . . . , n}. +Now, since λ1, λ2, . . . , λn are in GF(pl), we can infer that +{λ1, λ2, . . . , λn} ⊆ {h, h2, . . . , hd = 1}, +where h = g +pl−1 +d +with g a generator of the multiplicative group of GF(pl), and +d = gcd(pl−1, t−1). Therefore, n ≤ d and there exist {j1, j2, . . . , jn} ⊆ {1, 2, . . . , d} +such that λi = hji. Finally, we extract that q(hji) = hji for every i ∈ {1, 2, . . . , n} +with hd = 1, as expected. +The next example illustrates some concrete aspects of our calculating manipula- +tions. +Example 2.15. If in the previous Remark 2.14 we take λ1 = gi and λ2 = gi+1 +with i ∈ {1, 2, . . . , pl − 2}, then one inspects that +{gi, gi+1} ⊆ {g +pl−1 +d , g2 pl−1 +d , . . . , gd pl−1 +d }. +Therefore, +{i, i + 1} ⊆ {pl − 1 +d +, 2pl − 1 +d +, . . . , dpl − 1 +d +}. +Hence, the ordinary fraction pl−1 +d +is a common divisor of both i and i + 1, so it is +necessarily 1. We thus have now that pl − 1 = d = gcd(pl − 1, t − 1). In conclusion, +pl − 1 divides t − 1, as desired to demonstrate. +We are now ready to establish our second main result. +Theorem 2.16. Suppose n ≥ 1 is an integer and F is a field. If all companion +n×n matrices over F are the sum of a potent matrix and a square-zero matrix over +F which matrices commute each other, then F is a finite field of order greater than +n. + +COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS +11 +Proof. Suppose n ≥ 1 is an integer, F is a field and all companion n×n matrices over +F are the sum of a potent matrix and a square-zero matrix over F which commute. +Then, by Theorem 2.10, we have that F is a countable field of positive characteristic +(which is also an algebraic extension of a finite field). Assume now the contrary +that F is infinite. Thus, the second part in the statement of Proposition 2.8 applies +to write that F = ∪∞ +i=1GF(pli), where li divides li+1 for every positive integer i. +Therefore, there exists the sequence of integers greater than 1, say (ci)i≥1 such that +li+1 = li · ci and (li)i≥1 is a strictly increasing infinite sequence of positive integers. +Take g1 to be the generator of the multiplicative group of GF(pl1) and put +λ1 = g +[ pl1 −1 +n +] +1 +, λ2 = g +2·[ pl1 −1 +n +] +1 +, . . . , λn = g +n·[ pl1 −1 +n +] +1 +. +Consequently, Lemma 2.11 allows us to have the existence of a positive integer t > 1 +such that +λt−1 +1 += λt−1 +2 += · · · = λt−1 +n += 1. +Let C be the companion matrix with eigenvalues λ1, λ2, . . . , λn. +Note that the +integer t above has the property that C = P + N with P t = P, N 2 = 0 and +PN = NP. If, eventually, there are more than one such decompositions of C, then +we can take t to be the minimum of such integers t’s. So, t can be chosen to be a +fixed uniquely determined integer having the mentioned property. +Furthermore, since we are working in GF(pl1), Remark 2.14 leads us to the +relation +{g +[ pl1 −1 +n +] +1 +, g +2·[ pl1 −1 +n +] +1 +, . . . , g +n·[ pl1 −1 +n +] +1 +} ⊆ {g +pl1 −1 +d1 +1 +, g +2 pl1 −1 +d1 +1 +, . . . , g +d1 +pl1 −1 +d1 +1 +}, +where di = gcd(pli − 1, t − 1). Take ji ∈ {1, 2, . . . , di} such that +g +[ pl1 −1 +n +] +1 += g +j1· pl1 −1 +d1 +1 +, +and since the order of g1 in the multiplicative group of GF(pl1) is pl1 − 1, it follows +that +[pl1 − 1 +n +] = j1 · pl1 − 1 +d1 +. +Analogously, we obtain that +[pl1 − 1 +n +] = ji · pli − 1 +di +, +for any positive integer i. Therefore, +ji · pli − 1 +di += ji+1 · pli+1 − 1 +di+1, +and so +ji +ji+1 += +di +di+1 +· (pli)ci − 1 +pli − 1 +. +Take si = (pli)ci−1 + (pli)ci−2 + · · · + pli + 1. It thus follows that +ji +ji+1 += +gcd(pli − 1, t − 1) +gcd((pli)ci − 1, t − 1) · si. +Also, by Lemma 2.13 we have that there exists a strictly positive integer ki such +that +gcd((pli)ci − 1, t − 1) = gcd(pli − 1, t − 1) · gcd(si, t − 1)) +ki +. + +12 +P. DANCHEV AND A. POJAR +Now, we obtain that +ji +ji+1 += ki · +si +gcd(si, t − 1). +However, since gcd(si, t − 1) divides si, it follows that ji+1 divides ji for every +positive integer i ≥ 1. But ji ≥ 1 and thus there exists i0 ≥ 1 such that ji = ji+1 +for every i ≥ i0. So, we obtain +ki · +si +gcd(si, t − 1) = 1, +which forces si = gcd(si, t − 1). Hence, si divides t − 1 and so si ≤ t − 1. But +pli < si and then pli < t − 1, for every i ≥ i0, which is in sharp contradiction with +the fact that (li)i≥1 is a strictly increasing infinite sequence of positive integers. +In conclusion, one has that F is finite, as claimed. +We next once again apply +Theorem 2.10 to get that the order of the field is greater than n, as asserted. +□ +In the spirit of the last result, we pose the following conjecture. +Conjecture: Given n ≥ 1 is an integer and F is a field. Then all companion n × n +matrices over F are the sum of a potent matrix and a square-zero matrix over F +which matrices commute each other if, and only if, F is a finite (and hence potent) +field and n = 1. +On the other side, in regard to [1] and [6], we close our work with the following +question of some interest. +Problem 2.17. Suppose that D is a division ring and n ≥ 1 is an integer. Does +it follow that the matrix ring Mn(D) is weakly periodic if, and only if, D is a finite +(and hence potent) field? +Funding: The scientific work of Peter V. Danchev was partially supported by the +Bulgarian National Science Fund under Grant KP-06 No 32/1 of December 07, 2019 +and by the Junta de Andaluc´ıa, FQM 264. +References +[1] S. Breaz, G. Calugareanu, P. Danchev and T. Micu, Nil-clean matrix rings, Linear Algebra & +Appl. 439(10) (2013), 3115–3119. +[2] S. Breaz and G.C. Modoi, Nil-clean companion matrices, Linear Algebra & Appl. 489(2) +(2016), 50–60. +[3] J. Cui and P. Danchev, Some new characterizations of periodic rings, J. Algebra & Appl. +19(12) (2020). +[4] P. Danchev, E. Garc´ıa and M. G. Lozano, Decompositions of matrices into diagonalizable and +square-zero matrices, Linear & Multilinear Algebra 71(3) (2023). +[5] A.J. Diesl, Nil-clean rings, J. Algebra 383(11) (2013), 197–211. +[6] M.T. Kosan, T.-K. Lee and Y. Zhou, When is every matrix over a division ring a sum of an +idempotent and a nilpotent?, Linear Algebra & Appl. 450(11) (2014), 7–12. +[7] Y. Ye, Semi-clean rings, Commun. Algebra 31(11) (2003), 5609–5625. +Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad. +G. Bonchev” str., bl. 8, 1113 Sofia, Bulgaria +Email address: danchev@math.bas.bg; pvdanchev@yahoo.com +The Technical University of Cluj-Napoca, Departament of Mathematics, Str. Mem- +orandumului 28, 400114, Cluj-Napoca, Romania +Email address: andradapojar@gmail.com + diff --git a/YtE5T4oBgHgl3EQfdg-i/content/tmp_files/load_file.txt b/YtE5T4oBgHgl3EQfdg-i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b8ef423893b6c788472e219d9189644146b011c --- /dev/null +++ b/YtE5T4oBgHgl3EQfdg-i/content/tmp_files/load_file.txt @@ -0,0 +1,666 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf,len=665 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='05612v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='RA] 13 Jan 2023 COMPANION WEAKLY PERIODIC MATRICES OVER FINITE AND COUNTABLE FIELDS PETER DANCHEV AND ANDRADA POJAR Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We explore the situation where all companion n × n matrices over a field F are weakly periodic of index of nilpotence 2 and prove that this can be happen uniquely when F is a countable field of positive characteristic, which is an algebraic extension of its minimal simple (finite) subfield, with all subfields of order greater than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In particular, in the commuting case, we show even that F is a finite field of order greater than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Our obtained results somewhat generalize those obtained by Breaz-Modoi in Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Algebra & Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Introduction and Principalities Let F be a field and n an arbitrary non-negative integer, say n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We denote by Mn(F) the matrix ring consisting of all squared n × n matrices over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' For a non-negative integer m ≥ 2, we denote by τ the cycle τm = (1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' m) ∈ Sm, and by Pτm the m × m permutation matrix with only 0s and 1s over F, that is, Pτm = (aij)1≤i,j≤m, where (aij) = 1 if j = τm(i), and (aij) = 0 if j ̸= τm(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' The matrix M ∈ Mn(F) is called potent if there exists an integer k ≥ 1 such that Mk+1 = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, let STn the set of traces of all potent companion matrices C ∈ Mn(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' By a root of unity, we denote a root of the polynomial over F which is of the form g = Xi − 1, where i is an integer with i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, let SRn be the set of sums of at most n-th roots of unity which are not necessarily distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Likewise, we denote by Lm,n the set of polynomials having degree at most m and with non-negative integer multiples of unity coefficients such that their sum is not exceeding n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We also denote Wm = ∪f∈Lm,nSpec(f(Pτm)), and the symbol d(m) stands for the number of non-negative divisors of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' An element x ∈ R is said to be potent if there exists a non- negative integer q ≥ 1 with xq+1 = a, and x ∈ R is said to be weakly periodic with nilpotence index 2, provided x is the sum of a potent and a square-zero nilpotent of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Furthermore, we shall say that the ring R is weakly periodic with nilpotence index 2, provided every element of R is weakly periodic with nilpotence index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Historically, the concept of weak periodicity arisen quite normally in the existing on the subject literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In fact, it was showed in [3] that an element x of a ring R is periodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=', xn = xm for some two different positive integers m, n, if and only if x can be written as a sum of a potent element and a nilpotent element which commute each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, by removing the ”commuting property”, it is rather natural to consider the sum of such two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In addition, a ring R is called weakly periodic if all its elements are weakly periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' On the other hand, Diesl defined in [5] the notion of a nil-clean ring R as the ring for which, for every a ∈ R, there are an idempotent e and a nilpotent b such that 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Primary 15A23, 15B33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Secondary 16S50, 16U60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' companion matrices, fields, potents, square-zero nilpotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' DANCHEV AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' POJAR a = e+b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Moreover, Ye introduced in [7] the notion of semi-clean ring R as the ring for which, for each a ∈ R, there are a potent c and a unit u such that a = c + u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Henceforth, it is pretty obvious that the weakly periodic rings are situated between nil-clean and semi-clean rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, they really need a detailed exploration to which is devoted the present article being our basic motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Our major purpose is to decide what is the power structure of the base field, provided that all companion matrices over it are either commuting weakly periodic with index of nilpotence 2 or just weakly periodic with index of nilpotence 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In what follows, we shall show that such a field is either finite or countably infinite, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Main Results We divide our basic results into two subsections as follows: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' The general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We begin our work with the following plain but useful technicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let n ≥ 1 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, the inclusion STn ⊆ SRn holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let t ∈ STn be the trace of a potent companion matrix C such that Cm+1 = C for some non-negative integer m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let λ be an eigenvalue of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, λm+1 = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It follows that either λ = 0 or λm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Suppose k is the number of distinct non-zero distinct eigenvalues of C, say λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, it follows that there exist non-negative integers α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , αk, αk+1 with sum n, and non-zero integers α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , αk such that t = α1λ1 + α2λ2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αkλk + αk+1 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, t = α1λ1 + α2λ2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αkλk, such that the inequality α1 + α2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αk ≤ n is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, t is a sum of at most n m-th roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Consequently, t ∈ SRn, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ We continue with a few pivotal for our work statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let n ≥ 1 be a non-negative integer and C ∈ Mn(F) a companion matrix over a field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then the following two claims are true: (1) If C is weakly periodic with nilpotence index 2, then trace C ∈ SRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (2) If trace C ∈ STn, then C is weakly periodic with nilpotence index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (1) Assume C = M + N with M potent and N a square-zero nilpotent such that traceC /∈ SRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, it is obvious that traceC /∈ STn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since trace N = 0, it follows that trace C = trace M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, one deduces that M is not similar to a companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We shall now consider the next two cases: Case 1: M is similar to a direct sum of two companion matrices C1 of size l and C2 of size n − l, so that there exists an invertible matrix of size n over F with M = U(C1 ⊕ C2)U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, trace M = trace C = trace C1 + trace C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since the direct sum C1 ⊕ C2 is obviously potent, one sees that trace C1 ∈ STl and trace C2 ∈ STn−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' By virtue of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='1, one gets that trace C1 ∈ SRl and trace C2 ∈ SRn−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, trace C1+trace C2 ∈ SRn, and hence trace M ∈ SRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But trace C = trace M, and so we obtain that trace C ∈ SRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Case 2: M is similar to a direct sum of u companion matrices, where 3 ≤ u ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It can easily be seen that with induction on u we will have trace C ∈ SRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS 3 (2) Since trace C ∈ STn, there is a potent companion matrix C′ such that trace C′ = trace C, whence (C −C′)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In conclusion, C is weakly periodic with nilpotence index 2, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let n ≥ 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If all n × n companion matrices are weakly periodic with nilpotence index 2, then F ⊆ SRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' This follows immediately with the aid of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let n ≥ 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then the following containment is valid: ∪k≤n,d(m)≥kWm ⊆ ∪m∈N,m≥2Wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' This follows directly from the easy fact that, for any non-negative integer m ≥ 2, if m has at least k divisors, then m has at least one divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let n ≥ 1 be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then the following relation holds: SRn ⊆ ∪m∈N,m≥2Wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Letting t ∈ SRn, then there exist non-zero integers 1 ≤ k ≤ n and α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' αk with sum not exceeding n as well as there are roots of unity λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='λk and non- negative integers m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , mk such that λm1 1 = λm2 2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' = λmk k = 1 and such that t = α1λ1 + α2λ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' αkλk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If we take m to be the common multiple of m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , mk, then λm i = 1 for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Consequently, k ≤ m and {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λk} ⊆ {ǫa1 + ǫa2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' ǫak}, where {a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , ak} ⊆ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , m − 1} and ǫ is so that it generates all m-th roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Without loss of generality, we can assume that a1 < a2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' < ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, t = α1ǫa1 + α2ǫa2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αkǫak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Suppose now that f = α1Xa1 + α2Xa2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αkXak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , αk are non-negative integers with sum not exceeding n, it will follow that f ∈ Lm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We also will have the equality t = f(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since ǫ is a primitive m-th root of unity, one extracts that ǫ is an eigenvalue for Pτm, and hence f(ǫ) is an eigenvalue of f(Pτm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Furthermore, since m is a common multiple k non-zero integers greater than 1, m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , mk, it must be that m ≥ 2 is a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, as m is a non-negative integer, m is a common multiple of number k of its divisors if, and only if, m has at least k divisors and thus, in conjunction with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='4, one derives that SRn ⊆ ∪k≤n,d(m)≥kWm ⊆ ∪m∈N,m≥2Wm, as asserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let m ≥ 2 and n ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If ω ∈ Wm, then there exist two integer multiples of unity u = u(ω) and α = α(ω) such that (ω − α)m = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' DANCHEV AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' POJAR Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Given ω ∈ Wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, for every f ∈ Lm,n, we have ω ∈ Spec(f(Pτm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It follows now that ω is a root of the polynomial det(XIm − f(Pτm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But as f ∈ Lm,n, there exist 1 ≤ k ≤ m and non-zero integers α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , αk and non-negative pairwise distinct integers a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , ak with values at most n − 1 such that f = α1Xa1 + α2Xa2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αkXak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, f(Pτm) = α1(Pτm)a1 + α2(Pτm)a2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αk(Pτm)ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since τm is a cycle in Sm, and τm = (1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' m), it follows that τm is a product of m − 1 transpositions, and hence τm and m obviously have different parities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Furthermore, assuming m is even, it follows then that τm is odd, and thus τ ai m and ai have the same parity for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, let (bij)1≤i,j≤m = XIm − f(Pτm) and αi = f(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, det(f(Pτm)−XIm) = (αi−XIm)m+ � σ̸=e,σeven b1σ(1) · · · bmσ(m)− � σodd b1σ(1) · · · bmσ(m) = = (αi − X)m + (−1)a1+1αm 1 + (−1)a2+1αm 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + (−1)ak+1αm k − (−1)ai+1αm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Consequently, for even m, we obtain: (αi − ω)m = (−1)a1αm 1 + (−1)a2αm 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + (−1)akαm k − (−1)aiαm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Assuming that m is now odd, we then infer that τm is even, and so τ ai m is even for each i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, let (bij)1≤i,j≤m = f(Pτm) − XIm and αi = f(0) Therefore, det(f(Pτm)−XIm) = (αi −X)m + � σ̸=e,σeven b1σ(1) · · · bmσ(m) − � σodd b1σ(1) · · · bmσ(m) = = (αi − X)m + αm 1 + αm 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αm k − 0 − αm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Consequently, for odd m, we conclude: (ω − αi)m = αm 1 + αm 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αm k − αm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Finally, there exist multiple integers of unity, say α = αi and u, such that (ω − α)m = u, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ The next claim is closely related to assertions from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Every diagonalizable matrix over the finite Galois field GF(pl) for some l ≥ 1 with no multiple eigenvalues is a non-derogatory potent matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since every element x ∈ GF(pl) satisfies the equation xpl = x, it follows that each diagonal over GF(pl) is potent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, any diagonalizable matrix over GF(pl) is necessarily potent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since the matrix has no multiple eigenvalues, the algebraic multiplicity of every eigenvalue is exactly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' And since the matrix is diagonalizable, the geometric mul- tiplicity of any eigenvalue equals to the algebraic multiplicity, so will be the 1 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, the matrix is non-derogatory as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In conclusion, the matrix is a non-derogatory potent matrix, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ The following technical claim from field theory is our key to establish the chief result stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS 5 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Each infinite field, which is an algebraic extension of its minimal simple (finite) subfield, is an infinite countable union of its finite subfields, and thus it is countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In addition, these finite subfields can be taken of the sort GF(pli) such that GF(pli) is contained in GF(pli+1) and li divides li+1 for each i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Given F is an infinite field of characteristic p > 0, which is an algebraic extension of its simple subfield Fp of p elements, and u is an arbitrary element of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, each Fp(u) which means the extension of Fp generated by u, is a finite extension of Fp and so it is a finite subfield of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Precisely, all finite subfields of F are of this type, because for the finite extensions of Fp is valid the classical theorem for the ”primitive element”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We also know that Fp(u) ∼= Fp[X]/(gu(X)), where X is a transcendental element over Fp, and gu(X) is the minimal polynomial of u over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' To ensure an uniqueness of gu(X), as usual we will assume that it is normed, that is, its chief coefficient is exactly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Also, the degree [Fp(u) : Fp] of the extension Fp(u)/Fp coincides with the degree of the polynomial gu(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Furthermore, the extensions Fp(u)/Fp are normal with cyclic Galois group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' More- over, for every positive integer n, at most (exactly) one field of the form Fp(u) is an extension of the field Fp of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In other words, there is a bijection between the set of all subfields of F of the kind Fp(u) onto some subset of the set N consisting of all natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Concretely, this subset consists of those elements of N which are equal to the degree [Fp(u) : Fp] for some element u ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since any subset of N is either finite or infinitely countable, our arguments so far allow us to conclude that F can be presented as a finite or countably infinite union of finite (sub)fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' This union is properly countable uniquely when F is an infinitely countable field, as pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Next, by what we have already shown, the base field F is countable, and hence the elements of F can be linearly ordered as f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , fn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Set Fn = Fp(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , fn) for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It is now easily inspected that F is equal to the countable union of all its subfields Fn, where n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Besides, it easily follows that Fn/Fp is a finite extension, because it is simultaneously an algebraic extension and finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Likewise, Fn is obviously a subfield of Fn+1 for each natural n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, we arrive at the conclusion that Fn is a finite field of order pln, where ln = [Fn : Fp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since Fn ≤ Fn+1, one deduces that ln+1 is divided by ln and that the equality ln+1/ln = [Fn+1 : Fn] holds for all n ∈ N, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' As for the second part that such a field is necessarily is now an immediate con- sequence of the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ The next comments are worthwhile to explain that the conditions in the previous statement cannot be weakened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Knowing that Fp is the simple (finite) field of prime characteristic p, routine arguments show that the field Fp(X) of rational functions of the variable X with coefficients from Fp, which is actually a transcendental extension of Fp, is an example of a countable field of characteristic p which cannot be constructed as a countable union of finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Moreover, arguing in the same manner, one can see that the field of rational numbers, Q, also cannnot be presented as a countable union of finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We now arrive at our first main result in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Specifically, the following is true: 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' DANCHEV AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' POJAR Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let n ≥ 1 be an integer and let F be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then the following two conditions are equivalent: (1) Every n × n companion matrix over F is the sum of a potent and a square- zero nilpotent over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (2) F is a countable field of positive characteristic, which is an algebraic exten- sion of its minimal simple (finite) subfield, with all subfields of order greater than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (1) =⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Assume that each n × n companion matrix C over F is weakly periodic with nilpotence index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Referring to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='5, we have F ⊆ ∩m∈N,m≥2Wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But, for every ω ∈ Wm, there exist integer multiples of unity, say α and u, such that (ω − α)m = u by owing to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Assume in a way of contradiction that F has zero characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It then follows that each non-zero integer multiple of unity s is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Set ω = r·s−1, and let r and ω be integers multiple of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, r · s−1 ∈ ∩m∈N,m≥2Wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Fix an m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, there exist integer multiples of unity α and u such that (r · s−1 − α)m = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' By the classical Newton’s binomial formula, we find that (r · s−1)m + (−1)1 �m 1 � (r · s−1)m−1α + (−1)2 �m 2 � (r · s−1)m−2α2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' +(−1)m−1 � m m − 1 � (r · s−1)1αm−1 + αm = u and so rm + (−1)1 �m 1 � rm−1sα + (−1)2 �m 2 � rm−2s2α2+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + (−1)m−1r1sm−1αm−1 + αm · sm = u · sm, which is equivalent to rm = ((−1)0 �m 1 � rm−1 · α + (−1)1 �m 2 � rm−2α2s+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + (−1)m−2rαm−1sm−2 + αm · sm−1 + usm−1) · s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' As by assumption the field is of characteristic zero, we may consider with no harm of generality that the above equation is satisfied in Z, whence it follows that s divides rm for any s ∈ Z∗, r ∈ Z, which is manifestly false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' for example, by considering s = 2 and r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, the characteristic of the field has to be some non-zero prime, say p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Besides, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='3 ensures that the field is of necessity countable, because SRn is countable and F ⊆ SRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, assume that the containment GF(pl) ⊆ F is fulfilled for some integer l satisfying the inequalities 2t ̸= pl − 1 ≤ n − 1 for any non-negative integer t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, there exists an odd proper divisor d of pl − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Further, take q a non-negative odd integer such that q and pl − 1 are co-prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, m = d · q is odd and d = gcd(pl − 1, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let the equation xm − 1 = 0 holds over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It has d solutions in GF(pl), namely 1, h, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , hd−1, for h = g pl−1 d , where g is a generator of the multiplicative cyclic group of the field GF(pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since hd = 1, we have 1 + h + h2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + hd−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS 7 Therefore, −1 = h + h2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + hd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, −1 is the sum of d − 1 m-th roots of unity with each of them not equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since d − 1 < d ≤ pl − 1 ≤ n − 1, we detect that −1 = α1h + α2h2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αd−1hd−1 with α1 = α2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' = αd−1 = 1, and hence it follows from the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='6 that (−1 − 0)m = αm 1 + αm 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' + αm d−1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Also, (−1)m = d − 1, and since m is odd, we deduce that p divides d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But d divides pl − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, by the transitivity law, p must divide pl − 1 and thus p = 1, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, we have GF(pl) ⊆ F for some integer l with 2t = pl − 1 ≤ n − 1 for some non-negative integer t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It follows now that p−1 and r = pl−1+pl−2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='+p+1 are powers of 2 and p − 1 divides r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But since p − 1 divides r − l, we receive that p − 1 divides l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, since p ̸= 2, we deduce that l = 2u with u a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' That is why, (pu − 1)(pu + 1) = 2t and hence pu − 1 divides pu + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Finally, pu − 1 divides 2 and since p ̸= 2, we obtain p = 3 and u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, l = 2u = 2 and GF(pl) = GF(32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let g be a generator of the multiplicative group of GF(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since gcd(32 − 1, 12) = 4, there are four 12-th roots of unity in the field GF(32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' They are ǫ0, ǫ6, ǫa3 and ǫa4, where ǫ is a primitive 12-th root of unity, whereas a3 and a4 are positive distinct integers less then 12 and not equal to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' However, as 32−1 4 = 2, we have {1, −1, ǫa3, ǫa4} = {1, g2, (g2)2, (g2)3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since (g2)4 = 1, it follows that 1 + g2 + (g2)2 + (g2)3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Consequently, −1 = −1 + ǫa3 + ǫa4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We thus derive two things: ǫa3 = −ǫa4, and −1 is the sum of at most n 12−th roots of unity, because 3 < 32 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, imitating the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='6, taking into account that 12 is even, it follows that (0 − (−1))12 = (−1)6 · 112 + (−1)a3 · 112 + (−1)a4 · 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We thus inspect that the numbers a3 and a4 cannot be simultaneously odd or simultaneously even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In this way, without loss of generality, we can assume that a3 is even and a4 is odd, and since ǫa3 = −ǫa4, we obtain that (−ǫ)a3 = (−ǫ)a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' However, because ǫ is not lying in the set {−1, 0, 1}, while a3 and a4 are positive integers less then 12, it follows automatically that a3 = a4, which is a new contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, in order to demonstrate that F is an algebraic extension of its minimal simple (finite) subfield, we will prove that every non-zero element of F is a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In fact, owing to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='3, we have that F ∈ SRn, the set of all sums of at most n roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Also, according to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='5, we get that SRn ⊆ ∪m∈N,m≥2Wm, whereas Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='6 enables us that if x ∈ Wm, then there exist u ∈ Fp and α ∈ Fp such that (x − α)m = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Likewise, the proofs of the mentioned lemmas tell us that if x is the sum of at most n m-th roots of unity within there are exactly α values of 1, then (x − α)m ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, we may assume 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' DANCHEV AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' POJAR that x = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If x ̸= 0, then xp−1 = 1 and thus x is a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If, however, x is the sum of at most n roots of unity within there are no values of 1, then (x − α)m ∈ Fp − {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, x = α + y with αp−1 = 1 and ym(p−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If α ̸= 1, then x is the sum of two roots of unity not equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, xm(p−1) ∈ Fp − {0} and, therefore, xm(p−1)2 = 1 Thus, x is a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now take α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But as x = 1 · 1 + y = (p − 1) · (−1) + y = ((n − 1) · (−1) + y) + (p − n) · (−1), we have that (n − 1)(−1) + y is the sum of n − 1 roots of unity equal to (−1) and one m(p − 1) − th root of unity not equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It follows now that ((n − 1) · (−1) + y − 1 · 0)m(p−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Moreover, if (p − n) · (−1) = 1, then p divides n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But n − 1 ≤ p − 1, so that p ≤ p − 1 which is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' By this contradiction we derive that x is the sum of two m(p − 1)-th roots of unity not equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Consequently, xm(p−1) ∈ Fp − {0}, whence xm(p−1)2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Finally, x is a root of unity, as promised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In conclusion, F is a countable field of positive characteristic, which is an algebraic extension of its minimal simple (finite) subfield, with all subfields of order greater than n, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (2) =⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let GF(pl) be contained in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Take C1 to be an n × n com- panion matrix over GF(pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, as pl ≥ n + 1, it follows by the application of the main result from [4] that we may decompose C1 = D1 + N1, where D1 is a diagonalizable matrix over GF(pl) with no multiple eigenvalues and N1 is a nilpo- tent matrix of nilpotence index 2 over GF(pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='7 helps us to have that D1 is a non-derogatory potent matrix, and since trace N = 0 we can get that trace C1 = trace D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, any element in GF(pl) can be the trace of a non- derogatory potent matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, each element in GF(pl) can be the trace of a potent companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, according to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='8, every countable field of positive characteristic, which is an algebraic extension of its minimal simple (finite) subfield, is a countable union of its finite subfields, it follows at once that every element in F can be the trace of an n × n potent companion matrix over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, letting C be an arbitrary n × n companion matrix over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, trace C ∈ STn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In conclusion, applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='2(2), one infers that C can be written as a sum of a potent matrix and a nilpotent matrix of order 2 over F, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ We now come to the case when the potent and square-zero matrices commute each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' The commuting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We will attack here the commuting case of weakly periodic with nilpotence index 2 companion matrices, that is, the existing potent and square-zero nilpotent matrices commute each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' To that aim, we first need the following two technical conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let n be a positive integer and F a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If, for every n×n companion matrix C over F, there exist an integer t > 1, a potent matrix P such that P = P t and a square-zero nilpotent N such that C = P + N with PN = NP, then the following two points are true: (1) If C is invertible, then χC divides (Xt−1 − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (2) If λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn are non-zero elements of F, then there exists an integer t > 1 such that λt−1 i = 1 for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (1) Since C = P + N and PN = NP, it follows that CP = PC and CN = NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But we have (C − N)t = C − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Since CN = NC, we can apply COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS 9 Newton’s binomial formula to argue that there exists an n × n matrix over F such that Ct − tCN + MN 2 = C − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But as N 2 = 0, we obtain that C(Ct−1 − tN) = C − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Multiplying with Ct−1 + tN, we get that C(C2t−2 − t2N 2) = Ct + tCN − Ct−1N + tN 2, and using that N 2 = 0, we write the equality C2t−1 = Ct + (tC − Ct−1)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But C is invertible, and so C2t−2 − Ct−1 = (tIdn − Ct−2)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, bearing in mind that CN = NC and N 2 = 0, we infer ((Ct−1)2 − Ct−1)2 = 0, and since C is invertible, we conclude (Ct−1 − 1)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, the minimal polynomial of C divides (Xt−1 − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But the minimal polynomial of C is the characteristic polynomial, say χC, of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Summarizing all the information so far, χC divides (Xt−1 − 1)2, as promised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (2) Just take C to be the companion matrix with eigenvalues λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn and apply the preceding point (1) along with the fact that λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn are from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let n be a positive integer and F a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If, for every n×n companion matrix C over F, there exist an integer t > 1, a potent matrix P such that P = P t and a square zero nilpotent N such that C = P + N with PN = NP, then the following two conditions are valid: (1) If C is invertible, then there exists a polynomial q ∈ F[X] such that χC divides (q(X) − X)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (2) If λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn are non-zero elements of F, then there exists a polynomial q ∈ F[X] such that q(λi) = λi for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (1) Since C = P + N and PN = NP, it follows that CP = PC and CN = NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It is well known that all matrices that commute with a companion matrix C can be interpreted just as polynomials in C over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, there exists q ∈ F[X] with C = q(C) + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, since N 2 = 0, it follows that (q(C) − C)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, the minimal polynomial of C obviously divides (q(X) − X)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But the minimal polynomial of C is the characteristic polynomial, say χC, of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Summarizing all the information thus far, χC divides (q(X) − X)2, as asked for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' (2) Just take C to be the companion matrix with eigenvalues λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn and employ the preceding condition (1) together with the fact that λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn are from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ The following claim from number theory is a well-known folklore fact, but we state it here only for the sake of completeness and the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let a, b, c be three positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, gcd(bc, a) divides gcd(b, a)· gcd(c, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' DANCHEV AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' POJAR Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let we set f = gcd(a, bc), f1 = gcd(a, b) and f2 = gcd(a, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, there exist integers s1, s2, t1, t2 such that f1 = s1a + t1b and f2 = s2a + t2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, f1f2 = s1s2a2 + s1t2ac + t1s2ba + t1t2bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But f divides a and f divides bc, so f divides a2, f divides ac, f divides ba and f divides bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, f divides f1f2 whence gcd(bc, a) divides gcd(b, a) · gcd(c, a), as formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ The next comments are needed to explain the complicated situation in the com- muting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn be distinct non-zero elements of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='12(2), there exists q ∈ F[X] such that q(λi) = λi for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Take GF(pl1) such that λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn are in GF(pl1), GF(pl2) and such that q ∈ GF(pl2)[X] with l = max(l1, l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn are in GF(pl), while q ∈ GF(pl)[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Furthermore, since λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn are in F, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='11(2) that there exists a non-negative integer t > 1 such that λt−1 i = 1 for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Now, since λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn are in GF(pl), we can infer that {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn} ⊆ {h, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , hd = 1}, where h = g pl−1 d with g a generator of the multiplicative group of GF(pl), and d = gcd(pl−1, t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, n ≤ d and there exist {j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , jn} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , d} such that λi = hji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Finally, we extract that q(hji) = hji for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , n} with hd = 1, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' The next example illustrates some concrete aspects of our calculating manipula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If in the previous Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='14 we take λ1 = gi and λ2 = gi+1 with i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , pl − 2}, then one inspects that {gi, gi+1} ⊆ {g pl−1 d , g2 pl−1 d , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , gd pl−1 d }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, {i, i + 1} ⊆ {pl − 1 d , 2pl − 1 d , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , dpl − 1 d }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, the ordinary fraction pl−1 d is a common divisor of both i and i + 1, so it is necessarily 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We thus have now that pl − 1 = d = gcd(pl − 1, t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In conclusion, pl − 1 divides t − 1, as desired to demonstrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We are now ready to establish our second main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Suppose n ≥ 1 is an integer and F is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If all companion n×n matrices over F are the sum of a potent matrix and a square-zero matrix over F which matrices commute each other, then F is a finite field of order greater than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' COMPANION WEAKLY PERIODIC MATRICES OVER FIELDS 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Suppose n ≥ 1 is an integer, F is a field and all companion n×n matrices over F are the sum of a potent matrix and a square-zero matrix over F which commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='10, we have that F is a countable field of positive characteristic (which is also an algebraic extension of a finite field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Assume now the contrary that F is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Thus, the second part in the statement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='8 applies to write that F = ∪∞ i=1GF(pli), where li divides li+1 for every positive integer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, there exists the sequence of integers greater than 1, say (ci)i≥1 such that li+1 = li · ci and (li)i≥1 is a strictly increasing infinite sequence of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Take g1 to be the generator of the multiplicative group of GF(pl1) and put λ1 = g [ pl1 −1 n ] 1 , λ2 = g 2·[ pl1 −1 n ] 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn = g n·[ pl1 −1 n ] 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Consequently, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='11 allows us to have the existence of a positive integer t > 1 such that λt−1 1 = λt−1 2 = · · · = λt−1 n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Let C be the companion matrix with eigenvalues λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Note that the integer t above has the property that C = P + N with P t = P, N 2 = 0 and PN = NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' If, eventually, there are more than one such decompositions of C, then we can take t to be the minimum of such integers t’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, t can be chosen to be a fixed uniquely determined integer having the mentioned property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Furthermore, since we are working in GF(pl1), Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='14 leads us to the relation {g [ pl1 −1 n ] 1 , g 2·[ pl1 −1 n ] 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , g n·[ pl1 −1 n ] 1 } ⊆ {g pl1 −1 d1 1 , g 2 pl1 −1 d1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , g d1 pl1 −1 d1 1 }, where di = gcd(pli − 1, t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Take ji ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' , di} such that g [ pl1 −1 n ] 1 = g j1· pl1 −1 d1 1 , and since the order of g1 in the multiplicative group of GF(pl1) is pl1 − 1, it follows that [pl1 − 1 n ] = j1 · pl1 − 1 d1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Analogously, we obtain that [pl1 − 1 n ] = ji · pli − 1 di , for any positive integer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Therefore, ji · pli − 1 di = ji+1 · pli+1 − 1 di+1, and so ji ji+1 = di di+1 (pli)ci − 1 pli − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Take si = (pli)ci−1 + (pli)ci−2 + · · · + pli + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' It thus follows that ji ji+1 = gcd(pli − 1, t − 1) gcd((pli)ci − 1, t − 1) · si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Also, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='13 we have that there exists a strictly positive integer ki such that gcd((pli)ci − 1, t − 1) = gcd(pli − 1, t − 1) · gcd(si, t − 1)) ki .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' 12 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' DANCHEV AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' POJAR Now, we obtain that ji ji+1 = ki · si gcd(si, t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' However, since gcd(si, t − 1) divides si, it follows that ji+1 divides ji for every positive integer i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But ji ≥ 1 and thus there exists i0 ≥ 1 such that ji = ji+1 for every i ≥ i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' So, we obtain ki · si gcd(si, t − 1) = 1, which forces si = gcd(si, t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Hence, si divides t − 1 and so si ≤ t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' But pli < si and then pli < t − 1, for every i ≥ i0, which is in sharp contradiction with the fact that (li)i≥1 is a strictly increasing infinite sequence of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' In conclusion, one has that F is finite, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' We next once again apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='10 to get that the order of the field is greater than n, as asserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' □ In the spirit of the last result, we pose the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Conjecture: Given n ≥ 1 is an integer and F is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Then all companion n × n matrices over F are the sum of a potent matrix and a square-zero matrix over F which matrices commute each other if, and only if, F is a finite (and hence potent) field and n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' On the other side, in regard to [1] and [6], we close our work with the following question of some interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Suppose that D is a division ring and n ≥ 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Does it follow that the matrix ring Mn(D) is weakly periodic if, and only if, D is a finite (and hence potent) field?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Funding: The scientific work of Peter V.' metadata={'source': 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(2003), 5609–5625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Bonchev” str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=', bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' 8, 1113 Sofia, Bulgaria Email address: danchev@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='bas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='bg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' pvdanchev@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='com The Technical University of Cluj-Napoca, Departament of Mathematics, Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content=' Mem- orandumului 28, 400114, Cluj-Napoca, Romania Email address: andradapojar@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfdg-i/content/2301.05612v1.pdf'} diff --git a/ZdAzT4oBgHgl3EQfm_1e/content/tmp_files/2301.01572v1.pdf.txt b/ZdAzT4oBgHgl3EQfm_1e/content/tmp_files/2301.01572v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c87c7202cfc4fe788aabb3e07c6bd7ff02ad71b --- /dev/null +++ b/ZdAzT4oBgHgl3EQfm_1e/content/tmp_files/2301.01572v1.pdf.txt @@ -0,0 +1,1313 @@ +Multi-Task Learning with Prior Information +Mengyuan Zhang∗ +Kai Liu† +Abstract +Multi-task learning aims to boost the generalization perfor- +mance of multiple related tasks simultaneously by leveraging +information contained in those tasks. In this paper, we pro- +pose a multi-task learning framework, where we utilize prior +knowledge about the relations between features. We also +impose a penalty on the coefficients changing for each spe- +cific feature to ensure related tasks have similar coefficients +on common features shared among them. In addition, we +capture a common set of features via group sparsity. The +objective is formulated as a non-smooth convex optimization +problem, which can be solved with various methods, includ- +ing gradient descent method with fixed stepsize, iterative +shrinkage-thresholding algorithm (ISTA) with back-tracking, +and its variation – fast iterative shrinkage-thresholding algo- +rithm (FISTA). In light of the sub-linear convergence rate +of the methods aforementioned, we propose an asymptoti- +cally linear convergent algorithm with theoretical guarantee. +Empirical experiments on both regression and classification +tasks with real-world datasets demonstrate that our pro- +posed algorithms are capable of improving the generalization +performance of multiple related tasks. +1 +Introduction +Over past decades, Multi-Task Learning (MTL) [3] +has attracted great interest and has been applied +successfully in various research areas, including computer +vision [20], health informatics [17], and natural language +processing [6], etc. MTL aims to boost the generalization +performance of multiple related tasks simultaneously by +leveraging potentially useful information contained in +those related tasks. Generally speaking, the objective of +a MTL problem can be formulated as: +(1.1) +min +W +m +� +i=1 +∥Xiwi − yi∥2, +where W = [w1, . . . , wm] is the matrix formed by +the weight vectors of m tasks, Xi denotes the input +feature matrix for the i-th task, yi is the corresponding +response. Based on the above objective with the goal of +∗mengyuz@clemson.edu, School of Computing, Clemson Uni- +versity. +†kail@clemson.edu, School of Computing, Clemson University. +minimizing the overall error across all the tasks, previous +studies have been done to improve the performance by +identifying the intrinsic relationships among tasks [23], +such as a common set of features they share, and some +outlier tasks which are in fact not related with other +tasks. Existing methods focusing on finding common +feature sets can be classified into two major categories: +explicit parameter sharing, where all the tasks share +some common coefficients explicitly [1], and implicit +structure sharing, which captures the shared structure +among tasks implicitly by constraining a common low- +rank subspace [19]. Some studies also perform outlier +task detection [11] at the same time. +One key observation which is ignored by previous +studies is: +the prior knowledge regarding different +features relationship is not taken into account, which +can play an important role in feature selection. +For +instance, expertise knowledge may indicate two features +have a similar influence on the response, therefore the +correspondent coefficients should be close to each other, +while two features having opposite influences should have +significantly different coefficients. Besides, for common +features in related tasks, coefficients for the same feature +should not change dramatically across tasks. Such prior +information is reasonable and not difficult to obtain +when we deal with real-world data where each feature +denotes a certain property of the data and relationships +between different features can be inferred without much +cost. +Another shortcoming of previous MTL studies tar- +geting optimizing Eq.(1.1) is the slow convergence rate. +In short, gradient descent requires O( 1 +ϵ ) iterations to +achieve ϵ accuracy, while momentum based methods will +not exceed O( 1 +√ϵ), which is still sub-linear convergence +rate. To accelerate the convergence asymptotically and +in light of the objective by adding the regularization +terms described above, we propose a novel updating +algorithm that enjoys a linear convergence rate with +O(log( 1 +ϵ )) iterations. One can see its advantage over its +counterparts when ϵ becomes sufficiently small in terms +of fewer iterations to obtain ϵ precision. +We explicitly list the main contributions of this +paper as: +• We add a regularization term based on prior +information to obtain more accurate coefficients +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited +arXiv:2301.01572v1 [cs.LG] 4 Jan 2023 + +for related tasks. We impose an extra constraint +on the coefficients’ changing for the same features +among related tasks, which will lead the coefficients +for the same features to change smoothly across +similar tasks. +• We present three methods to optimize the MTL +with prior information objective, including the +vanilla gradient descent with a fixed stepsize, the +iterative shrinkage thresholding algorithm (ISTA) +with modified stepsize searching [2], and a novel +algorithm with linear convergence rate, which is +proved to have comparable speed with the fast +iterative shrinkage thresholding algorithm (FISTA) +with backtracking [2] in experiments. +We also +provide the linear convergence rate proof of our +proposed algorithm, validating the superiority of +our new algorithm in terms of speed accelerating in +the setting of MTL with prior information. +• We conduct empirical experiments with real-world +data, the experiment results demonstrate the effec- +tiveness of our proposed algorithm. +The remaining of this paper is organized as follows: +In Section 2 we introduce the problem formulation +of our proposed MTL with prior information. +In +Section 3, we present the three methods to solve the +optimization problem we formulated. Detailed proof of +the convergence rate of our proposed algorithm in the +MTL scenario is provided in Section 4. In Section 5 we +demonstrate the experimental results on both regression +and classification tasks, showing the good performance +of our proposed algorithm in MTL. +Before we start to formulate the problem, we first +introduce some notations throughout the paper in +advance for clarity and simplicity. +Scalars, vectors, +and matrices are denoted by lower case letters, bold +lower case letters, and bold capital letters, respectively. +xi denotes the i-th entry of a vector x, xij denotes +the (i, j)-th entry of a matrix X. +For the i-th row +in a matrix X, we use xi, and for the i-th column +we use xi. The lp,q-norm of a matrix X is defined as +∥X∥p,q = (� +i((� +j xp +ij)1/p)q)1/q. The inner product of +matrices X and Y is denoted as ⟨X, Y⟩. ∥·∥F represents +the Frobenius norm. Pk denotes the P matrix we obtain +in the k-th iteration, ηP k denotes the stepsize of P in the +k-th iteration, prox() denotes the proximal operator. +2 +Problem Formulation +In MTL, we are given m learning tasks associated with +the input data {X1, . . . , Xm} and the corresponding +responses {y1, . . . , ym}, where Xi ∈ Rni×d with each +row as a sample, and yi ∈ Rni×1. d is the number of +features in each task, and ni is the number of samples +in the ith task. +For each task, we aim to learn a +vector of coefficients pi such that yi ≈ Xipi, the +matrix P is formed by the m coefficient vectors as +P = [p1, . . . , pm] ∈ Rd×m. +Assume we have some prior knowledge about the +relationships between features, we are able to construct +a matrix D to contain such prior information, in this way +the regularization term ∥DP∥2 +F is able to force that the +learned coefficients are in accordance with the given prior +information. To give a straightforward illustration of +how it works, we provide an example as follows: suppose +we have some prior knowledge regarding features, say the +ith feature and the jth feature have a similar influence +on the response, then accordingly the corresponding +coefficients should be close, namely ∥pi − pj∥ should +be small. In practice, there can be various such feature +relationship constraints (say s), by following the above +we can formulate the constraint as: +s +� +t=1 +∥pi(t) − pj(t)∥2 = +����������� +� +�� +d11 +. . . +d1d +... +... +. . . +ds1 +. . . +dsd +� +�� +� +�� +� +D +· +� +�� +p11 +. . . +p1m +... +... +. . . +pd1 +. . . +pdm +� +�� +����������� +2 +F +, +(2.2) +where i(t) and j(t) denote the indices in t-th constraint +and each row in D all elements are 0’s except a pair +of {1, −1} indexed by i(t) and j(t). +Furthermore, +related tasks sharing a common set of features should +have similar coefficients for each feature, thus we can +integrate the term ∥pi−pi+1∥2 in the objective to ensure +the smoothness of coefficients’ changing between two +adjacent tasks (such as a temporal task). +Combined together, our multi-task learning with +prior information is formulated as follows: +min +P +1 +2 +m +� +i=1 +∥Xipi − yi∥2 + λ∥P∥2,1 ++ 1 +2θ∥DP∥2 +F + 1 +2ϵ +m−1 +� +i=1 +∥pi − pi+1∥2, +(2.3) +where all the parameters λ, θ, ϵ are nonnegative. Specif- +ically, the first term is the empirical loss, while the +following l2,1-norm regularization term is based on the +group Lasso penalty [13, 15], which is applied to the +rows of P to identify a common set of features 1. The +last two regularization terms are aiming to obtain a P +1One can also change the group Lasso to Nuclear norm (∥P∥∗ = +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +that is consistent with given prior information. From +the problem formulation, it’s easy to see we have the +beneficial fact that the smooth part in the objective +function is strongly-convex. +3 +Optimization Algorithm +In this section, we will show how to solve the problem +formulated in Eq.(2.3) with multiple methods. Obviously, +it is a nonsmooth convex problem due to the existence +of the group Lasso regularization term. To handle this, +we can decompose Eq.(2.3) into two parts: +f(P) = 1 +2 +m +� +i=1 +∥Xipi − yi∥2 + 1 +2θ∥DP∥2 +F ++ 1 +2ϵ +m−1 +� +i=1 +∥pi − pi+1∥2, +(3.4) +(3.5) +g(P) = λ∥P∥2,1, +thus +(3.6) +F(P) = f(P) + g(P), +where f(P) is a smooth differential strongly-convex +function, g(P) is a nondifferential convex function. +Without loss of generality, let si +min, si +max, dmin, dmax +denote the minimum and maximum singular value of +XT +i Xi and DT D, respectively, the Lipschitz constant LP +of f(P) can be calculated as maxi(si +max) + θ · dmax + 2ϵ, +and the strongly-convex constant σP can be calculated +as mini(si +min) + θ · dmin + ϵ. +Following the basic approximation model in [26], +given the Taylor expansion of f(P) at (A) is +(3.7) +TA,ηP (P) = f(A) + ⟨∇f(A), P − A⟩ + ηP +2 ∥P − A∥2 +F , +we can minimize F(P) via minimizing its quadratic +approximation MA,ηP (P): +(3.8) +arg min +P +MA,ηP (P) = arg min +P +TA,ηP (P) + g(P), +which admits a unique minimizer for any ηP +> 0. +Moreover, as long as ηP ≥ Lp, then MA,ηP (P) is a +majorization function w.r.t F(P), therefore we can +leverage majorize-minimization (MM ) to optimize P. +� +i σi(P)) to obtain the low-rank property of P, the whole general +process remains the same except the proximal solution changing +to SV T (Singular Value Thresholding) in Eq. (3.9). +Algorithm 1 Vanilla Gradient Descent Method with +Constant Stepsize +Input: ηP = LP . +Initialization: P0 +repeat +1. Set A = Pk−1. +2. Calculate Pk according to Eq.(3.10). +until convergence +Therefore we can get the following optimization +problem: +(3.9) +P = arg min +P +f(A) + ⟨∇f(A), P − A⟩ + ηP +2 ∥P − A∥2 +F + λ∥P∥2,1 += arg min +P +ηP +2 ∥P − (A − 1 +ηP +∇f(A))∥2 +F + λ∥P∥2,1, +which leads to a closed proximal operator of rows in P +with the following closed-form solution [14]: +(3.10) +U = A − 1 +ηP +∇f(A), +pi = proxηP (ui) = max(0, 1 − +λ +ηP ∥ui∥)ui. +We propose three methods to solve the problem +above, including the vanilla gradient descent method +with constant stepsize, ISTA with modified stepsize +searching, and an algorithm proposed by us with a linear +convergence rate. +In vanilla gradient descent with constant stepsize, +the optimal solution at the k-th iteration is obtained by +solving the following problem +(3.11) +Pk = arg min +P +MPk−1,ηP (P) +with an appropriate stepsize 1/ηP . We can verify that +the objective has a sufficient decrease when we set ηP +as the Lipschitz constant LP of f(P) with regards to P. +The algorithm is summarized in Algorithm 1. +While it is guaranteed that the objective in Eq.(2.3) +is monotonically non-increasing with vanilla gradient +descent, an obvious drawback of using 1/LP as the +constant stepsize is it is too small to achieve an optimal +result rapidly. To improve it, we can apply ISTA with +modified stepsize searching. In the previous work about +ISTA with backtracking, ηP 0 > 0 and βP > 1 are +initialized randomly and we need to find the smallest +non-negative integer ik such that with ηP = βik +P ηP k−1 +we have +(3.12) F(proxηP (Pk−1)) ≤ MPk−1,ηP (proxηP (Pk−1)). +One potential flaw in the ISTA with the conventional +backtracking method described above lies in the initial- +ization of η0. It is possible that the value of η0 at the +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +Algorithm 2 ISTA with Modified stepsize Searching +Input: ηP 0 = LP , 0 < βP < 1. +Initialization: P0 +repeat +1. Find the smallest integer ik such that with ηP = +βik +P ηP 0 Eq. (3.13) is satisfied. Set ηP k = ηP /βP . +3. +With A = Pk−1, ηP += ηP k, calculate Pk +according to Eq.(3.10). +until convergence +Algorithm 3 Fast Algorithm with Linear Convergence +rate +Input: ηP = LP , c = LP +σP . +Initialization: P0, set A0 = P0 +repeat +1. calculate Pk according to Eq.(3.10). +2. Update Ak = Pk + +√c−1 +√c+1(Pk − Pk−1) +until convergence +very first step is already larger than the actual Lipschitz +constant L, and the starting step size is already too small +to have fast convergence. And the stepsize ηk search- +ing in the kth iteration always starts from ηk−1, thus +there may be a larger stepsize available satisfying the +condition Eq.(3.12) in the kth iteration that cannot be +discovered by this searching process. For this reason, we +do the stepsize searching reversely starting from setting +η0 to its Lipschitz constant L, then we keep shrinking +it until Eq.(3.12) is not satisfied in terms of P in the +optimization process, which is +(3.13) F(proxηP (Pk−1)) > MPk−1,ηP (proxηP (Pk−1)), +in this way, we are able to find the largest stepsize +meeting the condition in each iteration. By updating +as Algorithm 2, the objective is decreasing much faster +than vanilla gradient descent with constant step size. +While ISTA converges faster than vanilla gradient +descent with constant stepsize, the convergence rate is +still sub-linear (including FISTA), therefore we propose +a new algorithm to solve the multi-task learning problem +in Eq.(3.9) with a linear convergence rate, utilizing the +strongly-convex property of f(P). As we said before, all +the parameters λ, θ, ϵ are nonnegative, thus we are able to +guarantee that f(P) in Eq.(3.4) is strongly-convex with +σP , and Lipschitz smooth with LP . The algorithm is +summarized as Algorithm 3. By updating as Algorithm +3, although the objective in Eq.(2.3) is not guaranteed +to be monotonically non-increasing, in general it can +achieve an optimal solution with a higher convergence +rate compared with Algorithm 1 and Algorithm 2. +Figure 1 shows the objective versus the number of +iterations in five algorithms with a synthetic dataset. We +can see that ISTA with our modified stepsize searching +converges faster than ISTA with backtracking in [2], +and the new algorithm we proposed generally converges +faster than FISTA with backtracking in [2]. +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +#iterations +4 +6 +8 +10 +12 +14 +16 +Objective +106 +Vanilla +FISTA +ISTA +ISTA modified +Ours +Figure 1: Objective plot in five algorithms. +4 +Convergence Analysis +In the previous section, we mentioned Algorithm 2 has +a sublinear convergence rate and Algorithm 3 has a +linear convergence rate. There are other studies showing +algorithms with a linear convergence rate for solving +such problem [25], but different from these studies, there +is no strong assumption required in our algorithm, and +we utilize the momentum trick following the Nesterov +accelerated gradient, which is proven to be unbeatable +in general cases. +The convergence proof of Algorithm 2 can be +easily adapted from the proof in [2] to modified step- +size searching and be extended from vector variables +to matrix variables due to the equivalence of matrix +Frobenius norm and vector Euclidean norm. +Here +we only present the key lemma and theorem of the +convergence rate: +Lemma 4.1. If +for +P +∈ +Rd×m, +we +have +F(proxηP (P)) +≤ +MP,ηP (proxηP (P)), +then +for +any +A +∈ +Rd×m, +F(A) − F(proxηP (P)) +≥ +ηP +2 ∥proxηP (P) − P∥2 +F + ηP ⟨P − A, proxηP (P) − P⟩. +We present the following theorem about the conver- +gence rate of solving Eq.(2.3) via Algorithm 2: +Theorem 4.1. Let Pk be the output generated by Al- +gorithm 2 in the k−th iteration, then for any k ≥ 1 we +have F(Pk) − F(P∗) = O( 1 +k), where P∗ is the optimal +solution in Eq.(2.3). +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +Before diving into the detailed proof of convergence +rate in Algorithm 3, we first provide two useful lemmas +that are important for the following proof process: +Lemma 4.2. For F(x) = f(x) + g(x), if g(x) is convex, +and f(x) is σ−strongly convex and L−smooth, then for +any x, y and α > 0 satisfying +f(proxα(y)) +≤f(y) + ⟨∇f(y), proxα(y) − y⟩ + α +2 ∥proxα(y) − y∥2 +(4.14) +the following inequality holds: +F(x) − F(proxα(y)) +≥α +2 ∥x − proxα(y)∥2 − α +2 ∥x − y∥2 ++ f(x) − f(y) − ⟨∇f(y), x − y⟩. +(4.15) +Proof. Consider function φ(u) = f(y) + ⟨∇f(y), u − +y⟩ + g(u) + α +2 ∥u − y∥2, it is obvious that such φ(u) +is α−strongly convex and proxα(y) = arg minu(φ(u)). +Then we have +(4.16) +φ(x) − φ(proxα(y)) ≥ α +2 ∥x − proxα(y)∥2. +According to Eq.(4.14): +φ(proxα(y)) +=f(y) + ⟨∇f(y), proxα(y) − y⟩ ++ α +2 ∥y − proxα(y)∥2 + g(proxα(y)) +≥f(proxα(y)) + g(proxα(y)) +=F(proxα(y)), +(4.17) +combine with Eq.(4.16), we obtain +(4.18) +φ(x) − F(proxα(y)) ≥ α +2 ∥x − proxα(y)∥2. +Therefore it’s easy to get the following inequality after +we plug in the formula of φ(x): +φ(x) − F(proxα(y)) +=F(x) − f(x) + f(y) + α +2 ∥x − y∥2 ++ ⟨∇f(y), x − y⟩ − F(proxα(y)) +≥α +2 ∥x − proxα(y)∥2, +(4.19) +which is the same as Eq.(4.15). +Lemma 4.3. For any vector a, b and constant β < 1, +we have the following equation: +(4.20) +∥a + b∥2 − β∥a∥2 = (1 − β)∥a + +1 +1 − β b∥2 − +β +1 − β ∥b∥2. +Here we introduce the theorem about the conver- +gence rate of Algorithm 3: +Theorem 4.2. For F(x) = f(x) + g(x) in Eq.(3.6), +g(x) is convex, and f(x) is σ−strongly convex and +L−smooth, let c = L +σ and t = √c. Let Pk be the kth +iteration’s output in Algorithm 3, P∗ be the optimal +solution,Vk = F(Pk) − F(P∗). Then for any k ≥ 1 we +have Vk ≤ (1 − 1 +t )k(V0 + σ +2 ∥P0 − P∗∥2). +Proof. According to Lemma 4.2 and the fact that f(x) +is σ−strongly convex and L−smooth, we obtain +F(x) − F(proxL(y)) +≥L +2 ∥x − proxL(y)∥2 − L +2 ∥x − y∥2 + σ +2 ∥x − y∥2. +(4.21) +Invoking the above inequality with x = 1 +t P∗ +(1− 1 +t )Pk +and y = Ak in Algorithm 3, we get +F(1 +t P∗ + (1 − 1 +t )Pk) − F(Pk+1) +≥L +2 ∥Pk+1 − (1 +t P∗ + (1 − 1 +t )Pk)∥2 +− L − σ +2 +∥Ak − (1 +t P∗ + (1 − 1 +t )Pk)∥2 += L +2t2 ∥tPk+1 − (P∗ + (t − 1)Pk)∥2 +− L − σ +2t2 ∥tAk − (P∗ + (t − 1)Pk)∥2. +(4.22) +Since f is a σ−strongly convex function, for α ∈ [0, 1], +we have f(αx + (1 − α)y) ≤ αf(x) + (1 − α)f(y) − +σα(1−α) +2 +∥x − y∥2, and obviously 1 +t ∈ [0, 1], so we have +F(1 +t P∗ + (1 − 1 +t )Pk) +≤1 +t F(P∗) + (1 − 1 +t )F(Pk) − σt−1(1 − t−1) +2 +∥Pk − P∗∥2. +(4.23) +With Vk = F(Pk) − F(P∗), we can get +F(1 +t P∗ + (1 − 1 +t )Pk) − F(Pk+1) +≤(1 − t−1)Vk − Vk+1 − σt−1(1 − t−1) +2 +∥Pk − P∗∥2. +(4.24) +Combine Eq.(4.24) and Eq.(4.22),we have +(4.25) +L − σ +2 +∥tAk − (P∗ + (t − 1)Pk)∥2 − σ(t − 1) +2 +∥Pk − P∗∥2 +≥ t2Vk+1 − t(t − 1)Vk + L +2 ∥tPk+1 − (P∗ + (t − 1)Pk)∥2. +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +With Lemma 4.3 and set a := Pk − P∗, b := t(Ak − +Pk), β := +σ(t−1) +L−σ , then for the left side in the above +inequality, we have +(4.26) +L − σ +2 +∥tAk − (P∗ + (t − 1)Pk)∥2 +− σ(t − 1) +2 +∥Pk − P∗∥2 +=L − σ +2 +{∥t(Ak − Pk) + (Pk − P∗)∥2 +− σ(t − 1) +L − σ ∥Pk − P∗∥2} +=L − σ +2 +{L − σt +L − σ ∥(Pk − P∗) + L − σ +L − σtt(Ak − Pk)∥2 +− σ(t − 1) +L − σt ∥t(Ak − Pk)∥2} +≤L − σt +2 +∥Pk − P∗ + L − σ +L − σtt(Ak − Pk)∥2 +Therefore we have the following inequality according to +Eq.(4.25): +(4.27) +t(t − 1)Vk + L − σt +2 +∥Pk − P∗ + L − σ +L − σtt(Ak − Pk)∥2 +≥ t2Vk+1 + L +2 ∥tPk+1 − (P∗ + (t − 1)Pk)∥2. +With the update rule Ak = Pk + +√c−1 +√c+1(Pk − Pk−1) in +Algorithm 3 and t = √c, +(4.28) +Pk − P∗ + L − σ +L − σtt(Ak − Pk) +=Pk − P∗ + L − σ +L − σt +t(t − 1) +t + 1 (Pk − Pk−1) +=tPk − (P∗ + (t − 1)Pk−1). +Since L = σt2, based on Eq.(4.27) and Eq.(4.28), divide +both sides of the inequality by t2, we have +(4.29) +Vk+1 + σ +2 ∥tPk+1 − (P∗ + (t − 1)Pk)∥2 +≤(1 − 1 +t )(Vk + σ +2 ∥tPk − (P∗ + (t − 1)Pk−1)∥2). +For k = 0, with the initialization setting A0 = P0, +(4.30) +P0 − P∗ + L − σ +L − σtt(A0 − P0) = P0 − P∗, +based on Eq.(4.29), naturally we have +(4.31) +Vk + σ +2 ∥tPk − (P∗ + (t − 1)Pk−1)∥2 +≤(1 − 1 +t )k(V0 + σ +2 ∥P0 − P∗∥2). +Thus we can get Vk ≤ (1 − 1 +t )k(V0 + σ +2 ∥P0 − P∗∥2). +The objective in Eq.(2.3) is not guaranteed to be +monotonically non-increasing due to the existence of +the σ +2 ∥tPk − (P∗ + (t − 1)Pk−1)∥2 term, but in general +we can expect it can achieve optimal solution with a +linear convergence rate. Also, one can see that when +the problem is well-conditioned, it converges faster, +otherwise, it can be rather slow. +5 +Experimental Results +5.1 +Experiment Setup We compare the experi- +mental results of the following MTL algorithms with +our method: +FedEM [18], DMTL [9], MTFL [10], +MMTFL [24], MTRL [28], RMTL [4], CLMT [5], +MKMTL [16]. +The empirical studies are conducted +on the following benchmark multi-task regression and +classification datasets: +School [7]: there are exam scores of 15362 students +from 139 schools in the dataset, each student is described +with 28 attributes. Thus there are 139 related tasks, each +sample has 28 features along with 1 output. We aim to +perform multi-task regression to predict exam scores. +Sarcos [22]: it is collected for an inverse dynamics +prediction problem for a seven degrees-of-freedom robot +arm. The number of related tasks is 7, and there are 21 +features for each sample. Following the work in [27] we +sample 2000 random samples for each task. +Yale: it contains 165 images from 15 subjects, each +image is scaled to 32 × 32 pixels. We use the first 8 +subjects from it to construct related tasks, each task is +defined as a binary classification problem of classifying +two subjects, there are 28 binary classification tasks. +MNIST [12]: we use a subset from the MNIST +dataset with 10000 samples of 10 handwritten digits. We +cast the multi-task learning as 45 binary classification +tasks to classify pairs of digits. +Letter [8]: it consists of handwritten letters from +different writers, we construct 8 binary classification +tasks from it to distinguish between pairs of letters. +ORL [21]: there are 10 different images of 40 distinct +subjects. What’s different for the ORL dataset is we +construct 40 one-vs-all multi-class classification tasks +from it rather than one-vs-one binary classification. +During the experiment process, each dataset is +randomly split into a training set and a test set. In +all classification tasks, the data is split according to a +rough 60%-40% training-test split ratio, in the School +regression task 20 random samples are used for training, +and in the Sarcos regression task, 50 random samples +are selected for training and the rest for testing. The +prior knowledge matrix D can be initialized based on +known prior, our common sense, and the correlation +among features obtained with statistical methods. In the +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +Table 1: Regression tasks: generalization performance measures over ten runs +Metric +Dataset +FedEM +DMTL +MTFL +MMTFL +OURS-NATURAL +VE (%) +School +38.7±3.2 +28.8±5.6 +29.7±2.1 +31.5±4.2 +39.8±2.2 +Sarcos +51.2±12.1 +42.6±7.3 +49.2±5.1 +49.9±2.7 +51.8±6.6 +nMSE (%) +School +61.2±0.9 +75.1±0.8 +72.9±1.1 +68.7±3.1 +60.2±0.5 +Sarcos +15.7±3.1 +20.9±0.8 +19.1±2.7 +17.1±2.2 +14.9±0.8 +Metric +Dataset +MTRL +RMTL +CLMT +MKMTL +OURS-ART +VE (%) +School +29.9±2.0 +33.6±5.7 +37.9±2.1 +37.9±1.9 +39.8±2.5 +Sarcos +42.5±8.2 +49.9±7.2 +50.1±9.9 +50.1±1.7 +51.9±6.2 +nMSE (%) +School +73.1±0.9 +68.9±2.7 +63.7±0.7 +62.7±0.9 +60.2±0.3 +Sarcos +17.9±0.7 +15.6±0.3 +16.2±0.6 +15.7±0.5 +14.7±0.9 +Table 2: Classification tasks: generalization performance measures over ten runs +Metric +Dataset +FedEM +DMTL +MTFL +MMTFL +OURS-NATURAL +AUC (%) +Yale +96.9±2.7 +95.7±3.2 +86.8±1.6 +92.7±1.1 +97.7±1.7 +MNIST +98.1±3.2 +90.9±3.1 +91.2±1.9 +91.5±2.3 +96.5±1.2 +Letter +63.2±2.9 +61.9±1.8 +62.1±2.2 +61.8±1.8 +63.5±1.1 +ORL +81.3±5.8 +72.9±3.7 +77.2±5.2 +77.9±3.2 +82.5±2.2 +Metric +Dataset +MTRL +RMTL +CLMT +MKMTL +OURS-ART +AUC (%) +Yale +96.1±3.5 +97.2±1.9 +96.7±2.3 +95.7±1.8 +97.7±1.7 +MNIST +93.7±7.1 +92.5±3.1 +94.7±3.8 +93.2±5.2 +97.6±1.2 +Letter +60.3±2.1 +62.7±3.6 +61.5±7.2 +60.9±7.1 +65.5±1.3 +ORL +77.6±3.9 +80.2±9.1 +78.9±6.3 +78.8±3.2 +84.3±1.8 +experiment, we have two methods to obtain the required +D: the first method is to utilize the natural correlation +among features, we calculate the covariance matrix for +all the features, and select the strongest ones as the +information contained in D, we call the prior knowledge +matrix obtained in this way the natural D; the second +method is we create the strong correlation among +features manually by appending features repeatedly on +purpose, for example, we transform the original feature +(x1, x2, x3) into (x1, x2, x3, x1) to enhance the correlation +among features (this may introduce multicollinearity, but +since we only care about the predictive result, it should +be fine), the matrix obtained is called as the artificial +D. Parameters in all the methods are tuned using 5- +fold cross-validation, and for each method, we stop the +experiment when the objective change is < 10−3. We use +Matlab R2019a on a laptop with a 1.4 GHz QuadCore +Intel Core i5 processor. +5.2 +Results Results of the experiment are presented +in Table 1 and Table 2, for regression tasks and +classification tasks respectively, our method with a +natural D is denoted as OURS-NATURAL, with an +artificial D is denoted as OURS-ART. In OURS-ART, +we manually repeat about 5% features to construct D. +In the case of regression tasks, we report the variance +explained (VE) and the normalized mean squared error +(nMSE) following previous studies [4,10], whereas the +receiver operating characteristic (ROC) curve and the +area under the ROC curve (AUC) are employed as +the classification performance measurements as used +in previous studies [4, 10]. +ROC curves show the +performance of a binary classification model at all +classification thresholds by plotting the true positive +rate against the false positive rate. Higher explained +variance and AUC indicate better performance, and the +opposite for nMSE reported. Each experiment with one +specific dataset is repeated 10 times and we report the +averaged performance and the standard deviation, the +best performances are in bold. +In Table 1 and Table 2, our method can achieve +the best performance in most cases, while the FedEM +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +False positive rate +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +True positive rate +(a) Yale +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +False positive rate +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +True positive rate +(b) MNIST +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +False positive rate +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +True positive rate +(c) ORL +Figure 2: The ROC curve for Yale, MNIST, ORL dataset with tuned parameters. +(a) AUC, Yale +(b) AUC, ORL +(c) VE, School +(d) nMSE, School +Figure 3: Ablation study – the influence of regularization parameters on learning performance. +method achieves better results with the MNIST dataset. +The reason perhaps is that neural networks still have +unique advantages when dealing with large-scale com- +plex image datasets, and prior knowledge about features +is hard to establish with complex image data. Also, in re- +gression tasks, our method works well with both natural +prior knowledge and artificial prior knowledge. In im- +age classification tasks, the improvement in performance +with artificial prior knowledge matrices is more signifi- +cant than that with natural ones, the reason may be due +to the fact that the natural correlations between features +in regression tasks are stronger than those involved in +image classification tasks. +We present the ROC curves with a natural prior +knowledge matrix for the classification tasks on Yale, +MNIST, and ORL datasets in Figure 2, the slim colorful +lines are the ROC curves for each task, and the weighted +black line is the averaged ROC curve over all the tasks, +and the green area represents the range between the +mean ± standard deviation. +While for each dataset +there is a small number of tasks with performances that +are not so perfect on some level, the overall performance +is satisfactory with a pretty high AUC, and the mean +− standard deviation curve is almost always over the +diagonal line, especially for the one-vs-one classification +tasks on Yale and MNIST. We also provide Figure 3 +to illustrate the influence of regularization parameters. +With a super wide tuning range for each parameter, +which is from 1 to 100, the effect of each parameter on the +performance is not significant until the value reaches a +threshold, there is a generous range for each parameter to +be able to provide stable and great performance. We can +roughly draw the conclusion that for classification tasks, +no matter whether it’s one-vs-one binary classification +tasks or one-vs-all binary classification tasks, the values +of regularization parameters have a marginal influence +on the results as long as the value is within a reasonable +range. While in a regression situation the performance +is much more sensitive to the value of parameters, +especially to the group Lasso penalty parameter, which +is in accordance with common sense that under-fitting +happens with large regularization term parameters. +6 +Conclusion +We propose a convex formulation of multi-task learning +problem utilizing prior information. A novel optimiza- +tion algorithm to solve the formulated problem with +a linear convergence rate is proposed with theoretical +guarantee instead of sub-linear rate of the counterparts. +Results on benchmark datasets with both regression and +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +1 +0.98 +AUC, Yale +0.96 +0.95 +0.94 +0.9 +1 +5 +0.92 +100 +10 +50 +20 +20 +10 +50 +5 +0.9 +100 +入 +1 +E0.85 +0.85 +AUC, ORL +0.8 +0.8 +0.75 +1 +5 +100 +10 +50 +20 +20 +10 +50 +5 +0.75 +100 +入 +1 +E0.4 +School +0.4 +0.38 +Explained, +0.36 +0.35 +0.34 +Variance +0.3 +1 +5 +0.32 +100 +10 +50 +20 +20 +10 +50 +5 +0.3 +100 +入 +1 +E0.65 +0.65 +School +0.6 +nMSE, $ +0.6 +0.55 +1 +5 +100 +10 +50 +20 +20 +10 +50 +5 +0.55 +100 +入 +1 +Eclassification tasks demonstrate the effectiveness and +advantages of our proposed multi-task learning formula- +tion and algorithm. 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A convex formulation +for learning task relationships in multi-task learning. +arXiv preprint arXiv:1203.3536, 2012. +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + diff --git a/ZdAzT4oBgHgl3EQfm_1e/content/tmp_files/load_file.txt b/ZdAzT4oBgHgl3EQfm_1e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cacad66aaf08c30a592b35382e3ae6e114d578b5 --- /dev/null +++ b/ZdAzT4oBgHgl3EQfm_1e/content/tmp_files/load_file.txt @@ -0,0 +1,644 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf,len=643 +page_content='Multi-Task Learning with Prior Information Mengyuan Zhang∗ Kai Liu† Abstract Multi-task learning aims to boost the generalization perfor- mance of multiple related tasks simultaneously by leveraging information contained in those tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In this paper, we pro- pose a multi-task learning framework, where we utilize prior knowledge about the relations between features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We also impose a penalty on the coefficients changing for each spe- cific feature to ensure related tasks have similar coefficients on common features shared among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In addition, we capture a common set of features via group sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The objective is formulated as a non-smooth convex optimization problem, which can be solved with various methods, includ- ing gradient descent method with fixed stepsize, iterative shrinkage-thresholding algorithm (ISTA) with back-tracking, and its variation – fast iterative shrinkage-thresholding algo- rithm (FISTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In light of the sub-linear convergence rate of the methods aforementioned, we propose an asymptoti- cally linear convergent algorithm with theoretical guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Empirical experiments on both regression and classification tasks with real-world datasets demonstrate that our pro- posed algorithms are capable of improving the generalization performance of multiple related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 1 Introduction Over past decades, Multi-Task Learning (MTL) [3] has attracted great interest and has been applied successfully in various research areas, including computer vision [20], health informatics [17], and natural language processing [6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' MTL aims to boost the generalization performance of multiple related tasks simultaneously by leveraging potentially useful information contained in those related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Generally speaking, the objective of a MTL problem can be formulated as: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1) min W m � i=1 ∥Xiwi − yi∥2, where W = [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' , wm] is the matrix formed by the weight vectors of m tasks, Xi denotes the input feature matrix for the i-th task, yi is the corresponding response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Based on the above objective with the goal of ∗mengyuz@clemson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='edu, School of Computing, Clemson Uni- versity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' †kail@clemson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='edu, School of Computing, Clemson University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' minimizing the overall error across all the tasks, previous studies have been done to improve the performance by identifying the intrinsic relationships among tasks [23], such as a common set of features they share, and some outlier tasks which are in fact not related with other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Existing methods focusing on finding common feature sets can be classified into two major categories: explicit parameter sharing, where all the tasks share some common coefficients explicitly [1], and implicit structure sharing, which captures the shared structure among tasks implicitly by constraining a common low- rank subspace [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Some studies also perform outlier task detection [11] at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' One key observation which is ignored by previous studies is: the prior knowledge regarding different features relationship is not taken into account, which can play an important role in feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' For instance, expertise knowledge may indicate two features have a similar influence on the response, therefore the correspondent coefficients should be close to each other, while two features having opposite influences should have significantly different coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Besides, for common features in related tasks, coefficients for the same feature should not change dramatically across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Such prior information is reasonable and not difficult to obtain when we deal with real-world data where each feature denotes a certain property of the data and relationships between different features can be inferred without much cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Another shortcoming of previous MTL studies tar- geting optimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1) is the slow convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In short, gradient descent requires O( 1 ϵ ) iterations to achieve ϵ accuracy, while momentum based methods will not exceed O( 1 √ϵ), which is still sub-linear convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' To accelerate the convergence asymptotically and in light of the objective by adding the regularization terms described above, we propose a novel updating algorithm that enjoys a linear convergence rate with O(log( 1 ϵ )) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' One can see its advantage over its counterparts when ϵ becomes sufficiently small in terms of fewer iterations to obtain ϵ precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We explicitly list the main contributions of this paper as: We add a regularization term based on prior information to obtain more accurate coefficients Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='01572v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='LG] 4 Jan 2023 for related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We impose an extra constraint on the coefficients’ changing for the same features among related tasks, which will lead the coefficients for the same features to change smoothly across similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We present three methods to optimize the MTL with prior information objective, including the vanilla gradient descent with a fixed stepsize, the iterative shrinkage thresholding algorithm (ISTA) with modified stepsize searching [2], and a novel algorithm with linear convergence rate, which is proved to have comparable speed with the fast iterative shrinkage thresholding algorithm (FISTA) with backtracking [2] in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We also provide the linear convergence rate proof of our proposed algorithm, validating the superiority of our new algorithm in terms of speed accelerating in the setting of MTL with prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We conduct empirical experiments with real-world data, the experiment results demonstrate the effec- tiveness of our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The remaining of this paper is organized as follows: In Section 2 we introduce the problem formulation of our proposed MTL with prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In Section 3, we present the three methods to solve the optimization problem we formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Detailed proof of the convergence rate of our proposed algorithm in the MTL scenario is provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In Section 5 we demonstrate the experimental results on both regression and classification tasks, showing the good performance of our proposed algorithm in MTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Before we start to formulate the problem, we first introduce some notations throughout the paper in advance for clarity and simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Scalars, vectors, and matrices are denoted by lower case letters, bold lower case letters, and bold capital letters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' xi denotes the i-th entry of a vector x, xij denotes the (i, j)-th entry of a matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' For the i-th row in a matrix X, we use xi, and for the i-th column we use xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The lp,q-norm of a matrix X is defined as ∥X∥p,q = (� i((� j xp ij)1/p)q)1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The inner product of matrices X and Y is denoted as ⟨X, Y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' ∥·∥F represents the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Pk denotes the P matrix we obtain in the k-th iteration, ηP k denotes the stepsize of P in the k-th iteration, prox() denotes the proximal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 2 Problem Formulation In MTL, we are given m learning tasks associated with the input data {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' , Xm} and the corresponding responses {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' , ym}, where Xi ∈ Rni×d with each row as a sample, and yi ∈ Rni×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' d is the number of features in each task, and ni is the number of samples in the ith task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' For each task, we aim to learn a vector of coefficients pi such that yi ≈ Xipi, the matrix P is formed by the m coefficient vectors as P = [p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' , pm] ∈ Rd×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Assume we have some prior knowledge about the relationships between features, we are able to construct a matrix D to contain such prior information, in this way the regularization term ∥DP∥2 F is able to force that the learned coefficients are in accordance with the given prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' To give a straightforward illustration of how it works, we provide an example as follows: suppose we have some prior knowledge regarding features, say the ith feature and the jth feature have a similar influence on the response, then accordingly the corresponding coefficients should be close, namely ∥pi − pj∥ should be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In practice, there can be various such feature relationship constraints (say s), by following the above we can formulate the constraint as: s � t=1 ∥pi(t) − pj(t)∥2 = ����������� � �� d11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' d1d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' ds1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' dsd � �� � �� � D � �� p11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' p1m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' pd1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' pdm � �� ����������� 2 F , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2) where i(t) and j(t) denote the indices in t-th constraint and each row in D all elements are 0’s except a pair of {1, −1} indexed by i(t) and j(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Furthermore, related tasks sharing a common set of features should have similar coefficients for each feature, thus we can integrate the term ∥pi−pi+1∥2 in the objective to ensure the smoothness of coefficients’ changing between two adjacent tasks (such as a temporal task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Combined together, our multi-task learning with prior information is formulated as follows: min P 1 2 m � i=1 ∥Xipi − yi∥2 + λ∥P∥2,1 + 1 2θ∥DP∥2 F + 1 2ϵ m−1 � i=1 ∥pi − pi+1∥2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3) where all the parameters λ, θ, ϵ are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Specif- ically, the first term is the empirical loss, while the following l2,1-norm regularization term is based on the group Lasso penalty [13, 15], which is applied to the rows of P to identify a common set of features 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The last two regularization terms are aiming to obtain a P 1One can also change the group Lasso to Nuclear norm (∥P∥∗ = Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited that is consistent with given prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' From the problem formulation, it’s easy to see we have the beneficial fact that the smooth part in the objective function is strongly-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 3 Optimization Algorithm In this section, we will show how to solve the problem formulated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3) with multiple methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Obviously, it is a nonsmooth convex problem due to the existence of the group Lasso regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' To handle this, we can decompose Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3) into two parts: f(P) = 1 2 m � i=1 ∥Xipi − yi∥2 + 1 2θ∥DP∥2 F + 1 2ϵ m−1 � i=1 ∥pi − pi+1∥2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='4) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='5) g(P) = λ∥P∥2,1, thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='6) F(P) = f(P) + g(P), where f(P) is a smooth differential strongly-convex function, g(P) is a nondifferential convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Without loss of generality, let si min, si max, dmin, dmax denote the minimum and maximum singular value of XT i Xi and DT D, respectively, the Lipschitz constant LP of f(P) can be calculated as maxi(si max) + θ · dmax + 2ϵ, and the strongly-convex constant σP can be calculated as mini(si min) + θ · dmin + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Following the basic approximation model in [26], given the Taylor expansion of f(P) at (A) is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7) TA,ηP (P) = f(A) + ⟨∇f(A), P − A⟩ + ηP 2 ∥P − A∥2 F , we can minimize F(P) via minimizing its quadratic approximation MA,ηP (P): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8) arg min P MA,ηP (P) = arg min P TA,ηP (P) + g(P), which admits a unique minimizer for any ηP > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Moreover, as long as ηP ≥ Lp, then MA,ηP (P) is a majorization function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='t F(P), therefore we can leverage majorize-minimization (MM ) to optimize P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' � i σi(P)) to obtain the low-rank property of P, the whole general process remains the same except the proximal solution changing to SV T (Singular Value Thresholding) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Algorithm 1 Vanilla Gradient Descent Method with Constant Stepsize Input: ηP = LP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Initialization: P0 repeat 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Set A = Pk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Calculate Pk according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' until convergence Therefore we can get the following optimization problem: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9) P = arg min P f(A) + ⟨∇f(A), P − A⟩ + ηP 2 ∥P − A∥2 F + λ∥P∥2,1 = arg min P ηP 2 ∥P − (A − 1 ηP ∇f(A))∥2 F + λ∥P∥2,1, which leads to a closed proximal operator of rows in P with the following closed-form solution [14]: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='10) U = A − 1 ηP ∇f(A), pi = proxηP (ui) = max(0, 1 − λ ηP ∥ui∥)ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We propose three methods to solve the problem above, including the vanilla gradient descent method with constant stepsize, ISTA with modified stepsize searching, and an algorithm proposed by us with a linear convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In vanilla gradient descent with constant stepsize, the optimal solution at the k-th iteration is obtained by solving the following problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='11) Pk = arg min P MPk−1,ηP (P) with an appropriate stepsize 1/ηP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We can verify that the objective has a sufficient decrease when we set ηP as the Lipschitz constant LP of f(P) with regards to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The algorithm is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' While it is guaranteed that the objective in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3) is monotonically non-increasing with vanilla gradient descent, an obvious drawback of using 1/LP as the constant stepsize is it is too small to achieve an optimal result rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' To improve it, we can apply ISTA with modified stepsize searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In the previous work about ISTA with backtracking, ηP 0 > 0 and βP > 1 are initialized randomly and we need to find the smallest non-negative integer ik such that with ηP = βik P ηP k−1 we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='12) F(proxηP (Pk−1)) ≤ MPk−1,ηP (proxηP (Pk−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' One potential flaw in the ISTA with the conventional backtracking method described above lies in the initial- ization of η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' It is possible that the value of η0 at the Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited Algorithm 2 ISTA with Modified stepsize Searching Input: ηP 0 = LP , 0 < βP < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Initialization: P0 repeat 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Find the smallest integer ik such that with ηP = βik P ηP 0 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='13) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Set ηP k = ηP /βP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' With A = Pk−1, ηP = ηP k, calculate Pk according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' until convergence Algorithm 3 Fast Algorithm with Linear Convergence rate Input: ηP = LP , c = LP σP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Initialization: P0, set A0 = P0 repeat 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' calculate Pk according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Update Ak = Pk + √c−1 √c+1(Pk − Pk−1) until convergence very first step is already larger than the actual Lipschitz constant L, and the starting step size is already too small to have fast convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' And the stepsize ηk search- ing in the kth iteration always starts from ηk−1, thus there may be a larger stepsize available satisfying the condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='12) in the kth iteration that cannot be discovered by this searching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' For this reason, we do the stepsize searching reversely starting from setting η0 to its Lipschitz constant L, then we keep shrinking it until Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='12) is not satisfied in terms of P in the optimization process, which is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='13) F(proxηP (Pk−1)) > MPk−1,ηP (proxηP (Pk−1)), in this way, we are able to find the largest stepsize meeting the condition in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' By updating as Algorithm 2, the objective is decreasing much faster than vanilla gradient descent with constant step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' While ISTA converges faster than vanilla gradient descent with constant stepsize, the convergence rate is still sub-linear (including FISTA), therefore we propose a new algorithm to solve the multi-task learning problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9) with a linear convergence rate, utilizing the strongly-convex property of f(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' As we said before, all the parameters λ, θ, ϵ are nonnegative, thus we are able to guarantee that f(P) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='4) is strongly-convex with σP , and Lipschitz smooth with LP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The algorithm is summarized as Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' By updating as Algorithm 3, although the objective in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3) is not guaranteed to be monotonically non-increasing, in general it can achieve an optimal solution with a higher convergence rate compared with Algorithm 1 and Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Figure 1 shows the objective versus the number of iterations in five algorithms with a synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We can see that ISTA with our modified stepsize searching converges faster than ISTA with backtracking in [2], and the new algorithm we proposed generally converges faster than FISTA with backtracking in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 0 50 100 150 200 250 300 350 400 450 500 #iterations 4 6 8 10 12 14 16 Objective 106 Vanilla FISTA ISTA ISTA modified Ours Figure 1: Objective plot in five algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 4 Convergence Analysis In the previous section, we mentioned Algorithm 2 has a sublinear convergence rate and Algorithm 3 has a linear convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' There are other studies showing algorithms with a linear convergence rate for solving such problem [25], but different from these studies, there is no strong assumption required in our algorithm, and we utilize the momentum trick following the Nesterov accelerated gradient, which is proven to be unbeatable in general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The convergence proof of Algorithm 2 can be easily adapted from the proof in [2] to modified step- size searching and be extended from vector variables to matrix variables due to the equivalence of matrix Frobenius norm and vector Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Here we only present the key lemma and theorem of the convergence rate: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' If for P ∈ Rd×m, we have F(proxηP (P)) ≤ MP,ηP (proxηP (P)), then for any A ∈ Rd×m, F(A) − F(proxηP (P)) ≥ ηP 2 ∥proxηP (P) − P∥2 F + ηP ⟨P − A, proxηP (P) − P⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We present the following theorem about the conver- gence rate of solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3) via Algorithm 2: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Let Pk be the output generated by Al- gorithm 2 in the k−th iteration, then for any k ≥ 1 we have F(Pk) − F(P∗) = O( 1 k), where P∗ is the optimal solution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited Before diving into the detailed proof of convergence rate in Algorithm 3, we first provide two useful lemmas that are important for the following proof process: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' For F(x) = f(x) + g(x), if g(x) is convex, and f(x) is σ−strongly convex and L−smooth, then for any x, y and α > 0 satisfying f(proxα(y)) ≤f(y) + ⟨∇f(y), proxα(y) − y⟩ + α 2 ∥proxα(y) − y∥2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='14) the following inequality holds: F(x) − F(proxα(y)) ≥α 2 ∥x − proxα(y)∥2 − α 2 ∥x − y∥2 + f(x) − f(y) − ⟨∇f(y), x − y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Consider function φ(u) = f(y) + ⟨∇f(y), u − y⟩ + g(u) + α 2 ∥u − y∥2, it is obvious that such φ(u) is α−strongly convex and proxα(y) = arg minu(φ(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Then we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='16) φ(x) − φ(proxα(y)) ≥ α 2 ∥x − proxα(y)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='14): φ(proxα(y)) =f(y) + ⟨∇f(y), proxα(y) − y⟩ + α 2 ∥y − proxα(y)∥2 + g(proxα(y)) ≥f(proxα(y)) + g(proxα(y)) =F(proxα(y)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='17) combine with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='16), we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='18) φ(x) − F(proxα(y)) ≥ α 2 ∥x − proxα(y)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Therefore it’s easy to get the following inequality after we plug in the formula of φ(x): φ(x) − F(proxα(y)) =F(x) − f(x) + f(y) + α 2 ∥x − y∥2 + ⟨∇f(y), x − y⟩ − F(proxα(y)) ≥α 2 ∥x − proxα(y)∥2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='19) which is the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' For any vector a, b and constant β < 1, we have the following equation: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='20) ∥a + b∥2 − β∥a∥2 = (1 − β)∥a + 1 1 − β b∥2 − β 1 − β ∥b∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Here we introduce the theorem about the conver- gence rate of Algorithm 3: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' For F(x) = f(x) + g(x) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='6), g(x) is convex, and f(x) is σ−strongly convex and L−smooth, let c = L σ and t = √c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Let Pk be the kth iteration’s output in Algorithm 3, P∗ be the optimal solution,Vk = F(Pk) − F(P∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Then for any k ≥ 1 we have Vk ≤ (1 − 1 t )k(V0 + σ 2 ∥P0 − P∗∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 and the fact that f(x) is σ−strongly convex and L−smooth, we obtain F(x) − F(proxL(y)) ≥L 2 ∥x − proxL(y)∥2 − L 2 ∥x − y∥2 + σ 2 ∥x − y∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='21) Invoking the above inequality with x = 1 t P∗ +(1− 1 t )Pk and y = Ak in Algorithm 3, we get F(1 t P∗ + (1 − 1 t )Pk) − F(Pk+1) ≥L 2 ∥Pk+1 − (1 t P∗ + (1 − 1 t )Pk)∥2 − L − σ 2 ∥Ak − (1 t P∗ + (1 − 1 t )Pk)∥2 = L 2t2 ∥tPk+1 − (P∗ + (t − 1)Pk)∥2 − L − σ 2t2 ∥tAk − (P∗ + (t − 1)Pk)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='22) Since f is a σ−strongly convex function, for α ∈ [0, 1], we have f(αx + (1 − α)y) ≤ αf(x) + (1 − α)f(y) − σα(1−α) 2 ∥x − y∥2, and obviously 1 t ∈ [0, 1], so we have F(1 t P∗ + (1 − 1 t )Pk) ≤1 t F(P∗) + (1 − 1 t )F(Pk) − σt−1(1 − t−1) 2 ∥Pk − P∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='23) With Vk = F(Pk) − F(P∗), we can get F(1 t P∗ + (1 − 1 t )Pk) − F(Pk+1) ≤(1 − t−1)Vk − Vk+1 − σt−1(1 − t−1) 2 ∥Pk − P∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='24) Combine Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='24) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='22),we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='25) L − σ 2 ∥tAk − (P∗ + (t − 1)Pk)∥2 − σ(t − 1) 2 ∥Pk − P∗∥2 ≥ t2Vk+1 − t(t − 1)Vk + L 2 ∥tPk+1 − (P∗ + (t − 1)Pk)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3 and set a := Pk − P∗, b := t(Ak − Pk), β := σ(t−1) L−σ , then for the left side in the above inequality, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='26) L − σ 2 ∥tAk − (P∗ + (t − 1)Pk)∥2 − σ(t − 1) 2 ∥Pk − P∗∥2 =L − σ 2 {∥t(Ak − Pk) + (Pk − P∗)∥2 − σ(t − 1) L − σ ∥Pk − P∗∥2} =L − σ 2 {L − σt L − σ ∥(Pk − P∗) + L − σ L − σtt(Ak − Pk)∥2 − σ(t − 1) L − σt ∥t(Ak − Pk)∥2} ≤L − σt 2 ∥Pk − P∗ + L − σ L − σtt(Ak − Pk)∥2 Therefore we have the following inequality according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='25): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='27) t(t − 1)Vk + L − σt 2 ∥Pk − P∗ + L − σ L − σtt(Ak − Pk)∥2 ≥ t2Vk+1 + L 2 ∥tPk+1 − (P∗ + (t − 1)Pk)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' With the update rule Ak = Pk + √c−1 √c+1(Pk − Pk−1) in Algorithm 3 and t = √c, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='28) Pk − P∗ + L − σ L − σtt(Ak − Pk) =Pk − P∗ + L − σ L − σt t(t − 1) t + 1 (Pk − Pk−1) =tPk − (P∗ + (t − 1)Pk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Since L = σt2, based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='27) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='28), divide both sides of the inequality by t2, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='29) Vk+1 + σ 2 ∥tPk+1 − (P∗ + (t − 1)Pk)∥2 ≤(1 − 1 t )(Vk + σ 2 ∥tPk − (P∗ + (t − 1)Pk−1)∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' For k = 0, with the initialization setting A0 = P0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='30) P0 − P∗ + L − σ L − σtt(A0 − P0) = P0 − P∗, based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='29), naturally we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='31) Vk + σ 2 ∥tPk − (P∗ + (t − 1)Pk−1)∥2 ≤(1 − 1 t )k(V0 + σ 2 ∥P0 − P∗∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Thus we can get Vk ≤ (1 − 1 t )k(V0 + σ 2 ∥P0 − P∗∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The objective in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3) is not guaranteed to be monotonically non-increasing due to the existence of the σ 2 ∥tPk − (P∗ + (t − 1)Pk−1)∥2 term, but in general we can expect it can achieve optimal solution with a linear convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Also, one can see that when the problem is well-conditioned, it converges faster, otherwise, it can be rather slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 5 Experimental Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 Experiment Setup We compare the experi- mental results of the following MTL algorithms with our method: FedEM [18], DMTL [9], MTFL [10], MMTFL [24], MTRL [28], RMTL [4], CLMT [5], MKMTL [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The empirical studies are conducted on the following benchmark multi-task regression and classification datasets: School [7]: there are exam scores of 15362 students from 139 schools in the dataset, each student is described with 28 attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Thus there are 139 related tasks, each sample has 28 features along with 1 output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We aim to perform multi-task regression to predict exam scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Sarcos [22]: it is collected for an inverse dynamics prediction problem for a seven degrees-of-freedom robot arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The number of related tasks is 7, and there are 21 features for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Following the work in [27] we sample 2000 random samples for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Yale: it contains 165 images from 15 subjects, each image is scaled to 32 × 32 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We use the first 8 subjects from it to construct related tasks, each task is defined as a binary classification problem of classifying two subjects, there are 28 binary classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' MNIST [12]: we use a subset from the MNIST dataset with 10000 samples of 10 handwritten digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We cast the multi-task learning as 45 binary classification tasks to classify pairs of digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Letter [8]: it consists of handwritten letters from different writers, we construct 8 binary classification tasks from it to distinguish between pairs of letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' ORL [21]: there are 10 different images of 40 distinct subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' What’s different for the ORL dataset is we construct 40 one-vs-all multi-class classification tasks from it rather than one-vs-one binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' During the experiment process, each dataset is randomly split into a training set and a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In all classification tasks, the data is split according to a rough 60%-40% training-test split ratio, in the School regression task 20 random samples are used for training, and in the Sarcos regression task, 50 random samples are selected for training and the rest for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The prior knowledge matrix D can be initialized based on known prior, our common sense, and the correlation among features obtained with statistical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In the Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited Table 1: Regression tasks: generalization performance measures over ten runs Metric Dataset FedEM DMTL MTFL MMTFL OURS-NATURAL VE (%) School 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 Sarcos 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='6±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='6 nMSE (%) School 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='5 Sarcos 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 20.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7 MNIST 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8 93.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='5±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3 ORL 77.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8 experiment, we have two methods to obtain the required D: the first method is to utilize the natural correlation among features, we calculate the covariance matrix for all the features, and select the strongest ones as the information contained in D, we call the prior knowledge matrix obtained in this way the natural D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' the second method is we create the strong correlation among features manually by appending features repeatedly on purpose, for example, we transform the original feature (x1, x2, x3) into (x1, x2, x3, x1) to enhance the correlation among features (this may introduce multicollinearity, but since we only care about the predictive result, it should be fine), the matrix obtained is called as the artificial D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Parameters in all the methods are tuned using 5- fold cross-validation, and for each method, we stop the experiment when the objective change is < 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We use Matlab R2019a on a laptop with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='4 GHz QuadCore Intel Core i5 processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 Results Results of the experiment are presented in Table 1 and Table 2, for regression tasks and classification tasks respectively, our method with a natural D is denoted as OURS-NATURAL, with an artificial D is denoted as OURS-ART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In OURS-ART, we manually repeat about 5% features to construct D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In the case of regression tasks, we report the variance explained (VE) and the normalized mean squared error (nMSE) following previous studies [4,10], whereas the receiver operating characteristic (ROC) curve and the area under the ROC curve (AUC) are employed as the classification performance measurements as used in previous studies [4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' ROC curves show the performance of a binary classification model at all classification thresholds by plotting the true positive rate against the false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Higher explained variance and AUC indicate better performance, and the opposite for nMSE reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Each experiment with one specific dataset is repeated 10 times and we report the averaged performance and the standard deviation, the best performances are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In Table 1 and Table 2, our method can achieve the best performance in most cases, while the FedEM Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9 1 True positive rate (b) MNIST 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9 1 False positive rate 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='9 1 True positive rate (c) ORL Figure 2: The ROC curve for Yale, MNIST, ORL dataset with tuned parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' (a) AUC, Yale (b) AUC, ORL (c) VE, School (d) nMSE, School Figure 3: Ablation study – the influence of regularization parameters on learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' method achieves better results with the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' The reason perhaps is that neural networks still have unique advantages when dealing with large-scale com- plex image datasets, and prior knowledge about features is hard to establish with complex image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Also, in re- gression tasks, our method works well with both natural prior knowledge and artificial prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' In im- age classification tasks, the improvement in performance with artificial prior knowledge matrices is more signifi- cant than that with natural ones, the reason may be due to the fact that the natural correlations between features in regression tasks are stronger than those involved in image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We present the ROC curves with a natural prior knowledge matrix for the classification tasks on Yale, MNIST, and ORL datasets in Figure 2, the slim colorful lines are the ROC curves for each task, and the weighted black line is the averaged ROC curve over all the tasks, and the green area represents the range between the mean ± standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' While for each dataset there is a small number of tasks with performances that are not so perfect on some level, the overall performance is satisfactory with a pretty high AUC, and the mean − standard deviation curve is almost always over the diagonal line, especially for the one-vs-one classification tasks on Yale and MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We also provide Figure 3 to illustrate the influence of regularization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' With a super wide tuning range for each parameter, which is from 1 to 100, the effect of each parameter on the performance is not significant until the value reaches a threshold, there is a generous range for each parameter to be able to provide stable and great performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' We can roughly draw the conclusion that for classification tasks, no matter whether it’s one-vs-one binary classification tasks or one-vs-all binary classification tasks, the values of regularization parameters have a marginal influence on the results as long as the value is within a reasonable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' While in a regression situation the performance is much more sensitive to the value of parameters, especially to the group Lasso penalty parameter, which is in accordance with common sense that under-fitting happens with large regularization term parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' 6 Conclusion We propose a convex formulation of multi-task learning problem utilizing prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' A novel optimiza- tion algorithm to solve the formulated problem with a linear convergence rate is proposed with theoretical guarantee instead of sub-linear rate of the counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Results on benchmark datasets with both regression and Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited 1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='55 100 入 1 Eclassification tasks demonstrate the effectiveness and advantages of our proposed multi-task learning formula- tion and algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Identifying outlier tasks using the framework will be left to our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' References [1] Rie Kubota Ando, Tong Zhang, and Peter Bartlett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' A framework for learning predictive structures from multiple tasks and unlabeled data.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Journal of the Operations Research Society of China, 1(2):163–186, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' [26] Mengyuan Zhang and Kai Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Rethinking symmetric matrix factorization: A more general and better clus- tering perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='02528, 2022.' metadata={'source': 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+page_content=' A convex formulation for learning task relationships in multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' arXiv preprint arXiv:1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content='3536, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} +page_content=' Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfm_1e/content/2301.01572v1.pdf'} diff --git a/aNFJT4oBgHgl3EQf8C11/content/2301.11681v1.pdf b/aNFJT4oBgHgl3EQf8C11/content/2301.11681v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e37e6a0d2b9d32fc911dd843fb05fd8de3bbf6d8 --- /dev/null +++ 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a/c9E1T4oBgHgl3EQfxwVP/content/tmp_files/2301.03425v1.pdf.txt b/c9E1T4oBgHgl3EQfxwVP/content/tmp_files/2301.03425v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..496b3c5490a9983962fb5175303f0a8a2515b7a0 --- /dev/null +++ b/c9E1T4oBgHgl3EQfxwVP/content/tmp_files/2301.03425v1.pdf.txt @@ -0,0 +1,544 @@ +Received: Added at production +Revised: Added at production +Accepted: Added at production +DOI: xxx/xxxx +ORIGINAL ARTICLE +Dynamic structure factor and excitation spectrum of the +one-component plasma: the case of weak to moderate +magnetization +Hanno Kählert* | Michael Bonitz +1Institut für Theoretische Physik und +Astrophysik, +Christian-Albrechts-Universität zu Kiel, +Germany +Correspondence +*Hanno Kählert, Institut für Theoretische +Physik und Astrophysik, Christian- +Albrechts-Universität zu Kiel, Germany. +Email: kaehlert@theo-physik.uni-kiel.de +Summary +Magnetized plasmas are well known to exhibit a rich spectrum of collective modes. +Here, we focus on the density modes in dense or cold plasmas, where strong coupling +effects alter the mode spectrum known from traditional weakly coupled plasmas. +In particular, we study the dynamic structure factor (DSF) of the magnetized one- +component plasma with molecular dynamics simulations. Extending our previous +results [H. Kählert and M. Bonitz, Phys. Rev. Research 2022, 4, 013197], it is shown +that Bernstein modes can be observed in the weakly magnetized regime, where they +are found below the upper hybrid frequency, provided the coupling strength is suf- +ficiently low. We investigate the DSF for a variety of different wave numbers and +plasma parameters and show that even small magnetization can give rise to a strong +zero-frequency mode perpendicular to the magnetic field and change the dispersion +as well as the damping of the upper hybrid mode. +KEYWORDS: +magnetized plasma, strongly coupled plasma, dense plasma, correlated plasma +1 +INTRODUCTION +Strong coupling effects can be observed in a variety of different plasmas, ranging from dusty plasmas[1] to confined charges[2–4] +and expanding ultra cold neutral plasmas[5–7]. Even though these systems have very different density and temperature, they can +all be characterized as strongly coupled, meaning that the Coulomb coupling parameter, +Γ = +푄2 +4휋휖0 푎 푘B푇 , +(1) +is close to or exceeds a value of one. The coupling is determined by the plasma density 푛 via the Wigner-Seitz radius, 푎 = +[3∕(4휋푛)]1∕3, the temperature 푇 , and the particle charge 푄, showing that a large Γ can be achieved with different combinations +of 푛, 푇 , and 푄. Plasmas with the same Γ can behave very similarly even when their physical parameters are vastly different. +While the properties of (classical) strongly coupled plasmas have been studied in great detail, mainly through simulations or +theory of reduced model systems such as the one-component plasma (OCP), e.g., Refs.[8–13], less attention has been paid to the +characteristics of magnetized strongly coupled plasmas. The dense Coulomb liquids and solids in the crust of neutron stars[14] +or laser-cooled ions in Penning traps[15,16] can easily reach a regime with very high magnetization. In addition, ultra cold neutral +plasmas have recently been studied under the influence of external magnetic fields[17–20]. The strongly coupled dust particles in +dusty plasmas are notoriously difficult to magnetize. However, setting the dusty plasma into rotation, such that the Coriolis force +arXiv:2301.03425v1 [physics.plasm-ph] 9 Jan 2023 + +2 +becomes appreciable, allows one to create an intense effective magnetization[21,22], which has been utilized to study waves[23] and +transport phenomena in two-dimensional dusty plasmas[24]. External magnetic fields also play an important role in the context +of inertial fusion concepts[25–28]. Previous theoretical work for strongly coupled magnetized plasmas has been performed, e.g., +for transport properties[29–33], stopping power and friction[34–37], and waves[38–45]. +In this work, we focus on density correlations in the time and spatial domain by studying the dynamic structure factor (DSF) +of the magnetized OCP. Specifically, we extend the results of Ref.[46] by mainly considering the DSF for weakly magnetized +systems. The results should be particularly useful for experiments in which magnetization is difficult to achieve. This is the case, +i.a., in very dense plasmas, where the high plasma frequency, 휔p = +√ +푛 푄2∕(휖0푚) (mass 푚), requires very intense magnetic +fields to increase the magnetization parameter 훽 = 휔c∕휔p, where 휔c = 푄퐵∕푚 is the cyclotron frequency (magnetic field 퐵), to +values on the order of one. Of particular interest is the excitation spectrum perpendicular to the magnetic field, which features +Bernstein modes in the weakly coupled domain and higher harmonics of the upper hybrid mode in strongly coupled systems. +It is shown that Bernstein modes exist also in weakly magnetized plasmas at moderate coupling, when they are located below +the upper hybrid mode. We further investigate the latter mode for a variety of coupling strengths in the Γ ∼ 1 regime, and the +transition to an unmagnetized plasma. +The work is organized as follows. In Sec. 2, we introduce the simulation method and detail the numerical parameters. The +results of the simulation are then presented in Sec. 3. We conclude with a brief summary and discussion in Sec. 4. +2 +SIMULATION METHOD +We briefly summarize the methodology of the simulations, which is the same as in our previous work[46]. We simulate the +classical one-component plasma, i.e., 푁 identical particles with mass 푚 and charge 푄 in a uniformly charged background, +subject to an external magnetic field B = 퐵 ̂푒푧. The particles are placed in a cubic box. Simulations are conducted with the +LAMMPS code[47,48], thereby integrating the equations of motion with an extension of the velocity Verlet algorithm[49]. We +study the density autocorrelation function in Fourier space, i.e., the dynamic structure factor, +푆(k, 휔) = 1 +2휋 +∞ +∫ +−∞ +퐹(k, 푡)푒푖휔푡푑푡, +(2) +where 퐹(k, 푡) = ⟨푛k(푡)푛∗ +k(0)⟩∕푁. The DSF is computed from a Fourier transform of the microscopic particle density, 푛k(푡) = +∑푁 +푖=1 푒−푖k⋅r푖(푡), where r푖(푡) (푖 = 1 … 푁) are the particle positions[46,50]. +As discussed above, the magnetization will be given as 훽 = 휔c∕휔p. The number of particles is set to either 푁 = 80, 000 or +푁 = 10, 000. In particular, in order to get access to (perpendicular) wave numbers 푘⟂ that are small compared to the inverse +Larmor radius, 푟−1 +L += 휔c∕푣th [thermal velocity 푣th = +√ +푘B푇 ∕푚], which is an important length scale for waves in the weakly +coupled domain, simulations with high particle numbers are required as the minimum wave number, 푘min = 2휋∕퐿, is dictated +by the size of the simulation cell 퐿, with 퐿∕푎 = (4휋 푁∕3)1∕3 (for cubic boxes). This is particularly important for small 훽, as +can be seen from 푘min푟L = 푘min푎 ⋅ (푟L∕푎) = 2휋 × [3∕(4휋 푁)]1∕3 × ( +√ +3Γ훽)−1, which should satisfy 푘min푟L < 1. The time step +varies from Δ푡 휔p = 0.0015 to Δ푡 휔p = 0.01, depending on the coupling strength. We conduct multiple new simulations for +weak magnetic fields but also analyze data produced in the simulation runs of Ref.[46]. +3 +RESULTS +Since the effect of the magnetic field on the DSF for wave vectors parallel to the external field is typically weak, in particular +for 훽 < 1, we focus on the DSF for wave vectors k perpendicular to B in the following. +Figure 1 depicts the transition of the DSF at a small wave number from weak to strong coupling in weakly magnetized +plasmas, thereby complementing results in Fig. 3 of Ref.[46] with much stronger magnetization. According to the RPA (random +phase approximation) description of collective modes in magnetized plasmas[51,52], Bernstein modes below the upper hybrid +frequency, 휔UH = +√ +휔2 +p + 휔2 +c, originate at 푛 휔c in the small wave number limit, with 푛 ≥ 2. This is clearly seen in Fig. 1(a) +for Γ = 0.125, where two maxima are found below the upper hybrid frequency. The lower peak is located slightly below 2 휔c, +which could be caused by thermal effects due to a finite 푘⟂푟L ≈ 0.57. The shift as well as the peak intensity become weaker +and eventually disappear as the coupling strength increases. The DSF for plasmas with very strong coupling, Γ ≳ 3, exhibits a + +3 +10−10 +10−8 +10−6 +10−4 +10−2 +0 +2 +4 +6 +8 +Γ = 0.125 +Γ = 30 +0 +2 +4 +6 +Γ = 0.125 +Γ = 30 +S(k, ω)ωc +ω/ωc +(a) β ≈ 0.258 +ωUH = 4 ωc +ω/ωc +(b) β ≈ 0.354 +ωUH = 3 ωc +Figure 1 DSF for k ⟂ B and coupling parameters Γ ∈ {0.125, 0.25, 0.5, 1, 3, 10, 30} (from top to bottom). The perpendicular +wave number is 푘⟂푎 = 0.0905. The magnetization is indicated in the figure. The vertical lines indicate harmonics of the cyclotron +frequency (long dashed, grey) and the upper hybrid frequency (short dashed, red). +rapid decay for frequency above 2 휔UH, as observed previously[46]. Very similar behavior can be seen in Fig. 1(b) with a slightly +larger magnetization. Here, only one Bernstein mode exists below the upper hybrid frequency at weak coupling. +The discussion above shows that the observations made in Ref.[46] remain valid in weakly magnetized plasmas, i.e., the +Bernstein modes that exist in weakly coupled systems disappear upon increase of Γ. When the plasma approaches the strongly +coupled regime, the DSF decays rapidly for 휔 ≳ 2휔UH, but, in the present simulations, the DSF does not (yet) display clear +peaks around the harmonics of 휔UH due to the weak magnetization. The harmonics are significantly more pronounced when +훽 ≳ 1 and Γ ≫ 1[46]. +We now consider a larger range of wave numbers in Fig. 2, for various coupling and magnetization strengths. Consider first +the top row with Γ = 0.25. For 훽 ≈ 0.258 [Fig. 2(a)], the DSF has a clear peak at small 푘⟂, which broadens and shifts to +higher frequencies as the wave number increases. A weak remnant of the lowest Bernstein mode is visible at 푘⟂푎 = 0.271 near +2 휔c, see the dashed vertical line. At the largest wave number, 푘⟂푎 = 1.18, the main peak has disappeared, and the DSF decays +monotonically, with a broad plateau region for 휔 ≲ 1.4 휔p. The lowest Bernstein mode becomes somewhat more pronounced as +the magnetization is increased to 훽 ≈ 0.354 [Fig. 2(b)]. At the largest magnetization [훽 ≈ 0.894, Fig. 2(c)], the Bernstein modes +are located above the upper hybrid frequency. In this case, they are clearly visible in the DSF. In particular, for the largest wave +number, the DSF no longer decays monotonically but is modulated by peaks around the harmonics of the cyclotron frequency. +The upper hybrid peak shifts to lower frequencies with increasing wave number. As the coupling parameter is raised to Γ = 1 +(middle row), any obvious signature of the Bernstein modes is lost for 훽 ≈ 0.258 and 훽 ≈ 0.354. Only for 훽 ≈ 0.894, they +remain detectable in the spectrum. At Γ = 3 (bottom row), the upper hybrid mode is the only visible excitation in the spectrum, +apart from a zero frequency peak, which generally becomes more dominant at larger 훽, see also Γ = 1 and Γ = 0.25. +Considering the position of the main peak with respect to the upper hybrid frequency, the results in Fig. 2 suggest that +the dispersion of the upper hybrid mode can change from positive at low 훽 to negative at high 훽, similar to the transition in +the unmagnetized OCP upon increase of Γ[12,53]. Before we discuss the simulation results in more detail, we first recall the +modification of the dispersion relation in the random-phase-approximation (RPA)[52], shown in Fig. 3(a). It is obtained from the +condition 휖RPA(k, 휔) = 0, where +휖RPA(k, 휔) = 1 + +1 +푘2휆2 +[ +1 + +∞ +∑ +푛=−∞ +휔 +휔 − 푛 휔c +퐼푛(휂)푒−휂 휁푛 푍(휁푛) +] +(3) + +4 +0.00 +0.05 +0.10 +0.15 +0.20 +0 +1 +2 +3 +0.633 +1.63 +2.44 +0 +1 +2 +3 +0.633 +1.63 +2.44 +0 +1 +2 +3 +0.633 +1.63 +2.44 +0.00 +0.05 +0.10 +0.15 +0.20 +0.452 +0.905 +2.08 +0.452 +0.905 +2.08 +0.905 +2.08 +0.00 +0.05 +0.10 +0.15 +0.20 +0.271 +0.633 +k⊥a = 1.18 +0.271 +0.633 +k⊥a = 1.18 +S(k, ω)ωp +ω/ωp +(g) +Γ = 3 +ω/ωp +(h) +ω/ωp +(i) +S(k, ω)ωp +(d) +Γ = 1 +(e) +(f) +0.452 +S(k, ω)ωp +(a) +β ≈ 0.258, ωUH = 4 ωc +Γ = 0.25 +β ≈ 0.354, ωUH = 3 ωc +(b) +(c) +β ≈ 0.894, ωUH = 1.5 ωc +0.271 +0.633 +k⊥a = 1.18 +Figure 2 DSF for k ⟂ B and coupling parameters Γ ∈ {0.25, 1, 3} (top row to bottom row) and 훽 ∈ {0.258, 0.354, 0.894} (left +column to right column). The perpendicular wave numbers 푘⟂푎 are indicated in the figure. The vertical lines show harmonics +of the cyclotron frequency (long dashed, grey) and the upper hybrid frequency (short dashed, red). +is the RPA dielectric function[54,55]. Here, 휆 = 푣th∕휔p is the usual Debye length, 휂 = 푘2 +⟂푟2 +L, 휁푛 = (휔 − 푛 휔c)∕( +√ +2|푘∥|푣th), and +퐼푛(휂) [푍(휁푛)] denotes the modified Bessel [plasma dispersion] function. For perpendicular wave propagation, the dispersion +relation is to be determined from the 푘∥ → 0 limit of Eq. (3)[52], resulting in +1 = 2 +∞ +∑ +푛=1 +퐼푛(휂)푒−휂 +휂 +푛2휔2 +p +휔2 − 푛2휔2 +c +. +(4) +We are primarily interested in the dispersion of the mode that starts at the upper hybrid frequency. The location of the latter +is indicated by the squares in Fig. 3(a). For 훽 < 1∕ +√ +3 ≈ 0.577 (훽 > 1∕ +√ +3), the upper hybrid frequency lies above (below) + +5 +0 +1 +2 +3 +4 +0 +1 +2 +3 +(a) +0.8 +0.9 +1 +1.1 +1.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(b) Γ = 0.25 +ω/ωc +k⊥rL +β = 0.3 +0.354 +0.4 +0.7 +0.894 +ω/ωUH +k⊥rL +β = 0.354 +0.4 +0.7 +0.894 +Figure 3 (a) RPA dispersion relation for the magnetized OCP for various levels of magnetization, as indicated in the figure. (b) +Comparison of the upper hybrid dispersion from the RPA (lines) with the main peak position from the DSF (symbols). Note the +different scaling of the vertical axis in (a) and (b). +the first Bernstein mode, which starts at 2 휔c. From the examples shown, one observes that the dispersion of the upper hybrid +mode is positive in the former case and negative in the latter. In fact, expanding Eq. (4) in powers of 휂 = 푘2 +⟂푟2 +L, one finds +휔2(푘⟂) ≈ 휔2 +UH + 3 휔2 +c 푘2 +⟂푟2 +L∕(1 − 3훽2), where the ∼ 푘2 +⟂ term becomes singular at 훽 = 1∕ +√ +3 and changes sign. In case the +upper hybrid frequency exactly coincides with one of the cyclotron harmonics, there are two modes originating from the same +frequency at 푘⟂ = 0. As discussed above, for 휔UH = 2휔c [훽 ≡ 1∕ +√ +3], the previous expression diverges and one finds, instead, +two modes with a linear dependence on 푘⟂, namely 휔2(푘⟂) ≈ (2휔c)2 ± 휔2 +p푘⟂푟L, see Ref.[46] for a comparison with simulations. +For 휔UH = 3휔c, the result is 휔2(푘⟂) = (3휔c)2 + 휔2 +p +( +3 +10 ± +√ +186 +20 +) +푘2 +⟂푟2 +L. For all other cases, 휔UH = 푛휔c, with 푛 ≥ 4, the general +expression for the upper hybrid dispersion remains valid, and the frequency of the second mode decreases ∼ (푘⟂푟L)2푛−2. +Figure 3(b) shows a comparison of the RPA dispersion and the peak position from the DSF for Γ = 0.25. For 훽 = 0.7 and +훽 = 0.894, there is good agreement between theory and simulation, as observed previously in Ref.[46] for 훽 = 1∕ +√ +3. However, +larger deviations occur for 훽 = 0.354 and, in particular, for 훽 = 0.4. In the latter case, the peak position from the simulations +increases monotonically with 푘⟂푟L, with a leap around 푘⟂푟L ≈ 0.75. In contrast, the frequency of the upper hybrid mode +from the RPA has a maximum at 푘⟂푟L ≈ 0.4 and generally lies well below the peak of the DSF, except for the smallest wave +numbers. Interestingly, the RPA Bernstein mode above the upper hybrid mode is in reasonable agreement with the simulations +for 푘⟂푟L ≳ 0.75. Even though 훽 = 0.354 is only marginally smaller than 훽 = 0.4, the RPA spectrum is quite different [shown is +the upper of the two modes that start at 휔UH = 3 휔c] and agrees much better with the simulation data, which, on the contrary, +are very similar to those for 훽 = 0.4. These observations and the small separation of the RPA modes in frequency space for +훽 = 0.4 suggest that both the upper hybrid mode and the nearby Bernstein mode may contribute significantly to the DSF. We +keep in mind, however, that the coupling is already outside the regime where the RPA is expected to apply. We also reiterate +that the peaks in the DSF do not necessarily correspond precisely to the frequencies of the collective modes[56,57]. +Finally, the evolution of the DSF upon magnetizing the plasma is studied in Fig. 4 for Γ = 1 and three different wave numbers. +At 푘⟂푎 = 0.724 [Fig. 4(a)], the plasmon (upper hybrid) peak for 훽 = 0 (훽 > 0) initially becomes broader as 훽 increases while, +at the same time, the peak height decreases. At larger magnetization, the trend is reversed. The peak becomes sharper, and the +peak intensity grows again (훽 = 0.894). Since the normalization of the DSF is independent of 훽, ∫ ∞ +−∞ 푆(k, 휔; 훽)푑휔 = 푆(푘), +where 푆(푘) is the static structure factor, the spectral weight contained in the plasmon merely becomes redistributed—the total + +6 +0.00 +0.05 +0.10 +0.15 +0.20 +0 +0.5 +1 +1.5 +β = 0 +0 +0.5 +1 +1.5 +2 +β = 0 +β ≈ 0.894 +0 +0.5 +1 +1.5 +2 +β = 0 +β ≈ 0.894 +S(k, ω)ωp +ω/ωp +(a) k⊥a = 0.724 +β ≈ 0.894 +0.577 +0.354 +ω/ωp +(b) k⊥a = 1.27 +0.577 +0.354 +ω/ωp +(c) k⊥a = 1.81 +0.354 +0.577 +Figure 4 DSF perpendicular to the magnetic field for Γ = 1 and various magnetizations 훽. Also shown is the unmagnetized +limit with 훽 = 0. The wave number 푘⟂푎 is indicated in the figure. +weight remains constant. In addition to the plasmon/upper hybrid feature, a very sharp zero-frequency peak develops, even for +relatively weak magnetization strengths, which is absent in the 훽 = 0 spectrum. For 푘⟂푎 = 1.27 [Fig. 4(b)] and 훽 = 0, the +plasmon peak is already much broader, which is well known from simulations of the unmagnetized OCP[58]. As the plasma +becomes magnetized, the peak shifts to lower frequencies. At 훽 ≈ 0.577, a strong zero-frequency peak has emerged. Increasing +훽 further to 훽 ≈ 0.894 leads to the formation of a well-pronounced upper hybrid peak. In case of the largest wave number, +푘⟂푎 = 1.81 [Fig. 4(c)], the plasmon peak practically vanishes for 훽 = 0. Similar to Fig. 4(b), increasing the magnetization leads +to the formation of a well-pronounced peak in the DSF, both at zero and finite frequency. Comparing the effect of the magnetic +field on the DSF for different wave numbers, one finds that the strongest modifications, compared to the zero magnetic field +case, are found at small wave numbers. We note that the peak position of the DSF is above (below) the upper hybrid frequency +for small (large) magnetization, as in Fig. 3(b). For a related discussion of the harmonics in the magnetized and unmagnetized +OCP, see also Fig. 4 in Ref.[46]. +4 +CONCLUSIONS +In summary, we have studied the DSF of the weakly magnetized OCP for wave vectors perpendicular to the magnetic field for +a variety of coupling strengths and wave numbers. At small wave numbers, the DSF shows signatures of Bernstein modes even +for weak magnetization, 0.25 ≲ 훽 ≲ 1, provided the coupling is sufficiently low, Γ ≲ 0.5. Bernstein modes disappear as soon as +Γ grows beyond Γ ≳ 1 − 3. In this strongly coupled regime, the DSF decays rapidly for 휔 > 2휔UH, but the magnetization is too +low to observe clear higher harmonics of the upper hybrid mode as in Ref.[46], where larger field strengths were considered. +The main peak in the DSF broadens as the wave number increases, similar to the plasmon peak for 훽 = 0. For large wave +numbers, the DSF decays monotonically in the case of weak magnetization. For stronger magnetization, Bernstein modes can +be observed that modulate the decay to high frequencies. In addition, a strong zero-frequency peak is observed, which appears +already for relatively mild magnetization. At finite frequencies, the dispersion of the main peak was studied in some detail for +Γ = 0.25. It changes from positive to negative upon increase of 훽. Since it is known that the dispersion of the plasmon becomes +negative for Γ ≳ 10 already in the unmagnetized OCP[12,53], this effect could vanish in the strongly coupled regime. Here, a +theory for the dispersion must include correlations[59]. The comparison between the RPA dispersion and the peak position from +the simulations shows that, under certain conditions, the main peak in the DSF could be influenced by the interplay of the upper +hybrid mode and a nearby Bernstein mode. 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Plasmas 2016, 23 (10), 104506. + diff --git a/c9E1T4oBgHgl3EQfxwVP/content/tmp_files/load_file.txt b/c9E1T4oBgHgl3EQfxwVP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..be395006aaa7b5074568954623725495470ae001 --- /dev/null +++ b/c9E1T4oBgHgl3EQfxwVP/content/tmp_files/load_file.txt @@ -0,0 +1,948 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf,len=947 +page_content='Received: Added at production Revised: Added at production Accepted: Added at production DOI: xxx/xxxx ORIGINAL ARTICLE Dynamic structure factor and excitation spectrum of the one-component plasma: the case of weak to moderate magnetization Hanno Kählert* | Michael Bonitz 1Institut für Theoretische Physik und Astrophysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Christian-Albrechts-Universität zu Kiel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Germany Correspondence Hanno Kählert,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Institut für Theoretische Physik und Astrophysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Christian- Albrechts-Universität zu Kiel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Email: kaehlert@theo-physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='uni-kiel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='de Summary Magnetized plasmas are well known to exhibit a rich spectrum of collective modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Here, we focus on the density modes in dense or cold plasmas, where strong coupling effects alter the mode spectrum known from traditional weakly coupled plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In particular, we study the dynamic structure factor (DSF) of the magnetized one- component plasma with molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Extending our previous results [H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Kählert and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bonitz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Research 2022, 4, 013197], it is shown that Bernstein modes can be observed in the weakly magnetized regime, where they are found below the upper hybrid frequency, provided the coupling strength is suf- ficiently low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We investigate the DSF for a variety of different wave numbers and plasma parameters and show that even small magnetization can give rise to a strong zero-frequency mode perpendicular to the magnetic field and change the dispersion as well as the damping of the upper hybrid mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' KEYWORDS: magnetized plasma, strongly coupled plasma, dense plasma, correlated plasma 1 INTRODUCTION Strong coupling effects can be observed in a variety of different plasmas, ranging from dusty plasmas[1] to confined charges[2–4] and expanding ultra cold neutral plasmas[5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Even though these systems have very different density and temperature, they can all be characterized as strongly coupled, meaning that the Coulomb coupling parameter, Γ = 푄2 4휋휖0 푎 푘B푇 , (1) is close to or exceeds a value of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The coupling is determined by the plasma density 푛 via the Wigner-Seitz radius, 푎 = [3∕(4휋푛)]1∕3, the temperature 푇 , and the particle charge 푄, showing that a large Γ can be achieved with different combinations of 푛, 푇 , and 푄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Plasmas with the same Γ can behave very similarly even when their physical parameters are vastly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' While the properties of (classical) strongly coupled plasmas have been studied in great detail, mainly through simulations or theory of reduced model systems such as the one-component plasma (OCP), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [8–13], less attention has been paid to the characteristics of magnetized strongly coupled plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The dense Coulomb liquids and solids in the crust of neutron stars[14] or laser-cooled ions in Penning traps[15,16] can easily reach a regime with very high magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In addition, ultra cold neutral plasmas have recently been studied under the influence of external magnetic fields[17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The strongly coupled dust particles in dusty plasmas are notoriously difficult to magnetize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' However, setting the dusty plasma into rotation, such that the Coriolis force arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='03425v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='plasm-ph] 9 Jan 2023 2 becomes appreciable, allows one to create an intense effective magnetization[21,22], which has been utilized to study waves[23] and transport phenomena in two-dimensional dusty plasmas[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' External magnetic fields also play an important role in the context of inertial fusion concepts[25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Previous theoretical work for strongly coupled magnetized plasmas has been performed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=', for transport properties[29–33], stopping power and friction[34–37], and waves[38–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In this work, we focus on density correlations in the time and spatial domain by studying the dynamic structure factor (DSF) of the magnetized OCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Specifically, we extend the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [46] by mainly considering the DSF for weakly magnetized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The results should be particularly useful for experiments in which magnetization is difficult to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' This is the case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=', in very dense plasmas, where the high plasma frequency, 휔p = √ 푛 푄2∕(휖0푚) (mass 푚), requires very intense magnetic fields to increase the magnetization parameter 훽 = 휔c∕휔p, where 휔c = 푄퐵∕푚 is the cyclotron frequency (magnetic field 퐵), to values on the order of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Of particular interest is the excitation spectrum perpendicular to the magnetic field, which features Bernstein modes in the weakly coupled domain and higher harmonics of the upper hybrid mode in strongly coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' It is shown that Bernstein modes exist also in weakly magnetized plasmas at moderate coupling, when they are located below the upper hybrid mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We further investigate the latter mode for a variety of coupling strengths in the Γ ∼ 1 regime, and the transition to an unmagnetized plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2, we introduce the simulation method and detail the numerical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The results of the simulation are then presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We conclude with a brief summary and discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2 SIMULATION METHOD We briefly summarize the methodology of the simulations, which is the same as in our previous work[46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We simulate the classical one-component plasma, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=', 푁 identical particles with mass 푚 and charge 푄 in a uniformly charged background, subject to an external magnetic field B = 퐵 ̂푒푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The particles are placed in a cubic box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Simulations are conducted with the LAMMPS code[47,48], thereby integrating the equations of motion with an extension of the velocity Verlet algorithm[49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We study the density autocorrelation function in Fourier space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=', the dynamic structure factor, 푆(k, 휔) = 1 2휋 ∞ ∫ −∞ 퐹(k, 푡)푒푖휔푡푑푡, (2) where 퐹(k, 푡) = ⟨푛k(푡)푛∗ k(0)⟩∕푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The DSF is computed from a Fourier transform of the microscopic particle density, 푛k(푡) = ∑푁 푖=1 푒−푖k⋅r푖(푡), where r푖(푡) (푖 = 1 … 푁) are the particle positions[46,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' As discussed above, the magnetization will be given as 훽 = 휔c∕휔p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The number of particles is set to either 푁 = 80, 000 or 푁 = 10, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In particular, in order to get access to (perpendicular) wave numbers 푘⟂ that are small compared to the inverse Larmor radius, 푟−1 L = 휔c∕푣th [thermal velocity 푣th = √ 푘B푇 ∕푚], which is an important length scale for waves in the weakly coupled domain, simulations with high particle numbers are required as the minimum wave number, 푘min = 2휋∕퐿, is dictated by the size of the simulation cell 퐿, with 퐿∕푎 = (4휋 푁∕3)1∕3 (for cubic boxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' This is particularly important for small 훽, as can be seen from 푘min푟L = 푘min푎 ⋅ (푟L∕푎) = 2휋 × [3∕(4휋 푁)]1∕3 × ( √ 3Γ훽)−1, which should satisfy 푘min푟L < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The time step varies from Δ푡 휔p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='0015 to Δ푡 휔p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='01, depending on the coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We conduct multiple new simulations for weak magnetic fields but also analyze data produced in the simulation runs of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 3 RESULTS Since the effect of the magnetic field on the DSF for wave vectors parallel to the external field is typically weak, in particular for 훽 < 1, we focus on the DSF for wave vectors k perpendicular to B in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Figure 1 depicts the transition of the DSF at a small wave number from weak to strong coupling in weakly magnetized plasmas, thereby complementing results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 3 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [46] with much stronger magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' According to the RPA (random phase approximation) description of collective modes in magnetized plasmas[51,52], Bernstein modes below the upper hybrid frequency, 휔UH = √ 휔2 p + 휔2 c, originate at 푛 휔c in the small wave number limit, with 푛 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' This is clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 1(a) for Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='125, where two maxima are found below the upper hybrid frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The lower peak is located slightly below 2 휔c, which could be caused by thermal effects due to a finite 푘⟂푟L ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The shift as well as the peak intensity become weaker and eventually disappear as the coupling strength increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The DSF for plasmas with very strong coupling, Γ ≳ 3, exhibits a 3 10−10 10−8 10−6 10−4 10−2 0 2 4 6 8 Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='125 Γ = 30 0 2 4 6 Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='125 Γ = 30 S(k, ω)ωc ω/ωc (a) β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='258 ωUH = 4 ωc ω/ωc (b) β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354 ωUH = 3 ωc Figure 1 DSF for k ⟂ B and coupling parameters Γ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='5, 1, 3, 10, 30} (from top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The perpendicular wave number is 푘⟂푎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='0905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The magnetization is indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The vertical lines indicate harmonics of the cyclotron frequency (long dashed, grey) and the upper hybrid frequency (short dashed, red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' rapid decay for frequency above 2 휔UH, as observed previously[46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Very similar behavior can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 1(b) with a slightly larger magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Here, only one Bernstein mode exists below the upper hybrid frequency at weak coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The discussion above shows that the observations made in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [46] remain valid in weakly magnetized plasmas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=', the Bernstein modes that exist in weakly coupled systems disappear upon increase of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' When the plasma approaches the strongly coupled regime, the DSF decays rapidly for 휔 ≳ 2휔UH, but, in the present simulations, the DSF does not (yet) display clear peaks around the harmonics of 휔UH due to the weak magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The harmonics are significantly more pronounced when 훽 ≳ 1 and Γ ≫ 1[46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We now consider a larger range of wave numbers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2, for various coupling and magnetization strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Consider first the top row with Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For 훽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='258 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2(a)], the DSF has a clear peak at small 푘⟂, which broadens and shifts to higher frequencies as the wave number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A weak remnant of the lowest Bernstein mode is visible at 푘⟂푎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='271 near 2 휔c, see the dashed vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At the largest wave number, 푘⟂푎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='18, the main peak has disappeared, and the DSF decays monotonically, with a broad plateau region for 휔 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='4 휔p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The lowest Bernstein mode becomes somewhat more pronounced as the magnetization is increased to 훽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At the largest magnetization [훽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2(c)], the Bernstein modes are located above the upper hybrid frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In this case, they are clearly visible in the DSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In particular, for the largest wave number, the DSF no longer decays monotonically but is modulated by peaks around the harmonics of the cyclotron frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The upper hybrid peak shifts to lower frequencies with increasing wave number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' As the coupling parameter is raised to Γ = 1 (middle row), any obvious signature of the Bernstein modes is lost for 훽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='258 and 훽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Only for 훽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894, they remain detectable in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At Γ = 3 (bottom row), the upper hybrid mode is the only visible excitation in the spectrum, apart from a zero frequency peak, which generally becomes more dominant at larger 훽, see also Γ = 1 and Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Considering the position of the main peak with respect to the upper hybrid frequency, the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2 suggest that the dispersion of the upper hybrid mode can change from positive at low 훽 to negative at high 훽, similar to the transition in the unmagnetized OCP upon increase of Γ[12,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Before we discuss the simulation results in more detail, we first recall the modification of the dispersion relation in the random-phase-approximation (RPA)[52], shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' It is obtained from the condition 휖RPA(k, 휔) = 0, where 휖RPA(k, 휔) = 1 + 1 푘2휆2 [ 1 + ∞ ∑ 푛=−∞ 휔 휔 − 푛 휔c 퐼푛(휂)푒−휂 휁푛 푍(휁푛) ] (3) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='20 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='633 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='44 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='633 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='44 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='633 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='452 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='905 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='452 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='905 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='905 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='633 k⊥a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='633 k⊥a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='18 S(k, ω)ωp ω/ωp (g) Γ = 3 ω/ωp (h) ω/ωp (i) S(k, ω)ωp (d) Γ = 1 (e) (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='452 S(k, ω)ωp (a) β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='258, ωUH = 4 ωc Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='25 β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354, ωUH = 3 ωc (b) (c) β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894, ωUH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='5 ωc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='633 k⊥a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='18 Figure 2 DSF for k ⟂ B and coupling parameters Γ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='25, 1, 3} (top row to bottom row) and 훽 ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='258, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894} (left column to right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The perpendicular wave numbers 푘⟂푎 are indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The vertical lines show harmonics of the cyclotron frequency (long dashed, grey) and the upper hybrid frequency (short dashed, red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' is the RPA dielectric function[54,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Here, 휆 = 푣th∕휔p is the usual Debye length, 휂 = 푘2 ⟂푟2 L, 휁푛 = (휔 − 푛 휔c)∕( √ 2|푘∥|푣th), and 퐼푛(휂) [푍(휁푛)] denotes the modified Bessel [plasma dispersion] function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For perpendicular wave propagation, the dispersion relation is to be determined from the 푘∥ → 0 limit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' (3)[52], resulting in 1 = 2 ∞ ∑ 푛=1 퐼푛(휂)푒−휂 휂 푛2휔2 p 휔2 − 푛2휔2 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' (4) We are primarily interested in the dispersion of the mode that starts at the upper hybrid frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The location of the latter is indicated by the squares in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For 훽 < 1∕ √ 3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='577 (훽 > 1∕ √ 3), the upper hybrid frequency lies above (below) 5 0 1 2 3 4 0 1 2 3 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='2 (b) Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='25 ω/ωc k⊥rL β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894 ω/ωUH k⊥rL β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894 Figure 3 (a) RPA dispersion relation for the magnetized OCP for various levels of magnetization, as indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' (b) Comparison of the upper hybrid dispersion from the RPA (lines) with the main peak position from the DSF (symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Note the different scaling of the vertical axis in (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' the first Bernstein mode, which starts at 2 휔c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' From the examples shown, one observes that the dispersion of the upper hybrid mode is positive in the former case and negative in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In fact, expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' (4) in powers of 휂 = 푘2 ⟂푟2 L, one finds 휔2(푘⟂) ≈ 휔2 UH + 3 휔2 c 푘2 ⟂푟2 L∕(1 − 3훽2), where the ∼ 푘2 ⟂ term becomes singular at 훽 = 1∕ √ 3 and changes sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In case the upper hybrid frequency exactly coincides with one of the cyclotron harmonics, there are two modes originating from the same frequency at 푘⟂ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' As discussed above, for 휔UH = 2휔c [훽 ≡ 1∕ √ 3], the previous expression diverges and one finds, instead, two modes with a linear dependence on 푘⟂, namely 휔2(푘⟂) ≈ (2휔c)2 ± 휔2 p푘⟂푟L, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [46] for a comparison with simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For 휔UH = 3휔c, the result is 휔2(푘⟂) = (3휔c)2 + 휔2 p ( 3 10 ± √ 186 20 ) 푘2 ⟂푟2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For all other cases, 휔UH = 푛휔c, with 푛 ≥ 4, the general expression for the upper hybrid dispersion remains valid, and the frequency of the second mode decreases ∼ (푘⟂푟L)2푛−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Figure 3(b) shows a comparison of the RPA dispersion and the peak position from the DSF for Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='7 and 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894, there is good agreement between theory and simulation, as observed previously in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [46] for 훽 = 1∕ √ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' However, larger deviations occur for 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354 and, in particular, for 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In the latter case, the peak position from the simulations increases monotonically with 푘⟂푟L, with a leap around 푘⟂푟L ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In contrast, the frequency of the upper hybrid mode from the RPA has a maximum at 푘⟂푟L ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='4 and generally lies well below the peak of the DSF, except for the smallest wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Interestingly, the RPA Bernstein mode above the upper hybrid mode is in reasonable agreement with the simulations for 푘⟂푟L ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Even though 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354 is only marginally smaller than 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='4, the RPA spectrum is quite different [shown is the upper of the two modes that start at 휔UH = 3 휔c] and agrees much better with the simulation data, which, on the contrary, are very similar to those for 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' These observations and the small separation of the RPA modes in frequency space for 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='4 suggest that both the upper hybrid mode and the nearby Bernstein mode may contribute significantly to the DSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We keep in mind, however, that the coupling is already outside the regime where the RPA is expected to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We also reiterate that the peaks in the DSF do not necessarily correspond precisely to the frequencies of the collective modes[56,57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Finally, the evolution of the DSF upon magnetizing the plasma is studied in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 4 for Γ = 1 and three different wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At 푘⟂푎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='724 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 4(a)], the plasmon (upper hybrid) peak for 훽 = 0 (훽 > 0) initially becomes broader as 훽 increases while, at the same time, the peak height decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At larger magnetization, the trend is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The peak becomes sharper, and the peak intensity grows again (훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Since the normalization of the DSF is independent of 훽, ∫ ∞ −∞ 푆(k, 휔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 훽)푑휔 = 푆(푘), where 푆(푘) is the static structure factor, the spectral weight contained in the plasmon merely becomes redistributed—the total 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='5 β = 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='5 2 β = 0 β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='5 2 β = 0 β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894 S(k, ω)ωp ω/ωp (a) k⊥a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='724 β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354 ω/ωp (b) k⊥a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354 ω/ωp (c) k⊥a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='577 Figure 4 DSF perpendicular to the magnetic field for Γ = 1 and various magnetizations 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Also shown is the unmagnetized limit with 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The wave number 푘⟂푎 is indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' weight remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In addition to the plasmon/upper hybrid feature, a very sharp zero-frequency peak develops, even for relatively weak magnetization strengths, which is absent in the 훽 = 0 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For 푘⟂푎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='27 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 4(b)] and 훽 = 0, the plasmon peak is already much broader, which is well known from simulations of the unmagnetized OCP[58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' As the plasma becomes magnetized, the peak shifts to lower frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At 훽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='577, a strong zero-frequency peak has emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Increasing 훽 further to 훽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='894 leads to the formation of a well-pronounced upper hybrid peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In case of the largest wave number, 푘⟂푎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='81 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 4(c)], the plasmon peak practically vanishes for 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 4(b), increasing the magnetization leads to the formation of a well-pronounced peak in the DSF, both at zero and finite frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Comparing the effect of the magnetic field on the DSF for different wave numbers, one finds that the strongest modifications, compared to the zero magnetic field case, are found at small wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' We note that the peak position of the DSF is above (below) the upper hybrid frequency for small (large) magnetization, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For a related discussion of the harmonics in the magnetized and unmagnetized OCP, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 4 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 4 CONCLUSIONS In summary, we have studied the DSF of the weakly magnetized OCP for wave vectors perpendicular to the magnetic field for a variety of coupling strengths and wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At small wave numbers, the DSF shows signatures of Bernstein modes even for weak magnetization, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='25 ≲ 훽 ≲ 1, provided the coupling is sufficiently low, Γ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bernstein modes disappear as soon as Γ grows beyond Γ ≳ 1 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In this strongly coupled regime, the DSF decays rapidly for 휔 > 2휔UH, but the magnetization is too low to observe clear higher harmonics of the upper hybrid mode as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [46], where larger field strengths were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The main peak in the DSF broadens as the wave number increases, similar to the plasmon peak for 훽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For large wave numbers, the DSF decays monotonically in the case of weak magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' For stronger magnetization, Bernstein modes can be observed that modulate the decay to high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In addition, a strong zero-frequency peak is observed, which appears already for relatively mild magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At finite frequencies, the dispersion of the main peak was studied in some detail for Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' It changes from positive to negative upon increase of 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Since it is known that the dispersion of the plasmon becomes negative for Γ ≳ 10 already in the unmagnetized OCP[12,53], this effect could vanish in the strongly coupled regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Here, a theory for the dispersion must include correlations[59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' The comparison between the RPA dispersion and the peak position from the simulations shows that, under certain conditions, the main peak in the DSF could be influenced by the interplay of the upper hybrid mode and a nearby Bernstein mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' This, however, requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 7 The study of the DSF at a fixed wave number and intermediate coupling (Γ = 1) shows that the plasmon/upper hybrid peak shifts and changes its shape upon increase of the magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At small 푘⟂, the peak initially broadens and its intensity decreases after the trend is reversed, and the peak sharpens again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' At larger 푘⟂, the plasmon peak for 훽 = 0 is much less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Increasing the magnetization, however, leads to the formation of well-developed upper hybrid features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' In addition, a strong zero-frequency mode emerges at all considered wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' ACKNOWLEDGMENTS The simulations were performed at the Norddeutscher Verbund für Hoch- und Höchstleistungsrechnen (HLRN) under Grant No.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Kalman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Silvestri, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Donkó, Contrib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2015, 55 (5), 421–427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [13] Setsuo Ichimaru, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 1982, 54 (4), 1017–1059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [14] Alexander Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Potekhin, José A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Pons, Dany Page, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2015, 191 (1), 239–291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Anderegg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Dubin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' O’Neil, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Driscoll, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2009, 102, 185001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [16] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Anderegg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Driscoll, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Dubin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' O’Neil, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Plasmas 2010, 17 (5), 055702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [17] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Fletcher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rolston, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Guzdar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Swisdak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2008, 100, 235002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Gorman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Warrens, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bradshaw, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Killian, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2021, 126, 085002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Tucker Sprenkle, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bergeson, Luciano G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Silvestri, Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Murillo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E 2022, 105, 045201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [20] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Gorman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Warrens, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Kählert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Carstensen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bonitz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Löwen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Greiner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Piel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Kählert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bonitz, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Korolov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Donkó, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E 2019, 99, 013203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [25] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Schmit, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Knapp, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Hansen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Gomez, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Hahn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Sinars, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Peterson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Slutz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Sefkow, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Awe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Harding, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Jennings, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Chandler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Cooper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Cuneo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Geissel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Harvey-Thompson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Herrmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Hess, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Johns, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lamppa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Martin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' McBride, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Porter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Robertson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rochau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rovang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Ruiz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Savage, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Smith, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Stygar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Vesey, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2014, 113, 155004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Gomez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Slutz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' B.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Harding, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Knapp, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Schmit, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Jennings, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Awe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Geissel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rovang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Chandler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Cooper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Cuneo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Harvey-Thompson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Herrmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Hess, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Johns, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lamppa, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Savage, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Smith, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Stygar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Vesey, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2014, 113, 155003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Gomez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Slutz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Jennings, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Ampleford, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Weis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Myers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Yager-Elorriaga, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Hahn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Hansen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Harding, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Harvey-Thompson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lamppa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Mangan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Knapp, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Awe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Chandler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Cooper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Fein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Geissel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Glinsky, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lewis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Ruiz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Ruiz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Savage, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Schmit, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Styron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Porter, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Jones, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Mattsson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Peterson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rochau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Sinars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2020, 125, 155002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bose, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Peebles, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Walsh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Frenje, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Kabadi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Adrian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Sutcliffe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Gatu Johnson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Frank, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Davies, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Betti, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Glebov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Marshall, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Regan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Stoeckl, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Campbell, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Sio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Moody, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Crilly, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Appelbe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Chittenden, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Atzeni, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Barbato, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Forte, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Seguin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Petrasso, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2022, 128, 195002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [29] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Ott, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bonitz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2011, 107, 135003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [30] Yan Feng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Goree, Bin Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Intrator, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Murillo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E 2014, 90, 013105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [31] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Ott, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bonitz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Hartmann, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Donkó, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E 2017, 95, 013209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [32] Scott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Baalrud, Jérôme Daligault, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E 2017, 96, 043202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [33] Keith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Vidal, Scott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Baalrud, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Plasmas 2021, 28 (4), 042103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [34] David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bernstein, Trevor Lafleur, Jérôme Daligault, Scott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Baalrud, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E 2020, 102, 041201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [35] Louis Jose, Scott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Baalrud, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Plasmas 2020, 27 (11), 112101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [36] David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bernstein, Scott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Baalrud, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Plasmas 2021, 28 (6), 062101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [37] Louis Jose, Scott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Baalrud, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Plasmas 2021, 28 (7), 072107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [38] Lu-Jing Hou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Shukla, Alexander Piel, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Mišković, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Plasmas 2009, 16 (7), 073704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Baiko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' E 2009, 80, 046405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [40] D A Baiko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' : Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2014, 496 (1), 012010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [41] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Bonitz, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Donkó, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Ott, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Kählert, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Hartmann, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' 2010, 105, 055002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' [42] Xue-Feng Yang, Zheng-Xiong Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfxwVP/content/2301.03425v1.pdf'} +page_content=' Plasmas 2012, 19 (7), 073704.' metadata={'source': 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Zilinskas1, Y. Miguel1, 2, C.P.A. van Buchem1, and I. A. G. Snellen1. +1 Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333CA Leiden, the Netherlands +2 SRON Netherlands Institute for Space Research , Niels Bohrweg 4, 2333 CA Leiden, the Netherlands +e-mail: zilinskas@strw.leidenuniv.nl +Received June XX, 2021; accepted July XX, 2021 +ABSTRACT +Many of the confirmed short period super-Earths and smaller sub-Neptunes are sufficiently irradiated for the surface silicates to be sus- +tained in a long-lasting molten state. While there is no direct evidence of magma ocean influence on exoplanets, theory suggests that +due to outgassing and diverse evolution paths, a wide range of resulting atmospheric compositions should be possible. Atmospheric +contamination caused by the outgassing of the underlying magma ocean is potentially detectable using low resolution spectroscopy. +The James Webb Space Telescope provides the necessary spectral coverage and sensitivity to characterise smaller planets, including +lava worlds. In this light, we assess observability of outgassed silicates submerged in volatile atmospheres on the edge of the evapora- +tion valley. By placing a hypothetical 2 R⊕ planet around a Sun-like star, we self-consistently model, in 1-D, a wide range of potential +atmospheric compositions, including thermal structure and outgassing. We focus on atmospheres rich in H, C and N. We assess di- +verse chemistry of silicates and volatiles, and what features of outgassed species could be detected via emission spectroscopy using +MIRI LRS. Results indicate that even for substantial volatile envelopes, strong in infrared opacity, the presence of silicates causes +deep thermal inversions, affecting emission. Similar to pure lava worlds, SiO remains the only outgassed species with major infrared, +5 and 9 µm, bands. However, even a small amount of volatiles, especially of H2O and H–, may hinder its observability. We also find +that the C/O ratio plays a large role in determining the abundance of SiO. Detecting SiO on a strongly irradiated planet could indicate +an atmosphere with high metallicity and a low C/O ratio, which may be a result of efficient interaction between the atmosphere and +the underlying melt. +Key words. Planets and satellites: atmospheres – Planets and satellites: terrestrial planets – Techniques: spectroscopic +1. Introduction +Ever since their discovery, there has been great interest in try- +ing to characterise an unravel the mysteries of the seemingly +ambiguous super-Earths and sub-Neptunes. While more mas- +sive, Neptune-like, planets are expected to retain most of the +primordial H/He, intermediate (1.5-2.5 R⊕) and smaller worlds +are likely to be extremely diverse in their atmospheric composi- +tion and structure. Figure 1 showcases the population of close-in +planets. +Occupying the edge of the evaporation desert, rocky worlds +are shaped by erosion, accretion, degassing and volcanism, +with some possibly forming long-lasting secondary atmospheres +(Elkins-Tanton & Seager 2008). Though because of such close +proximity to the star, many of the planets likely end up as bare +rocks, with no visible atmospheric component. Observations of +several temperate and hot super-Earths seem to favour this theory +(Kreidberg et al. 2019; Zieba et al. 2022; Crossfield et al. 2022). +That said, even without an insulating atmosphere, these have +temperatures high enough to engulf the dayside of the planet +with magma oceans, which should result in tenuous, but observ- +able silicate envelopes (Schaefer & Fegley 2009; Miguel et al. +2011; Ito et al. 2015; Kite et al. 2016; Zilinskas et al. 2022). +For such worlds, SiO and SiO2 have been proposed to be the pri- +mary species that could be probed via infrared emission (Ito et al. +2015; Nguyen et al. 2020; Zilinskas et al. 2022). It is also fea- +sible that high-mean-molecular-weight species can survive ero- +sion, leaving denser, CO or N2 atmospheres intact (Zilinskas +et al. 2020). 55 Cnc e may be a primary example of this (De- +mory et al. 2016; Angelo & Hu 2017; Hammond & Pierrehum- +bert 2017). While studies indicate that planets below ≲ 2.0 R⊕ +are likely stripped of H2 (Rogers & Owen 2021), new interior +models show that magma-atmosphere interaction during evolu- +tion could lead to large reservoirs of H2O, buffering H2O atmo- +spheres, which, due to thermal and photochemical dissociation, +should result in abundant H2 as a by-product (Kite & Schaefer +2021; Dorn & Lichtenberg 2021). The only outstanding weak- +ness of the proposed theory is the efficiency of the interaction +between the melt and the atmosphere. +Going to larger radii (above 2.0 R⊕), the observed discrep- +ancy of densities may indicate the existence of water/ocean plan- +ets that are shrouded with dense steam atmospheres (Zeng et al. +2019; Mousis et al. 2020; Nixon & Madhusudhan 2021; Bean +et al. 2021). Insulation on sub-Neptunes is also expected to al- +low for deep magma oceans to be sustained indefinitely (Kite +et al. 2020). Just as with smaller planets, depending on the effi- +ciency of magma-vapour interaction and atmospheric mixing, it +could result in H2 or H2O-rich envelopes that are heavily con- +Article number, page 1 of 14 +arXiv:2301.05190v1 [astro-ph.EP] 12 Jan 2023 + +A&A proofs: manuscript no. main +Fig. 1: Short period exoplanets with radii < 4 R⊕. Coloured +markers indicate confirmed planets, grey markers are candi- +date planets from Kepler, K2 and TESS missions. Colour value +of confirmed planets represents the density of the occurrence +rate, which peaks at two distinct radii of 1.5 and 2.4 R⊕, +seemingly separating the population into super-Earths and sub- +Neptunes. The highlighted region roughly encompasses the pa- +rameter space applicable to our modelled cases. Marked on the +figure is the evaporation desert and one of the most well stud- +ied super-Earths - 55 Cnc e. The data is taken from the NASA +exoplanet archive. +taminated by silicate species (Schlichting & Young 2022; Kite +et al. 2020). +Observations with JWST will provide necessary constraints +for the ongoing theoretical work. Extended H2 atmospheres +of larger super-Earths and sub-Neptunes with substantial scale +heights are easily probed via transmission spectroscopy (Hu +et al. 2021b). For intermediate and smaller planets, measuring +emission of the dayside may prove to be the only viable char- +acterisation method. However, even with JWST, characterising +the chemistry of potential atmospheres will be challenging; de- +tecting silicates even more so. From an observer’s standpoint, +finding whether these planets have atmospheres at all is a major +stepping stone in the field of exoplanets. +In this work we explore the chemistry and observability +of outgassed silicates in volatile envelopes of irradiated rocky +worlds. The highlighted region in Figure 1 roughly indicates the +parameter space applicable to this work. In contrast to studies +done by Kite et al. (2020); Kite & Schaefer (2021); Schlicht- +ing & Young (2022), we do not model substantial atmospheres, +but focus on cases where the surface pressure is relatively low +in comparison to Neptune-like planets (< 10 bar). We make use +of consistent outgassing equilibrium and radiative-transfer mod- +els to predict what silicate features are potentially characteris- +able through infrared emission spectroscopy, especially at wave- +lengths relevant for JWST’s MIRI instrument. Finding signs of +silicates could hint at an underlying magma ocean, allowing us to +put better constraints on the proposed diversity of super-Earths +and sub-Neptunes. +The paper is structured as follows. Section 2 contains the +description of our approach in constructing 1-D, self-consistent +atmospheric models, including chemistry and thermal structure. +The analysis of the results is given in Section 3. We discuss some +of the important factors that may affect observability in Section +4, and finally conclude in Section 5. +2. Methods +2.1. Setting up the system +To explore observability of silicates in volatile atmospheres we +set up a grid of models that would represent a typical super-Earth +or a sub-Neptune orbiting a Sun-like star. We focus on intermedi- +ate pressure envelopes, ranging from 1-10 bar. Our models focus +on cases where the surface temperature is higher than 1400 K, +which is enough for magma oceans to form and influence the at- +mospheric composition. For a G-type star and dayside confined +heat redistribution, this typically results in a maximum orbital +distance of 0.06 AU. We make use of several different codes that +are set up to consistently calculate outgassing, chemical abun- +dances and temperature profiles, inclusive of the surface temper- +ature. The models are then used to simulate emission spectra and +expected JWST noise levels. Below we describe the approach for +each of the components in more detail. +2.2. Determining the chemistry +A major assumption made in this work is that the overlying +atmosphere equilibrates with the molten surface, allowing out- +gassing to control the abundances of all silicates, including oxy- +gen. Since general atmospheric compositions of super-Earths +and sub-Neptunes are unknown, we take the freedom to explore a +grid of possible outcomes. These are drastically varied in metal- +licity, C/O ratio, volatile content, atmospheric pressure and even +internal temperature. +For volatiles, we take the solar composition (Lodders et al. +2009) and adjust its metallicity (M/H), where all of the elements +except for H and O are linearly scaled. While we normally as- +sume that oxygen abundance is determined via outgassing, to ex- +plore differences between solar and outgassed atmospheres, we +also model cases where oxygen is set by the metallicity parame- +ter. In addition to this, we vary the C/O ratio (via carbon adjust- +ment) together with the abundance of H/He, which allows us to +carefully dictate the major molecular constituents in the atmo- +sphere. This allows us to explore cases where strong irradiation +and large sinks of light elements (e.g., photoevaporation, dis- +solution) may result in high-mean-molecular-weight envelopes, +dominated by either CO, CO2 or even N2. +The outgassing budget is determined by an open-source code +LavAtmos1 (van Buchem et al. 2022), which calculates the melt- +vapour equilibrium for a given surface temperature and melt +composition. To accurately determine the activity of the oxides +in the melt, LavAtmos makes use of the liquid-solidus code +MELTS (Ghiorso & Sack 1995). The package solves for the ox- +ides containing the following elements: Al, Ca, Fe, K, Mg, Na, +Si, Ti. The resulting outgassed partial pressures are added to +the volatile atmospheres while keeping the total surface pressure +constant. This is equivalent to reducing the relative abundances +of volatiles. As in (Zilinskas et al. 2022), we take the magma to +be composed as Bulk Silicate Earth (BSE). It contains 45.97 % +SiO2, 36.66 % MgO, 8.24 % FeO, 4.77 % Al2O3, 3.78 % CaO, +0.35 % Na2O, 0.18 % TiO and 0.04 % K2O (wt%). The surface +temperature and the outgassing are consistently calculated using +a radiative-transfer code, as explained in Section 2.3. Important +to note that currently LavAtmos does not account for possible +deposition of volatiles into the magma. As shown in the work of +Kite et al. (2020), this can have substantial consequences on the +atmospheric composition. The detailed analysis of this is out of +scope for this paper and will be a focus of a future study. +1 https://github.com/cvbuchem/LavAtmos +Article number, page 2 of 14 + +4.0 +3.5 +Evaporation Deser +3.0 +2.5 +nce +2.0 +1.5 +1.0 +0.01 +0.05 +0.10 +Orbital Distance (AU)M. Zilinskas et al.: Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes +With the elemental budget determined, atmospheric chem- +istry is solved using the thermochemical equilibrium code +FastChem2 (Stock et al. 2018; Stock et al. 2022). We take into +account over 200 relevant species, inclusive of neutral and ion +chemistry. The thermal data used is compiled from the Burcat +NASA thermodynamics database3. +2.3. Computing thermal profiles +The temperature structure is solved using the radiative-transfer +code HELIOS4 (Malik et al. 2017; Malik et al. 2019b). We allow +convective adjustment to take place using an adiabatic coefficient +of κ = 2/7, applicable to diatomic atmospheres. The profiles +are treated with a rocky surface boundary, the implementation of +which is described in detail in Malik et al. (2019a); Whittaker +et al. (2022). All of our models are reiterated until convergence +such that the attained surface temperature is in good agreement +with the atmospheric chemistry. For the purposes of showing +possible spectral features, the heat redistribution is confined to +the dayside of the planet (f=2/3). In specific cases it is approxi- +mated using the longwave optical depth of the atmosphere, based +on equations from Koll (2022). +We use a total of 50 opacity sources, including all of the +major volatile and silicate species. The entire list and descrip- +tions of all the opacities used in this study are listed in Table A.1 +of Appendix A. All of the atomic opacities are obtained from +the DACE5 database, with the majority using the Vienna Atomic +Line Database (VALD3) line lists (Ryabchikova et al. 2015). For +molecular opacities we make use of both the DACE database and +the opacity calculator HELIOS-K6 (Grimm & Heng 2015; Grimm +et al. 2021). Following Grimm et al. (2021), such are approxi- +mated using a Voigt fitting profile, wing cutting length of 100 +cm−1 and, where line lists allow, a temperature of up to 4000 K. +In terms of observability, SiO is expected to be a key species +of irradiated atmospheres (Ito et al. 2015; Zilinskas et al. 2022). +In this work, we use the new ExoMol7 SiOUVenIR line list +(Yurchenko et al. 2022), which covers the entire UV-infrared +wavelength range and is applicable to high temperatures ex- +pected to occur on hot super-Earths. Figure 2 shows key un- +weighted opacities considered in this work. SiO shortwave opac- +ity (< 1 µm) is a strong contributor towards occurring tempera- +ture inversions, while the longwave bands peak at 5 and 10 µm +and are potentially detectable spectral features. While not dis- +played, there are many other potential species that are significant +absorbers in silicate and volatile atmospheres. +For short period planets shortwave stellar flux becomes an +important factor in shaping the thermal structure of the atmo- +sphere. Using simple blackbody stellar models results in incor- +rect UV flux. Thus all stellar irradiation models used in this work +are generated via HELIOS using the PHOENIX (Husser et al. +2013) and MUSCLES (France et al. 2016; Youngblood et al. +2016; Loyd et al. 2016) databases. Spectra and opacities are sam- +pled at a resolution of λ/∆λ = 2000 and cover the range of 0.1 - +200 µm. +2 https://github.com/exoclime/FastChem +3 http://garfield.chem.elte.hu/Burcat/burcat.html +4 https://github.com/exoclime/HELIOS +5 https://dace.unige.ch/ +6 https://github.com/exoclime/HELIOS-K +7 https://www.exomol.com/ +Fig. 2: Comparison of SiO, H2O, TiO and H– opacities, shown +at a resolution of λ/∆λ = 2000 for a temperature of 3000 K and +atmospheric pressure of 10−2 bar. The description and sources of +all used opacities can be found in Table A.1. +2.4. Simulating emission spectra +On hot super-Earths, silicate atmospheres are expected to be +confined to the tidally locked dayside of the planet, generally +making them poor candidates for low-resolution transmission +spectroscopy (Zilinskas et al. 2022). Due to large atmospheric +temperatures, spectral features may instead be probed through +emission of the secondary eclipse. If, however, such planets do +possess global, volatile atmospheres, transmission could be pos- +sible, but its viability will depend strongly on the scale height +(Zilinskas et al. 2020). While in this work we focus on emis- +sion spectroscopy, we note that for a number of known targets +low-resolution transmission spectroscopy with JWST may also +be feasible. +We generate emission spectra using the radiative-transfer +code petitRADTRANS8 (Mollière et al. 2019, 2020). We use +the same atomic and molecular opacities described in Section +2.3, including H2, H2O, O2 Rayleigh scattering, and H2 –H2, +H2 –He, O2 –O2 and H– continuum opacities. The spectra are +calculated at a resolution of λ/∆λ = 1000 for a wavelength +range of 0.3 - 28 µm, encompassing the coverage of all JWST +instruments. In all figures, the spectra are convolved to a lower +resolution for better readability. +For notable targets, we assess JWST noise using PANDEXO9 +(Batalha et al. 2017), which is built on the Pandeia10 engine. We +only simulate MIRI Low Resolution Spectroscopy (MIRI LRS +with λ/∆λ ≈ 100) in slitless mode, as it is likely to be the most +suitable mode for characterisation of silicate features. The wave- +lengths covered by the instrument are 5 - 12 µm. In each case, we +use the corresponding stellar and planetary parameters obtained +from the NASA exoplanets archive. For corresponding stellar +models we use PHOENIX generated spectra. +3. Results +3.1. Outgassed silicates in hydrogen atmospheres +For a given initial composition, the thermal structure and the re- +sulting chemistry of an atmosphere is determined by the stellar +8 http://gitlab.com/mauricemolli/petitRADTRANS +9 https://exoctk.stsci.edu/pandexo/ +10 http://jwst.etc.stsci.edu +Article number, page 3 of 14 + +8 +10 +Sio +H20 +6 +10 +Tio +10 +2 +10 +10 +2 +10 +4 +10 +6 +10 +0.1 +10 +100 +Wavelength (micron)A&A proofs: manuscript no. main +flux that the planet receives. In Figure 3, we showcase a hypo- +thetical world placed around a Sun-like star of T = 5750 K. The +only free parameter varied between the cases is the orbital dis- +tance. The 2 R⊕ planet is assumed to have a volatile-rich, solar- +like, 1 bar atmosphere that is in equilibrium with an outgassed +silicate component. In each model, silicate abundances are com- +puted via outgassing of a BSE melt of a numerically converged +surface temperature. Naturally, with increasing orbital distance, +the temperatures fall and the abundance of silicates decreases. +At close orbits the surface temperature can reach over 3000 +K, which results in a substantial amount of outgassed O, allow- +ing for plentiful formation of oxides, including SiO and H2O. +Our models show that 1 bar atmospheres with a surface tempera- +ture higher than 2300-2500 K produce super-solar abundances of +silicates, causing drastic changes in thermal structure. Shortwave +absorbers heat the atmosphere causing deep thermal inversions, +affecting even the photosphere. Below the photosphere, around +10−2 bar, the atmosphere becomes optically thick to radiation, +resulting in isothermal regions where no heat transport occurs +(Blue and two faded TP profiles in the top left panel of Fig. 3). +Similar thermal structure is observed in volatile-free, pure sili- +cate atmospheres (Zilinskas et al. 2022), implying that silicate +opacities may largely be responsible in shaping the atmosphere. +Looking at the chemistry we find that even at the highest +modelled temperatures many molecules survive thermal dissoci- +ation. Abundances of major absorbers are shown in the top right +panel of Fig. 3. While these atmospheres are filled with atomic +species (H, O, Fe, Mg and others), oxides, such as SiO, H2O +and TiO dominate its opacity. At high pressures, H and O form +H2O, making it a strong absorber (see Fig. 4). Near the surface, +H2O and SiO have similar volume mixing ratios. Moving to the +upper, low pressure regions, H2O begins to dissociate into atoms +and ions, while SiO remains in its molecular form, making it one +of the most abundant species throughout the entire atmosphere. +It should be expected that SiO is a major constituent in atmo- +spheres with an underlying magma ocean. For these cases we +find that chemically, the abundance of SiO is only weakly af- +fected by pre-existing volatiles. In addition to SiO, vaporisation +of magma at large temperatures also results in high abundances +of TiO, which can become one of the most influential shortwave +absorbers. +In the bottom panel of the figure we show the correspond- +ing emission spectra. While in this case the small planet-to-star +contrast results in a relatively low emission signal, the emer- +gence of the 5 and 9 µm SiO features is clear (Blue curve). Due +to occurring inversions SiO increases the observed flux at these +wavelengths. Unlike silicate atmospheres modelled in Zilinskas +et al. (2022), these show no significant sign of SiO2 absorption +at 7 µm. This is partly attributed to oxygen being chemically +favoured to bond with volatiles, such as hydrogen. At shorter +wavelengths, TiO is one of the dominant absorbers, causing a +broad feature below 1 µm. For BSE compositions, its presence +is only important at high surface temperatures, typically larger +than 2500 K. It is worth noting that many different molecules and +atoms contribute to the shortwave opacity, some of which may +be detectable in more extended atmospheres using transmission +spectroscopy. Shortwave opacities are discussed in more detail +at the end of this section. In addition to molecular opacities H– +becomes an important factor throughout the entire JWST wave- +length range. Not only does it have a strong shortwave compo- +nent, but its strong continuum at 10 µm may hinder observability +of SiO. Overall, out of all the outgassed silicates, the two SiO +features are likely to be the easiest to characterise using MIRI +LRS covering the 5-12 µm range. +Moving to colder cases, the abundance of all silicates de- +creases rapidly, becoming sub-solar at 0.04 AU (Pink curve). +The total outgassed pressure of just silicates at temperatures be- +low 2500 K is comparable to a millibar (Zilinskas et al. 2022). +Assuming melt-vapour equilibrium is attained, the volatile com- +ponent is likely to dominate, making species such as SiO or TiO +unobservable with low resolution spectroscopy. Another conse- +quence of this is drastic reduction of outgassed oxygen, which +raises the C/O ratio, allowing hydrocarbons to efficiently form. +Most of the species in cooler atmospheres are heavily weighted +towards infrared opacity, resulting in a lack of any significant +inversions that may impact observability. The spectrum here is +dominated by molecules such as CH4, C2H2 and HCN, all show- +ing deep absorption features. Detecting silicates in emission at +relatively large orbits could indicate that either the temperature +of the melt is much higher then the planetary equilibrium temper- +ature, or that silicates are not in equilibrium with the underlying +melt. +3.2. Contribution function +In Figure 4, we take one of strongly irradiated cases and show +its emission contribution function. In the right panel, the high- +lighted region represents the emitting photosphere. For wave- +lengths > 1 µm, this mostly coincides with pressures between +10−4 and 10−2 bar. A major contributing molecule for longwave +opacity is H2O. Its dominance is a general occurrence in our +models. Plentiful hydrogen and oxygen assure that even at high +temperatures, it is one of the most optically dominating species. +Additional leftover hydrogen results in a strong H– continuum, +pushing the general opacity higher up. The tail of the contin- +uum can be seen at wavelengths > 10 µm. Since the abundance +of SiO is not strongly affected by increasing temperatures and +lower pressures, its opacity has large contributions from inverted +regions. If atmospheres of super-Earths are prone to thermal in- +versions, it is likely that SiO will show up as increased flux. The +9 µm feature is and should be visible even with strong volatile +opacities present. If no volatiles are present, enough SiO2 may +form to appear at 7 µm, complimenting the SiO feature (Zilin- +skas et al. 2022). +There are many different species contributing to the total +shortwave opacity (< 1 µm). SiO, AlO, MgO, TiO, Mg and Fe, +all have very strong opacities. Some lesser, but important species +are: SiH, MgH, VO, Al, Ca, K, Na, Si and Ti. TiO, having broad +wavelength coverage, is perhaps the most important for observa- +tions, as well as in its influence in shaping the thermal structure. +Its presence is known to strongly affect atmospheres even in gas +giants (Serindag et al. 2021). Note that on rocky planets, TiO is +typically sustained in significant abundances only above 2500- +2800 K. Aside from TiO, the SiO UV band and Fe opacity have +major influence on the strength and depth of the occurring in- +versions. These species are also much more volatile and readily +vaporised from the magma. Important to note that because of +the large number of shortwave absorbers, even atmospheres that +are missing major oxides such as SiO or TiO can still have deep +occurring inversions. +Previous studies have shown that pure silicate atmospheres +have similar total shortwave opacity (Zilinskas et al. 2022). This +is unsurprising since the majority of shortwave absorbers come +from silicate outgassing. While there are additional shortwave +absorbers due to the presence of volatiles, namely SiH, MgH +and VO, these are relative minor in comparison to silicates. Note +that, due to a lack of thermal data, our models do not include +FeH, the opacity of which peaks at 1 µm. For atomic species +Article number, page 4 of 14 + +M. Zilinskas et al.: Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +Temperature (K) +10 +8 +10 +6 +10 +4 +10 +2 +10 +0 +Pressure (bar) +Dayside +0.01 AU +0.04 AU +0.01 AU +0.04 AU +10 +12 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +10 +0 +Volume Mixing Ratio +10 +8 +10 +6 +10 +4 +10 +2 +10 +0 +Pressure (bar) +SiO +H2O +TiO +SiO +H2O +TiO +1 +5 +10 +30 +Wavelength (micron) +0 +50 +100 +150 +200 +250 +Fplanet/Fstar ppm +SiO +SiO +TiO +Hydrocarbons (CH4, C2H2, HCN) +Fig. 3: The figure has been updated. Extra legend in the top left panel has been added. Atmospheric models a super-Earth of 2 R⊕ +orbiting at Sun-like star. In all cases, dayside confined heat redistribution (f=2/3) and a surface pressure of 1 bar are assumed. The +top left panel shows the temperature-pressure profiles at orbital distances of 0.01, 0.015, 0.02, 0.03, 0.04, 0.05 and 0.06 AU, with +two highlighted cases being 0.01 AU (Blue) and 0.04 AU (Pink). In the top right panel, the highlighted curves indicate abundances +of SiO, H2O and TiO for the case of 0.01 AU with an effective planetary temperature of 3174 K. In the same panel, the faded curves +represent the chemistry of the same species at 0.04 AU (Te f f = 1771 K). The bottom panel contains the corresponding synthetic +emission spectra, with the flat, thinner curves representing blackbody emission (assuming computed surface temperature). Major +absorbers for highlighted cases are indicated via shaded areas, with SiO, TiO and hydrocarbons (CH4, C2H2 and HCN) being the +primary species of interest. Spectra are shown at a resolution of λ/∆λ = 600. +we also do not use pressure-caused broadening, likely leading to +some underestimation of the line widths. It is possible that many +of the atomic species, especially alkali metals, are a lot more +dominant in shaping the atmosphere. +The inherent complexity of the shortwave region makes it +difficult to correctly model temperature profiles. Many of the +mentioned opacities here are often overlooked, leading to the- +oretically incorrect temperatures. After the chemistry, shortwave +opacities are likely to be a major source of uncertainties which +can greatly affect interpretations of observed spectra. +3.3. Impact of metallicity and C/O ratio +The process of formation for short-period rocky planets is un- +known, but it is often assumed that such are heavily enriched in +metals (Weiss et al. 2013; Moses et al. 2013). In Figure 5, we +use varied metallicity to explore what effect it may have on ob- +servability of silicates. The blue and pink curves represent the +original solar models showcased in the previous section, while, +for each of the orbits, the overplotted curves show atmospheres +with 10, 100 and 1000 times increased metallicity. Note that the +metallicity here does not control the abundances of outgassed +silicates or oxygen, but only of all volatiles. The main impact of +it is thus increase of C, N and the C/O ratio. The corresponding +chemistry of the close orbit cases is shown in Figure 7. +For close orbits an x10 increase in metallicity has minimum +effect on atmospheric chemistry or thermal structure. With SiO +and H2O remaining as dominant oxides, the spectral features are +mostly unchanged. This slight increase in metallicity does al- +low for CO to form more efficiently, very slightly boosting its +opacity at 5 µm. The unweighted opacity of CO, along with a +few other species that are discussed later, are shown in Figure 6. +When the metallicity is increased to x100, the C/O ratio crosses +unity and the chemistry starts prioritising the formation of CO, +heavily diminishing other oxides (see Fig. 7). Atmospheres that +do not outgas or retain enough oxygen are likely to suffer this +Article number, page 5 of 14 + +A&A proofs: manuscript no. main +Fig. 4: Emission contribution function of a strongly irradiated super-Earth orbiting a Sun-like star at 0.015 AU. The temperature +profile in the left panel is taken from the models showcased in Fig. 3. Marked in dashed is the effective temperature of the planet T = +2950 K. The right panel showcases the emitting region of the atmosphere as a function of wavelength. Major contributing molecules +are marked in their respective regions. Lesser contributing opacities are discussed in the text. Spectra are shown at a resolution of +λ/∆λ = 600. +Fig. 5: Synthetic emission spectra for an atmosphere of in- +creased metallicity. The main cases, blue and pink, represent +models with solar metallicity at two different orbital distances +(0.015 and 0.03 AU). For each orbit, atmospheres of 10, 100 +and 1000 times metallicity are shown. Some of the contribut- +ing opacities are shown for their respective wavelengths. The in- +set displays the corresponding temperature temperature profiles. +Note that metallicity here does not control the abundance of out- +gassed silicates or oxygen. Spectra are shown at a resolution of +λ/∆λ = 600. +effect, erasing opacities of SiO or H2O in the spectrum. With no +SiO, Si either remains in atomic form or bonds with H to form +SiH. Though, due to abundant N and high C/O ratio, H priori- +tises bonding with CN to form HCN (rightmost panel of Fig. 7). +This chemistry is now reflected in the thermal structure as inver- +sions become significantly weaker. Pushing metallicity higher, +further increases the C/O ratio, resulting in a mostly CO and +hydrocarbon-dominated atmosphere, even at high temperatures. +Because the emitting photosphere resides mostly in the isother- +mal region, the spectrum becomes largely featureless. +With increasing orbital distance (Pink curve of Fig. 5), the +trends in the chemistry and spectra remain similar, but more +severe. Since the abundance of oxygen from outgassing is low +Fig. 6: Abundance unweighted opacities of CO, HCN, SiH and +PS, shown at a resolution of λ/∆λ = 2000 for a temperature of +3000 K and atmospheric pressure of 10−2 bar. Detailed descrip- +tion of all opacities used in this work can be found in Table A.1 +of Appendix A. +at these temperatures, the C/O ratio at x1 metallicity is already +near unity. Even at x10 the C/O ratio becomes much larger than +unity causing efficient formation of hydrocarbons. The dominant +molecules become HCN, C2H2 and CO, while H2O and any po- +tential SiO are erased from the atmosphere. This results in opac- +ity heavily weighted towards infrared wavelengths, thus a lack +of deep inversions. +3.4. Keeping the C/O ratio constant with metallicity +The balance between carbon and oxygen is a major factor in de- +termining atmospheric chemistry and whether SiO is allowed to +thrive. While in Figures 5 and 7 we allow outgassing to control +abundances of oxygen and, therefore, the C/O ratio, in Figure 8 +we set a constant C/O ratio. The value of oxygen is now scaled +with metallicity. The blue and pink curves represent models at +same orbital distances as in the previous figures with cases of +10, 100 and 1000 times increased metallicity shown for each. +Article number, page 6 of 14 + +-8 +-8 +10 +10 +Sio +TiO +H20 +Sio +Sio +H20 +10 +(bar) +(bar) +Pressure +Pressure ( +-4 +10 +10 +10 +iTeff +0 +10 +10 +2750 +3000 +3250 +3500 +3750 +4000 +5 +10 +30 +7 +Temperature (K) +Wavelength (micron)250 +-8 +SiO +SiO +10 +5 +200 +10 +udd +10 +150 +1200 2000 +3000 3800 +Temperature (K) +100 +x10 +H20 +L +x100 +50 +x1000 +H20 +Hydrocarbons +5 +10 +30 +Wavelength (micron)8 +CO +6 +HCN +10 +SiH +10 +PS +2 +10 +10 +2 +10 +10 +6 +10 +0.1 +10 +100 +Wavelength (micron)M. Zilinskas et al.: Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes +Fig. 7: Figure has been updated. Moved labels. Volume mixing ratios of SiO, CO and HCN (from left to right). The models shown +here are for the close orbit (0.015 AU) cases of Fig. 5. Each panel contains the original solar metallicity (Blue) as well as 10, 100 +and 1000 times increased metallicity curves. +1 +5 +10 +30 +Wavelength (micron) +0 +50 +100 +150 +200 +250 +Fplanet/Fstar ppm +SiO +SO2 +CO2 +H +H2O +C/O = 0.46 +x10 +x100 +x1000 +1200 2000 +3000 3800 +Temperature (K) +10 +2 +10 +5 +10 +8 +Pressure (bar) +x10 +x100 +x1000 +Fig. 8: Synthetic emission spectra as in Fig. 5, but with constant +C/O ratio of 0.46. The oxygen abundance here is scaled with +metallicity. Dark blue and and pink curves represent cases with +solar metallicity. Respective blackbody emission is represented +with thin curves of the same colour. Major contributing opac- +ities for solar cases are indicated with shaded regions. The in- +set shows corresponding temperature profiles. Note that surface +temperature increases with metallicity, shifting temperature- +pressure profiles to the right. Spectra are shown at a resolution +of λ/∆λ = 600. +The abundance of oxygen at solar value is an order of mag- +nitude lower than what would be outgassed in our close orbit +case (Blue curve). This directly results in decreased abundances +of all oxides, inclusive of SiO. Additionally, the solar C/O ratio +is high enough for CO to form and become an abundant oxy- +gen carrier in the atmosphere. Other oxides, such as CO2, H2O +or SiO follow closely behind. The resulting infrared opacity is +dominated by the continuum of H–, with some contribution from +H2O. Buried beneath are opacities of SiO and CO2. The atmo- +sphere in this case becomes less opaque, allowing radiation to +penetrate deeper. This causes inversions to extend nearly all the +way to the surface. To conserve energy, the surface temperature +is consequently decreased. Increasing metallicity to 100 results +in oxygen abundance similar to what a melt-vapour equilibrium +would produce. This is reflected in the drastic change of the tem- +perature profile, which now resembles cases presented in previ- +ous figures (Blue curve of Fig. 5). The high surface temperature +also produces more Si, which is why we see its features begin to +emerge. Increasing metallicity further has similar effect. Regard- +less of metallicity in these models, a solar C/O ratio makes CO a +prolific species, which absorbs much of the oxygen and reduces +the possibility of SiO being a dominant species. +With outgassing lessened at larger orbits (Pink curve), H2O +and CO are the main oxygen carriers. In contrast to previous +cases, increasing metallicity here results in large abundances of +CO2, which has several important bands in the infrared. The ma- +jor two being at 4.5 and 14 µm. While with high metallicity, SiO +reaches a volume mixing ratio of nearly 10−3, the opacity of H2O +completely overshadows it, making it invisible. We additionally +see emergence of some lesser species, such as SO2, which does +have strong opacity at 7.5 µm. +In summary, our models indicate that it is mainly the C/O ra- +tio that significantly affects atmospheric chemistry, including the +final abundance of SiO, with metallicity of volatiles, being much +less important. Therefore, lava worlds enveloped in volatiles are +likely to depend heavily on the balance between carbon and oxy- +gen. High C/O ratios drive oxygen atoms away from SiO, poten- +tially making SiH a species of interest. As shown in Fig. 6, SiH +has a strong feature at 0.4 µm and several other bands between 1- +10 µm. Nonetheless, in hydrogen-rich worlds, regardless of the +C/O ratio, H2O and H– may further hinder observability of sili- +cate species. +3.5. Effect of increased surface pressure and internal heating +Down from millibar silicate atmospheres to volatile-shrouded +sub-Neptunes, the envelopes of rocky worlds may vary orders +of magnitude in surface pressure. Larger, thermally-thick atmo- +spheres, can act as insulators, allowing the molten state of a sur- +face to exist indefinitely, even if the planet is weakly irradiated +(Lopez & Fortney 2014; Kite et al. 2020). One resulting con- +sequence of this may be that insulation and trapped heat near +the surface allow for much greater melt temperatures, making +silicates more dominant over volatiles. Of course, it is also pos- +sible that due to an increase of volatiles, the outgassing becomes +heavily suppressed. In previous sections we have assumed the +Article number, page 7 of 14 + +-8 +-8 +8 +10 +10 +10 +sio +CO +HCN +-6 +-6 +10 +10 +10 +Pressure (bar) + (bar) +(bar) +Pressure ( +Pressure ( +4 +10 +4 +10 +10 +x10 +-2 +-2 +10 +10 +x100 +x1000 +19° +10 +-7 +-4 +-7 +-4 +-1 +-4 +10 +-1 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +Volume Mixing Ratio +Volume Mixing Ratio +Volume Mixing RatioA&A proofs: manuscript no. main +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Temperature (K) +10 +8 +10 +6 +10 +4 +10 +2 +10 +0 +Pressure (bar) +Tint = 300K +3 +5 +10 +20 +4 +6 +7 +8 +9 +Wavelength (micron) +130 +160 +190 +220 +250 +Fplanet/Fstar ppm +SiO +1 bar;Tint = 0K +10 bar;Tint = 0K +10 bar;Tint = 300K +Fig. 9: Temperature-pressure profiles and emission spectra of +varied atmospheric pressure and internal heating. The top panel +contains profiles of atmospheres with 1 and 10 bar at two dif- +ferent orbital distances (0.01 and 0.04 AU). For the close or- +bit, an additional case with Tint=300 K is shown. The lower +panel shows the resulting emission spectrum of the close or- +bit cases. The chosen wavelength region is where SiO features +are expected to appear. Spectra are shown at a resolution of +λ/∆λ = 600. +volatiles to be capped at 1 bar, while also keeping the internal +temperature of the planet at 0 K. While the exact dynamics of +this are out of scope for this paper, in Figure 9 we show how an +increase in pressure and internal heating might affect the observ- +ability of silicon-bearing species. +The bright cyan curves in the top panel of the figure represent +atmospheres with a surface pressure increased to 10 bar. With- +out internal temperature enabled, there is no additional heat to +transport, making convection unnecessary, resulting in a simple +extension of the isothermal region. However, if the atmosphere +is supplied with energy from bellow, the optically and thermally +thick portion becomes unstable to convection, dramatically in- +creasing the temperature of the melt. Because of the exponen- +tial scaling of outgassing, this can often lead to an immense rise +of silicate abundances. If such atmospheres are well mixed, one +should expect to see substantial silicate features. +The bottom panel of Figure 9 shows the spectra for the close +orbit models. A 10 bar atmosphere with no internal heating re- +sults in a greater dominance of the volatile component. The H– +continuum becomes more effective, concealing the presence of +SiO. With Tint=300 K, even at 10 bar of volatile content, SiO +becomes the most abundant species in the atmosphere, caus- +ing its features to dominate the spectrum. Note that our arbi- +trary use of unusually high Tint is purely for demonstrative pur- +poses. Close-in rocky planets are susceptible to various heating +mechanisms outside stellar irradiation. With an insulating, opti- +cally thick atmosphere, it is possible that trapping of heat allows +global magma oceans to be sustained at much hotter tempera- +tures than expected. Observations of silicates can therefore pro- +vide important constraints on the properties of the melt and the +interior (Zilinskas et al. 2022). +3.6. Depletion of hydrogen +Atmospheres of increasingly shorter orbital period are expected +to be affected by stellar wind erosion (Owen & Wu 2013; Lopez +& Fortney 2014; Lopez 2017; Fulton et al. 2017). For less mas- +sive worlds, such as ultra-short period super-Earths, this likely +results in extreme loss of volatile species and even extensive tails +of escaping particles (including silicates) (Mura et al. 2011; Cas- +tan & Menou 2011; Léger et al. 2011). If the internal reservoir +is unable to counteract depletion, the atmosphere will increase +in its mean-molecular weight. With depletion of hydrogen, CO, +CO2, N2 or SO2 atmospheres are not unusual outcomes. In Fig- +ure 10, we investigate models that are severely depleted of hy- +drogen. As before, planets at two different orbital distances are +shown with the oxygen abundance being determined via out- +gassing. In addition to hydrogen depletion, by varying the carbon +mixing ratio, we explore the added impact of the C/O parameter. +For the close orbit cases the major difference caused by the +depletion of hydrogen is the lack of H2O in the atmosphere. For +a C/O ratio of 0.2 (Faint TP profile and spectrum), the thermal +structure is similar to the cases discussed in Section 3.1. How- +ever, with no H2O the excess oxygen makes SiO the dominant +constituent, followed by CO and O2. Depletion of hydrogen is +yet another pathway in which SiO-dominant atmospheres are +possible. Due to the lack of H, the H– continuum is exchanged +for the opacities of CO and CO2, with a major band of CO2 ap- +pearing at 4.5 µm (faint upper spectrum in the bottom panel). +SiO still remains the only absorber at 9 µm. If the C/O ratio is +increased to 1.0 (Blue curves), the formation of CO becomes +even more efficient, leaving SiO nearly two orders of magnitude +behind in volume mixing ratio (highlighted abundance curves in +the top right panel). From a theoretical perspective, for C/O ra- +tios close to unity, CO atmospheres are easy to attain. Due to +this, observability of the 5 µm SiO feature may not be feasible in +such atmospheres. For comparison we show a pure, outgassing- +produced, silicate atmosphere (Cyan). While between volatile +and silicate cases, the 9 µm SiO feature remains, the presence +of it at 5 µm can be lacking. Detecting a combination of carbon +oxides and SiO could indicate that the atmosphere is of a high +C/O ratio with a significant outgassing component. +In the less irradiated cases, with C/O = 0.2, the tempera- +ture profile is only inverted in very the upper region, similar to +what was found in the original models (Fig. 3). However, the +chemistry here is vastly different (see Figure 11). The lack of +outgassed oxygen allows for N2 to take over as the dominant +species. Its abundance is also closely followed by sulphur, no- +tably SO2. While N2 is a weak absorber, the opacity of SO2 +is significant, peaking at 4 and 7-9 µm (Faded lower spectrum +of 10, features are not marked). Sulphur, hot Venus-like atmo- +spheres may be possible on irradiated rocky worlds and should +be taken into consideration for observations with MIRI LRS +(Schaefer & Fegley 2011; Kane et al. 2014; Zolotov 2018; Lig- +gins et al. 2022). Other, slightly diminished, yet still visible spec- +tral features include that of CO and CO2. If the C/O ratio is +increased to 1 (Pink curves), N2 still remains as the dominant +Article number, page 8 of 14 + +M. Zilinskas et al.: Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Temperature (K) +10 +8 +10 +6 +10 +4 +10 +2 +10 +0 +Pressure (bar) +Depleted H +C/O = 1.0 +C/O = 0.2 +Silicate +C/O = 1.0 +C/O = 0.2 +Silicate +10 +8 +10 +6 +10 +4 +10 +2 +10 +0 +Volume Mixing Ratio +10 +8 +10 +6 +10 +4 +10 +2 +10 +0 +Pressure (bar) +0.01 AU +SiO +CO +SiO +CO +1 +5 +10 +30 +Wavelength (micron) +0 +50 +100 +150 +200 +250 +Fplanet/Fstar ppm +SiO +CO +CO +Silicates +PS +PS +CO +VO +Fig. 10: Figure has been updated. Moved labels. Atmospheric models with severely depleted hydrogen at orbital distances of +0.01 and 0.04 AU. In the top panel we show TP profiles for C/O ratios of 1.0 (Pink and Blue) and 0.2 (Faded) as well as an +additional model of a pure silicate atmosphere (Cyan). All, cases assume dayside heat redistribution (f=2/3) with volatile-containing +atmospheres having 1 bar surface pressure. The total pressure in the pure silicate case is computed using the outgassing code. The top +right panel contains abundances of SiO and CO for 0.01 AU case (Blue TP profile), with the faint curves representing a pure silicate +atmosphere. The bottom panel displays emission spectra colour-coded for their respective TP profiles. Marked regions denote major +opacity contributions for the cases with a C/O ratio of 1.0. Spectral features for other cases are explained in the text. Spectra are +shown at a resolution of λ/∆λ = 600. +component, however, previously seen oxygen-rich species, such +as SO2 are no longer abundant. Instead, PS rises as the second +most abundant molecule, causing increase in shortwave opacity. +The photosphere becomes severely inverted, resulting in large +emission features. Aside from its shortwave component, PS has +potentially observable IR bands at 7 and 15 µm (see Fig. 6 for +its full opacity). Because of the strong VO at 10 µm, this species +may be interesting for observers. +3.7. Observability of currently known targets +Despite numerous observations with low and high resolution in- +struments, no definite detections of molecules have been made +on any lava world. JWST with its broad spectral coverage and +high sensitivity, is expected to shed new light on the subject. +So far, the performance of the telescope has surpassed all ex- +pectations (Rigby et al. 2022; The JWST Transiting Exoplanet +Community Early Release Science Team et al. 2022). If short +period rocky planets do possess atmospheres, it is likely that +JWST will be able to pick these up with greater confidence than +anything before. Targets such as 55 Cnc e, or K2-141 b are ex- +cellent labs to test silicate outgassing, and have been chosen to +be observed during JWST’s first cycle. If the observations are +successful, many more targets are likely to follow. +In Figure 12, we showcase currently confirmed planets, with +a radius of 1-4 R⊕, as a function of stellar magnitude and emis- +sion flux at 9 µm. Note that here the flux ratio represents the +baseline emission, not affected by possible absorption of present +species. The simulated error bars are for 20 hours of observing +time with MIRI LRS binned to a resolution of R = 10. Observ- +ability of a target will depend drastically not only on the con- +trast between the star and the planet but also on the brightness +of the system. As the surface temperature defines outgassed sili- +cate abundances, the equilibrium temperature of the planet is an +additional factor that needs to be considered. For less tenuous +atmospheres, this condition may be somewhat relaxed, as insu- +lation of heat can allow for global magma oceans to be sustained +at greater temperatures. Despite the ample number of discov- +Article number, page 9 of 14 + +A&A proofs: manuscript no. main +Fig. 11: Figure has been updated. Moved labels. Five most abun- +dant molecules in an atmosphere corresponding to a hydrogen- +depleted case with C/O = 0.2 at 0.04 AU. These are, in decreas- +ing abundance, N2, SO2, CO, CO2 and SO. The relevant spec- +trum for this case is shown in Fig. 10 (Pink curve). +Fig. 12: The figure has been updated with new noise estimates. +Currently confirmed super-Earths and sub-Neptunes plotted as +a function of stellar magnitude (K) and emission flux of the +planet at 9 µm. The flux ratio here represents baseline emission +modelled with dayside redistribution (f = 0.667), which is not +affected by present absorbing species. 9 µm is the wavelength +where the largest SiO feature is expected to manifest. A selec- +tion of favourable targets show PANDEXO simulated noise for the +MIRI LRS instrument with 20 hours observation time, binned to +R=10. +ered worlds, most orbit stars that are too dim to be good targets +for atmospheric characterisation with JWST’s MIRI instrument. +While the brightest systems, like 55 Cnc, have simulated noise +of just a few ppm, at a stellar magnitude (K) of 9, it increases +close to 50 ppm. Considering such atmospheres only reach a few +hundred ppm at the 9 µm range, characterisation of any present +features may be extremely difficult. That said, there are a num- +ber of planets that are potentially good follow up candidates for +short duration programs. +One of the brightest and most studied super-Earths is 55 Cnc +e, which will be observed with NIRCam and MIRI LRS instru- +ments during JWST’s first cycle (Hu et al. 2021a; Brandeker +et al. 2021). Whether this planet possess an atmosphere is cur- +rently debated, with a general consensus leading to an existing +high-mean-molecular-weight atmosphere, possibly with an out- +gassed silicate component. Compositions dominated by CO, N2 +or O2 are all possible, with low metallicity, H2-rich atmospheres +being less probable (Demory et al. 2016; Angelo & Hu 2017; +Zilinskas et al. 2020, 2021). Though there have been claims of +detected HCN, which would indicate abundant H2 and a high +C/O ratio (Tsiaras et al. 2016). Recent reanalysis of Spitzer +phase curve data of 55 Cnc e also suggest a high average dayside +temperature of T⋆= 3771 K, which may be caused by the pres- +ence of SiO (Mercier et al. 2022). Still, without observations of +broad spectral coverage, conclusions for this planet cannot yet +be drawn. +For high equilibrium temperature, several other targets are +of various radii are of notable interest. Smaller rocky worlds, +such as K2-141 b, TOI-1807 b, TOI-561 b are ideal for ob- +serving silicates (Hedges et al. 2021; Dang et al. 2021; Zieba +et al. 2022). TOI-561 b is reported to have unusually low den- +sity, which could imply a large volatile component (Lacedelli +et al. 2022). Larger worlds include K2-100 b, TOI-849 b or TOI- +2260 b, all of which may also host volatile, H2-rich atmospheres +with underlying magma oceans. Indicated via arrows are some +of the highest contrast planets, including GJ-1214 b, GJ 436 b, +LHS-3844 b and several others. While the equilibrium tempera- +ture of these planets is generally too low to host magma oceans, +an insulating atmosphere could force larger surface temperatures +and thus visible contamination of silicates. +The search for outgassed silicates is certainly not hindered +by the lack of suitable targets, but rather by how small spectral +features are expected to be. While we do not model full spectra +for any of the suggested targets, the emission features for most +of these should be expected to closely mimic our presented 2 +R⊕ test cases. Generally, planets will not be observed for more +than a few orbits, making spectral noise a considerable issue. +With JWST’s MIRI LRS it should be possible to characterise the +presence of SiO on short-period rocky worlds and sub-Neptunes, +but to do so will certainly be a grand challenge. +4. Discussion +4.1. Importance of heat redistribution and stellar type +Characterisation of lava worlds depends as greatly on the struc- +ture of the atmosphere as it does on the chemistry. While the +spectral energy distribution of the host star largely determines +the features of the temperature profile and the emission spectrum +of the planet, the atmosphere itself is prone to physical processes +that impact observability. Namely, for 1-D models, the efficiency +of heat redistribution determines the total given energy budget, +which subsequently decides much of the occurring chemistry. +Approximations of this effect are made through a single factor +f. Tenuous, silicate atmospheres are expected to be inefficient in +transporting heat, confining it to the dayside of the planet (f = +0.667-1.0). More volatile, optically thick atmospheres are more +efficient, to point where a significant fraction of the incoming +stellar radiation can be deposited on the night side. A maximum, +full redistribution results in f = 0.25. Using analytical heat re- +distribution scaling theory from Koll (2022), in Figure 13 we +demonstrate the possible effect that this may have on observabil- +ity of silicates. +In general, we assume that our modelled planets do not re- +distribute heat efficiently. Most of the models are set to use +f = 0.667, however using a scaling theory derived for tidally +locked worlds, we find that strongly irradiated atmospheres with +volatiles can become very efficient at transporting heat, in cases +Article number, page 10 of 14 + +10 +C/O = 0.2: 0.04 AU +-6 +10 +Pressure (bar) +N2 +10 +SO2 +CO +CO2 +10 +SO +3 +10 +10 +10 +10 +Volume Mixing Ratio700 +3000 +GJ-1214 b (1335 ppm) +K2-320 b (920 ppm +K2-266 b (767 ppm) +TOl-2406 b (840 ppm) +LHS-3844 b (812 ppm) +TOl-1696 b (781 ppm)l +600 +GJ 436 b (710 ppm) +TO1-849 b +2500 +500 +2000 +Dayside +400 +K2-100 b +300 +K2-141 b +1500 +TOl-2260 b +eq +TOl-1807 b +200 +55 Cnc e +TOI-431 b +1000 +HD 213885 b +100 +0 +500 +14 +12 +10 +6 +3 +8 +4 +Magnitude of Stellar Target (K)M. Zilinskas et al.: Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes +1 +5 +10 +30 +Wavelength (micron) +0 +50 +100 +150 +200 +250 +Fplanet/Fstar ppm +SiO +SiO +F = 0.363 +F = 0.581 +F = 0.667 +1200 2000 +3000 3800 +Temperature (K) +10 +2 +10 +5 +10 +8 +Pressure (bar) +F = 0.363 +F = 0.581 +F = 0.667 +Fig. 13: Temperature profiles and emission spectra of models +with computed heat redistribution factor f at orbital distances +of 0.015 and 0.04 AU. The atmospheres with f = 0.667 (Both +cyan curves) are arbitrarily set to represent dayside-confined re- +distribution of irradiation. Factors of f = 0.363 and f = 0.581 +are calculated using the formulation of heat redistribution for +rocky planets in Koll (2022). The parameters determining heat +transport efficiency are planetary equilibrium temperature, atmo- +spheric surface pressure and longwave optical depth. Spectra are +shown at a resolution of λ/∆λ = 600. +reaching as high as f = 0.363. While not attaining global redistri- +bution (f = 0.25), this severely impacts the total energy budget, +reducing the surface temperature and thus silicate observability. +It is no surprise that volatile atmospheres increase the efficiency, +but the large magnitude of it for just 1 bar of an atmosphere +does indicate that even a small amount of volatiles can have a +severe impact. On the contrary, planets at larger orbital distances +show no large increase in redistribution efficiency, keeping most +of the energy confined to the dayside. Since the primary mech- +anism of heat transport is the atmosphere, phase curve measure- +ments can indicate its significance. Detecting temperature offsets +or high nightside flux could indicate that the planet has retained +a substantial, volatile-rich atmosphere. An unusual super-Earth +55 Cnc e has been found to show signs of this (Demory et al. +2016). +Since emission of the planet probes its thermal profile, ob- +servability is also greatly impacted by the spectrum of the host +star. In Figure 14, we take one of our solar cases and compare it +to models of planets placed around stars of different type, but at +an equivalent irradiation distance. With increasing stellar tem- +perature, its spectrum is shifted towards shorter wavelengths, +causing greater incident UV flux. Going from a G to a typical +K type star (T⋆= 4305 K) inversions weaken drastically. Atmo- +spheres around cool M-dwarfs are likely to contain no deep in- +version that strongly affect the emitting photosphere. For charac- +terisation of lava worlds through emission spectroscopy this may +present a slight issue, since strong inversions are one of the key +characteristics of silicate-rich atmospheres that may help iden- +tify them. +5. Conclusion +In preparation for JWST observations, we have used consistent +outgassing equilibrium chemistry and radiative-transfer models +to assess the possibility of detecting silicates in volatile atmo- +spheres of super-Earths and smaller sub-Neptunes. Placing a hy- +2500 +3000 +3500 +4000 +4500 +5000 +Temperature (K) +10 +8 +10 +6 +10 +4 +10 +2 +10 +0 +Pressure (bar) +Tstar = 5750 K +Tstar = 3026 K +Tstar = 4305 K +Tstar = 6407 K +Tstar = 5750 K +Tstar = 3026 K +Tstar = 4305 K +Tstar = 6407 K +Fig. 14: Temperature profiles computed consistently with atmo- +spheric chemistry for different stellar type hosts. In each case +the planet is placed to an equivalent irradiation distance from +the star. T⋆= 3026 K spectrum represents GJ 1214 and is taken +from the MUSCLES database. The other three are modelled us- +ing PHOENIX spectra of the denoted temperature. +pothetical 2 R⊕ planet at varying orbital distances around a Sun- +like star, we have explored a wide variety of viable atmospheric +compositions, rich in H, C and N, that also include an outgassed +silicate component, e.g., Si, O, Ti. We modelled atmospheres of +up to 10 bar surface pressure, varied in metallicity and C/O ratio. +A major assumption made in this work is that the atmosphere is +in complete equilibrium with the underlying melt. Our results +are intended to guide observers towards potentially detectable +species that would help characterise worlds with exposed or con- +cealed lava oceans. Below are our key findings. +1. For emission spectroscopy with JWST, SiO is likely to +remain the only characterisable species that could indicate +strong outgassing from an underlying magma ocean. How- +ever, on atmospheres containing volatiles, the main opacity +bands of SiO at 5 and 9 µm can be heavily suppressed. +Only the most irradiated worlds, with melt temperatures > +2500-2800 K are expected to show super-solar abundances +of silicates (Assuming 1 bar of volatiles). Unlike for pure +lava worlds, we do not find features of SiO2 to be of +significance. Very high temperature melts may also result +in broad shortwave features, arising mainly from TiO and +several other outgassed species. Ultimately, the visibility of +silicates in volatile atmospheres will largely depend on the +atmospheric properties and the efficiency of its interaction +with the melt. +2. Thermal inversions are prominent in atmospheres con- +taminated with silicates. We find that numerous outgassed +silicates cause deep inversions that affect the photosphere, +even if the atmosphere has strong infrared absorption origi- +nating from volatiles. The largest contributors to shortwave +opacity are: TiO, SiO, MgO, Fe and Mg. We also find that +alkali metals, metal hydrates and, in certain cases, PS or +VO can become a source of inversions. Because outgassing +scales exponentially with the temperature of the melt, +planets with no insulating atmospheres and larger orbital +distances are not likely to show strong inversions originating +due to silicates. +Article number, page 11 of 14 + +A&A proofs: manuscript no. main +3. The presence of hydrogen can affect observability of +silicates, including of SiO. Our models show that added +hydrogen results in formation of ample hydrocarbons and +H2O. Chemically, H2O and SiO are direct competitors for +the outgassed oxygen, however, SiO is much less prone +to thermal dissociation, making it more prominent in the +lower-pressure, inversion-affected regions. On strongly +irradiated worlds, the continuum of H– can also heavily +obscure the 5 and 9 µm SiO features. +4. The C/O ratio has a large effect on SiO. Even in H2-rich +atmospheres, formation of CO due to a C/O ratio ≳ 1.0 can +result in a drastic reduction of silicate oxides. Chemically, +CO takes priority for oxygen, affecting all other oxides. This +can consequently result in Si bonding with H instead of O, +forming SiH, potentially making it a species of interest for +observations. Detecting CO at 4.5 µm and SiO at 9 µm could +indicate an atmosphere with no hydrogen and a high C/O +ratio. +5. Atmospheric pressure, insulation and redistribution of heat +are major factors in deciding whether volatile atmospheres +are contaminated with silicates. Our models show that atmo- +spheres of 10 bar with internal temperature enabled become +convective, resulting in a large increase of surface tempera- +ture. Exponential scaling of outgassing can lead to SiO be- +coming a dominant species, even in substantial volatile at- +mospheres. In contrary, effective heat redistribution can re- +duce surface temperatures. Using a scaling theory for tidally +locked planets, we find that even small volatile atmospheres +are efficient at transporting heat, lowering surface tempera- +tures. Observations of silicates can, therefore, provide im- +portant constraints on the underlying melt properties and the +balance between insulation and redistribution of heat. +Acknowledgements. +We acknowledge funding from the European Research Council under the +European Union’s Horizon 2020 research and innovation program under grant +agreement No 694513. We thank Matej Malik for the insightful discussion on +HELIOS. We are also grateful for the comments of the editor and the anonymous +referee, which helped improve the quality of this paper. +Software used in this work: HELIOS-K (Grimm & Heng 2015; Grimm +et al. 2021); HELIOS(Malik et al. 2017; Malik et al. 2019b); FASTCHEM (Stock +et al. 2018); LavAtmos (van Buchem et al. 2022); petitRADTRANS (Mollière +et al. 2019, 2020); PANDEXO Batalha et al. (2017); numpy (Harris et al. 2020); +matplotlib (Hunter 2007); seaborn (Waskom 2021); astropy (Astropy +Collaboration et al. 2022). +Supplementary material is available on request from the author. +References +Adam, A. Y., Yachmenev, A., Yurchenko, S. N., & Jensen, P. 2019, Journal of +Physical Chemistry A, 123, 4755 +Angelo, I. & Hu, R. 2017, The Astronomical Journal, 154, 232 +Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, +167 +Azzam, A. A. A., Tennyson, J., Yurchenko, S. N., & Naumenko, O. V. 2016, +MNRAS, 460, 4063 +Barber, R. J., Strange, J. 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(2015) +CaH +HELIOS-K +MoLLIST +Li et al. (2012); Bernath (2020) +CaO +HELIOS-K +VBATHY +Yurchenko et al. (2016) +Fe +DACE +VALD +Ryabchikova et al. (2015) +K +DACE +VALD +Ryabchikova et al. (2015) +Mg +DACE +Kurucz +Kurucz (1992) +MgH +HELIOS-K +MoLLIST +GharibNezhad et al. (2013); Bernath (2020) +MgO +HELIOS-K +LiTY +Li et al. (2019) +Na +DACE +VALD +Ryabchikova et al. (2015) +NaH +HELIOS-K +Rivlin +Rivlin et al. (2015) +Si +DACE +VALD +Ryabchikova et al. (2015) +SiH +HELIOS-K +Sightly +Yurchenko et al. (2018b) +SiO +HELIOS-K +SiOUVenIR +Yurchenko et al. (2022) +SiO2 +DACE +OYT3 +Owens et al. (2020) +Ti +DACE +VALD +Ryabchikova et al. (2015) +TiH +HELIOS-K +MoLLIST +Burrows et al. (2005); Bernath (2020) +TiO +HELIOS-K +Toto +McKemmish et al. (2019) +Volatiles +CO +DACE +Li2015 +Li et al. (2015) +CO2 +DACE +HITEMP & UCL-4000c +Rothman et al. (2010); Yurchenko et al. (2020) +CH3 +DACE +AYYJ +Adam et al. (2019) +CH4 +DACE +YT34to10 +Yurchenko et al. (2017) +C2H2 +DACE +aCeTY +Chubb et al. (2020) +C2H4 +DACE +MaYTY +Mant et al. (2018) +CN +HELIOS-K +Trihybrid +Syme & McKemmish (2021) +H2O +DACE +POKAZATEL +Polyansky et al. (2018) +HCN +HELIOS-K +Harris +Barber et al. (2014) +HS +HELIOS-K +GYT +Gorman et al. (2019) +H2S +DACE +AYT2 +Azzam et al. (2016) +NH3 +DACE +CoYuTe +Coles et al. (2019) +OH +DACE +HITEMP +Rothman et al. (2010) +S +DACE +VALD +Ryabchikova et al. (2015) +CS +ExoMold +JnK +Paulose et al. (2015) +NS +ExoMol +SNaSH +Yurchenko et al. (2018a) +SO2 +ExoMol +ExoAmes +Underwood et al. (2016a) +SO3 +ExoMol +UYT2 +Underwood et al. (2016b) +PH3 +DACE +SAlTY +Sousa-Silva et al. (2015) +PS +HELIOS-K +POPS +Prajapat et al. (2017) +VO +HELIOS-K +VOMYT +McKemmish et al. (2016) +Scattering and Continuum +H, H2, H2O, He, O2 +Scattering +H– +Continuum (bf & ff) +John (1988); Gray (2008) +H2 –H2 +HELIOS-K +HITRAN +Gordon et al. (2017) +H2 –He, O2 –O2 +petitRADTRANS +Mollière et al. (2019, 2020) +a DACE database https://dace.unige.ch/; b Opacities are computed with HELIOS-K https://github.com/exoclime/HELIOS-K (Grimm & +Heng 2015; Grimm et al. 2021); c We use HITEMP2010 for temperature profiles and UCL-4000 for emission spectra; d For ExoMol +opacities we make use of the conversion work done by Chubb et al. (2021), which are only used for generating emission spectra. In +general, we make heavy use of the DACE (Grimm & Heng 2015; Grimm et al. 2021), ExoMol (Chubb et al. 2021), Kurucz (Kurucz 1992), +VALD3 (Ryabchikova et al. 2015) and HITRAN (Gordon et al. 2017) databases. +Article number, page 14 of 14 + diff --git a/c9E4T4oBgHgl3EQfpQ0e/content/tmp_files/load_file.txt b/c9E4T4oBgHgl3EQfpQ0e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d9ae37387d45f51e934410e21632c43dc2597bc --- /dev/null +++ b/c9E4T4oBgHgl3EQfpQ0e/content/tmp_files/load_file.txt @@ -0,0 +1,1441 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf,len=1440 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' main ©ESO 2023 January 13, 2023 Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes Exploring the edge of the evaporation desert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Miguel1, 2, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' van Buchem1, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Snellen1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 1 Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333CA Leiden, the Netherlands 2 SRON Netherlands Institute for Space Research , Niels Bohrweg 4, 2333 CA Leiden, the Netherlands e-mail: zilinskas@strw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='leidenuniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='nl Received June XX, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' accepted July XX, 2021 ABSTRACT Many of the confirmed short period super-Earths and smaller sub-Neptunes are sufficiently irradiated for the surface silicates to be sus- tained in a long-lasting molten state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While there is no direct evidence of magma ocean influence on exoplanets, theory suggests that due to outgassing and diverse evolution paths, a wide range of resulting atmospheric compositions should be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Atmospheric contamination caused by the outgassing of the underlying magma ocean is potentially detectable using low resolution spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The James Webb Space Telescope provides the necessary spectral coverage and sensitivity to characterise smaller planets, including lava worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In this light, we assess observability of outgassed silicates submerged in volatile atmospheres on the edge of the evapora- tion valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' By placing a hypothetical 2 R⊕ planet around a Sun-like star, we self-consistently model, in 1-D, a wide range of potential atmospheric compositions, including thermal structure and outgassing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We focus on atmospheres rich in H, C and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We assess di- verse chemistry of silicates and volatiles, and what features of outgassed species could be detected via emission spectroscopy using MIRI LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Results indicate that even for substantial volatile envelopes, strong in infrared opacity, the presence of silicates causes deep thermal inversions, affecting emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Similar to pure lava worlds, SiO remains the only outgassed species with major infrared, 5 and 9 µm, bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' However, even a small amount of volatiles, especially of H2O and H–, may hinder its observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We also find that the C/O ratio plays a large role in determining the abundance of SiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Detecting SiO on a strongly irradiated planet could indicate an atmosphere with high metallicity and a low C/O ratio, which may be a result of efficient interaction between the atmosphere and the underlying melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Planets and satellites: atmospheres – Planets and satellites: terrestrial planets – Techniques: spectroscopic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Introduction Ever since their discovery, there has been great interest in try- ing to characterise an unravel the mysteries of the seemingly ambiguous super-Earths and sub-Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While more mas- sive, Neptune-like, planets are expected to retain most of the primordial H/He, intermediate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5 R⊕) and smaller worlds are likely to be extremely diverse in their atmospheric composi- tion and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Figure 1 showcases the population of close-in planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Occupying the edge of the evaporation desert, rocky worlds are shaped by erosion, accretion, degassing and volcanism, with some possibly forming long-lasting secondary atmospheres (Elkins-Tanton & Seager 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Though because of such close proximity to the star, many of the planets likely end up as bare rocks, with no visible atmospheric component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Observations of several temperate and hot super-Earths seem to favour this theory (Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zieba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Crossfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' That said, even without an insulating atmosphere, these have temperatures high enough to engulf the dayside of the planet with magma oceans, which should result in tenuous, but observ- able silicate envelopes (Schaefer & Fegley 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Miguel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Kite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For such worlds, SiO and SiO2 have been proposed to be the pri- mary species that could be probed via infrared emission (Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' It is also fea- sible that high-mean-molecular-weight species can survive ero- sion, leaving denser, CO or N2 atmospheres intact (Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 55 Cnc e may be a primary example of this (De- mory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Angelo & Hu 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Hammond & Pierrehum- bert 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While studies indicate that planets below ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 R⊕ are likely stripped of H2 (Rogers & Owen 2021), new interior models show that magma-atmosphere interaction during evolu- tion could lead to large reservoirs of H2O, buffering H2O atmo- spheres, which, due to thermal and photochemical dissociation, should result in abundant H2 as a by-product (Kite & Schaefer 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Dorn & Lichtenberg 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The only outstanding weak- ness of the proposed theory is the efficiency of the interaction between the melt and the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Going to larger radii (above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 R⊕), the observed discrep- ancy of densities may indicate the existence of water/ocean plan- ets that are shrouded with dense steam atmospheres (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Mousis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Nixon & Madhusudhan 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Bean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Insulation on sub-Neptunes is also expected to al- low for deep magma oceans to be sustained indefinitely (Kite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Just as with smaller planets, depending on the effi- ciency of magma-vapour interaction and atmospheric mixing, it could result in H2 or H2O-rich envelopes that are heavily con- Article number, page 1 of 14 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='05190v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='EP] 12 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 1: Short period exoplanets with radii < 4 R⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Coloured markers indicate confirmed planets, grey markers are candi- date planets from Kepler, K2 and TESS missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Colour value of confirmed planets represents the density of the occurrence rate, which peaks at two distinct radii of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='4 R⊕, seemingly separating the population into super-Earths and sub- Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The highlighted region roughly encompasses the pa- rameter space applicable to our modelled cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Marked on the figure is the evaporation desert and one of the most well stud- ied super-Earths - 55 Cnc e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The data is taken from the NASA exoplanet archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' taminated by silicate species (Schlichting & Young 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Kite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Observations with JWST will provide necessary constraints for the ongoing theoretical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Extended H2 atmospheres of larger super-Earths and sub-Neptunes with substantial scale heights are easily probed via transmission spectroscopy (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For intermediate and smaller planets, measuring emission of the dayside may prove to be the only viable char- acterisation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' However, even with JWST, characterising the chemistry of potential atmospheres will be challenging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' de- tecting silicates even more so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' From an observer’s standpoint, finding whether these planets have atmospheres at all is a major stepping stone in the field of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In this work we explore the chemistry and observability of outgassed silicates in volatile envelopes of irradiated rocky worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The highlighted region in Figure 1 roughly indicates the parameter space applicable to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In contrast to studies done by Kite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Kite & Schaefer (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Schlicht- ing & Young (2022), we do not model substantial atmospheres, but focus on cases where the surface pressure is relatively low in comparison to Neptune-like planets (< 10 bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We make use of consistent outgassing equilibrium and radiative-transfer mod- els to predict what silicate features are potentially characteris- able through infrared emission spectroscopy, especially at wave- lengths relevant for JWST’s MIRI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Finding signs of silicates could hint at an underlying magma ocean, allowing us to put better constraints on the proposed diversity of super-Earths and sub-Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Section 2 contains the description of our approach in constructing 1-D, self-consistent atmospheric models, including chemistry and thermal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The analysis of the results is given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We discuss some of the important factors that may affect observability in Section 4, and finally conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Setting up the system To explore observability of silicates in volatile atmospheres we set up a grid of models that would represent a typical super-Earth or a sub-Neptune orbiting a Sun-like star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We focus on intermedi- ate pressure envelopes, ranging from 1-10 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Our models focus on cases where the surface temperature is higher than 1400 K, which is enough for magma oceans to form and influence the at- mospheric composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For a G-type star and dayside confined heat redistribution, this typically results in a maximum orbital distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='06 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We make use of several different codes that are set up to consistently calculate outgassing, chemical abun- dances and temperature profiles, inclusive of the surface temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The models are then used to simulate emission spectra and expected JWST noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Below we describe the approach for each of the components in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Determining the chemistry A major assumption made in this work is that the overlying atmosphere equilibrates with the molten surface, allowing out- gassing to control the abundances of all silicates, including oxy- gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Since general atmospheric compositions of super-Earths and sub-Neptunes are unknown, we take the freedom to explore a grid of possible outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' These are drastically varied in metal- licity, C/O ratio, volatile content, atmospheric pressure and even internal temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For volatiles, we take the solar composition (Lodders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2009) and adjust its metallicity (M/H), where all of the elements except for H and O are linearly scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While we normally as- sume that oxygen abundance is determined via outgassing, to ex- plore differences between solar and outgassed atmospheres, we also model cases where oxygen is set by the metallicity parame- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In addition to this, we vary the C/O ratio (via carbon adjust- ment) together with the abundance of H/He, which allows us to carefully dictate the major molecular constituents in the atmo- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This allows us to explore cases where strong irradiation and large sinks of light elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=', photoevaporation, dis- solution) may result in high-mean-molecular-weight envelopes, dominated by either CO, CO2 or even N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The outgassing budget is determined by an open-source code LavAtmos1 (van Buchem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022), which calculates the melt- vapour equilibrium for a given surface temperature and melt composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' To accurately determine the activity of the oxides in the melt, LavAtmos makes use of the liquid-solidus code MELTS (Ghiorso & Sack 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The package solves for the ox- ides containing the following elements: Al, Ca, Fe, K, Mg, Na, Si, Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The resulting outgassed partial pressures are added to the volatile atmospheres while keeping the total surface pressure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This is equivalent to reducing the relative abundances of volatiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' As in (Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022), we take the magma to be composed as Bulk Silicate Earth (BSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' It contains 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='97 % SiO2, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='66 % MgO, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='24 % FeO, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='77 % Al2O3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='78 % CaO, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='35 % Na2O, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='18 % TiO and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 % K2O (wt%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The surface temperature and the outgassing are consistently calculated using a radiative-transfer code, as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Important to note that currently LavAtmos does not account for possible deposition of volatiles into the magma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' As shown in the work of Kite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2020), this can have substantial consequences on the atmospheric composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The detailed analysis of this is out of scope for this paper and will be a focus of a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='com/cvbuchem/LavAtmos Article number, page 2 of 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5 Evaporation Deser 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5 nce 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='10 Orbital Distance (AU)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' : Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes With the elemental budget determined, atmospheric chem- istry is solved using the thermochemical equilibrium code FastChem2 (Stock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Stock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We take into account over 200 relevant species, inclusive of neutral and ion chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The thermal data used is compiled from the Burcat NASA thermodynamics database3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Computing thermal profiles The temperature structure is solved using the radiative-transfer code HELIOS4 (Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We allow convective adjustment to take place using an adiabatic coefficient of κ = 2/7, applicable to diatomic atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The profiles are treated with a rocky surface boundary, the implementation of which is described in detail in Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2019a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Whittaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' All of our models are reiterated until convergence such that the attained surface temperature is in good agreement with the atmospheric chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For the purposes of showing possible spectral features, the heat redistribution is confined to the dayside of the planet (f=2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In specific cases it is approxi- mated using the longwave optical depth of the atmosphere, based on equations from Koll (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We use a total of 50 opacity sources, including all of the major volatile and silicate species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The entire list and descrip- tions of all the opacities used in this study are listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' All of the atomic opacities are obtained from the DACE5 database, with the majority using the Vienna Atomic Line Database (VALD3) line lists (Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For molecular opacities we make use of both the DACE database and the opacity calculator HELIOS-K6 (Grimm & Heng 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Following Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2021), such are approxi- mated using a Voigt fitting profile, wing cutting length of 100 cm−1 and, where line lists allow, a temperature of up to 4000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In terms of observability, SiO is expected to be a key species of irradiated atmospheres (Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In this work, we use the new ExoMol7 SiOUVenIR line list (Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022), which covers the entire UV-infrared wavelength range and is applicable to high temperatures ex- pected to occur on hot super-Earths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Figure 2 shows key un- weighted opacities considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' SiO shortwave opac- ity (< 1 µm) is a strong contributor towards occurring tempera- ture inversions, while the longwave bands peak at 5 and 10 µm and are potentially detectable spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While not dis- played, there are many other potential species that are significant absorbers in silicate and volatile atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For short period planets shortwave stellar flux becomes an important factor in shaping the thermal structure of the atmo- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Using simple blackbody stellar models results in incor- rect UV flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Thus all stellar irradiation models used in this work are generated via HELIOS using the PHOENIX (Husser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2013) and MUSCLES (France et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Youngblood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Loyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2016) databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Spectra and opacities are sam- pled at a resolution of λ/∆λ = 2000 and cover the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1 - 200 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='com/exoclime/FastChem 3 http://garfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='hu/Burcat/burcat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='html 4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='com/exoclime/HELIOS 5 https://dace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='ch/ 6 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='com/exoclime/HELIOS-K 7 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='exomol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='com/ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2: Comparison of SiO, H2O, TiO and H– opacities, shown at a resolution of λ/∆λ = 2000 for a temperature of 3000 K and atmospheric pressure of 10−2 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The description and sources of all used opacities can be found in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Simulating emission spectra On hot super-Earths, silicate atmospheres are expected to be confined to the tidally locked dayside of the planet, generally making them poor candidates for low-resolution transmission spectroscopy (Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Due to large atmospheric temperatures, spectral features may instead be probed through emission of the secondary eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' If, however, such planets do possess global, volatile atmospheres, transmission could be pos- sible, but its viability will depend strongly on the scale height (Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While in this work we focus on emis- sion spectroscopy, we note that for a number of known targets low-resolution transmission spectroscopy with JWST may also be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We generate emission spectra using the radiative-transfer code petitRADTRANS8 (Mollière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We use the same atomic and molecular opacities described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='3, including H2, H2O, O2 Rayleigh scattering, and H2 –H2, H2 –He, O2 –O2 and H– continuum opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The spectra are calculated at a resolution of λ/∆λ = 1000 for a wavelength range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='3 - 28 µm, encompassing the coverage of all JWST instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In all figures, the spectra are convolved to a lower resolution for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For notable targets, we assess JWST noise using PANDEXO9 (Batalha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2017), which is built on the Pandeia10 engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We only simulate MIRI Low Resolution Spectroscopy (MIRI LRS with λ/∆λ ≈ 100) in slitless mode, as it is likely to be the most suitable mode for characterisation of silicate features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The wave- lengths covered by the instrument are 5 - 12 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In each case, we use the corresponding stellar and planetary parameters obtained from the NASA exoplanets archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For corresponding stellar models we use PHOENIX generated spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Outgassed silicates in hydrogen atmospheres For a given initial composition, the thermal structure and the re- sulting chemistry of an atmosphere is determined by the stellar 8 http://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='com/mauricemolli/petitRADTRANS 9 https://exoctk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='edu/pandexo/ 10 http://jwst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='edu Article number, page 3 of 14 8 10 Sio H20 6 10 Tio 10 2 10 10 2 10 4 10 6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1 10 100 Wavelength (micron)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' main flux that the planet receives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In Figure 3, we showcase a hypo- thetical world placed around a Sun-like star of T = 5750 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The only free parameter varied between the cases is the orbital dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The 2 R⊕ planet is assumed to have a volatile-rich, solar- like, 1 bar atmosphere that is in equilibrium with an outgassed silicate component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In each model, silicate abundances are com- puted via outgassing of a BSE melt of a numerically converged surface temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Naturally, with increasing orbital distance, the temperatures fall and the abundance of silicates decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' At close orbits the surface temperature can reach over 3000 K, which results in a substantial amount of outgassed O, allow- ing for plentiful formation of oxides, including SiO and H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Our models show that 1 bar atmospheres with a surface tempera- ture higher than 2300-2500 K produce super-solar abundances of silicates, causing drastic changes in thermal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Shortwave absorbers heat the atmosphere causing deep thermal inversions, affecting even the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Below the photosphere, around 10−2 bar, the atmosphere becomes optically thick to radiation, resulting in isothermal regions where no heat transport occurs (Blue and two faded TP profiles in the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Similar thermal structure is observed in volatile-free, pure sili- cate atmospheres (Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022), implying that silicate opacities may largely be responsible in shaping the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Looking at the chemistry we find that even at the highest modelled temperatures many molecules survive thermal dissoci- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Abundances of major absorbers are shown in the top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While these atmospheres are filled with atomic species (H, O, Fe, Mg and others), oxides, such as SiO, H2O and TiO dominate its opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' At high pressures, H and O form H2O, making it a strong absorber (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Near the surface, H2O and SiO have similar volume mixing ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Moving to the upper, low pressure regions, H2O begins to dissociate into atoms and ions, while SiO remains in its molecular form, making it one of the most abundant species throughout the entire atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' It should be expected that SiO is a major constituent in atmo- spheres with an underlying magma ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For these cases we find that chemically, the abundance of SiO is only weakly af- fected by pre-existing volatiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In addition to SiO, vaporisation of magma at large temperatures also results in high abundances of TiO, which can become one of the most influential shortwave absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In the bottom panel of the figure we show the correspond- ing emission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While in this case the small planet-to-star contrast results in a relatively low emission signal, the emer- gence of the 5 and 9 µm SiO features is clear (Blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Due to occurring inversions SiO increases the observed flux at these wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Unlike silicate atmospheres modelled in Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2022), these show no significant sign of SiO2 absorption at 7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This is partly attributed to oxygen being chemically favoured to bond with volatiles, such as hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' At shorter wavelengths, TiO is one of the dominant absorbers, causing a broad feature below 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For BSE compositions, its presence is only important at high surface temperatures, typically larger than 2500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' It is worth noting that many different molecules and atoms contribute to the shortwave opacity, some of which may be detectable in more extended atmospheres using transmission spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Shortwave opacities are discussed in more detail at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In addition to molecular opacities H– becomes an important factor throughout the entire JWST wave- length range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Not only does it have a strong shortwave compo- nent, but its strong continuum at 10 µm may hinder observability of SiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Overall, out of all the outgassed silicates, the two SiO features are likely to be the easiest to characterise using MIRI LRS covering the 5-12 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Moving to colder cases, the abundance of all silicates de- creases rapidly, becoming sub-solar at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU (Pink curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The total outgassed pressure of just silicates at temperatures be- low 2500 K is comparable to a millibar (Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Assuming melt-vapour equilibrium is attained, the volatile com- ponent is likely to dominate, making species such as SiO or TiO unobservable with low resolution spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Another conse- quence of this is drastic reduction of outgassed oxygen, which raises the C/O ratio, allowing hydrocarbons to efficiently form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Most of the species in cooler atmospheres are heavily weighted towards infrared opacity, resulting in a lack of any significant inversions that may impact observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The spectrum here is dominated by molecules such as CH4, C2H2 and HCN, all show- ing deep absorption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Detecting silicates in emission at relatively large orbits could indicate that either the temperature of the melt is much higher then the planetary equilibrium temper- ature, or that silicates are not in equilibrium with the underlying melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Contribution function In Figure 4, we take one of strongly irradiated cases and show its emission contribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In the right panel, the high- lighted region represents the emitting photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For wave- lengths > 1 µm, this mostly coincides with pressures between 10−4 and 10−2 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' A major contributing molecule for longwave opacity is H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Its dominance is a general occurrence in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Plentiful hydrogen and oxygen assure that even at high temperatures, it is one of the most optically dominating species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Additional leftover hydrogen results in a strong H– continuum, pushing the general opacity higher up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The tail of the contin- uum can be seen at wavelengths > 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Since the abundance of SiO is not strongly affected by increasing temperatures and lower pressures, its opacity has large contributions from inverted regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' If atmospheres of super-Earths are prone to thermal in- versions, it is likely that SiO will show up as increased flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The 9 µm feature is and should be visible even with strong volatile opacities present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' If no volatiles are present, enough SiO2 may form to appear at 7 µm, complimenting the SiO feature (Zilin- skas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' There are many different species contributing to the total shortwave opacity (< 1 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' SiO, AlO, MgO, TiO, Mg and Fe, all have very strong opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Some lesser, but important species are: SiH, MgH, VO, Al, Ca, K, Na, Si and Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' TiO, having broad wavelength coverage, is perhaps the most important for observa- tions, as well as in its influence in shaping the thermal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Its presence is known to strongly affect atmospheres even in gas giants (Serindag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Note that on rocky planets, TiO is typically sustained in significant abundances only above 2500- 2800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Aside from TiO, the SiO UV band and Fe opacity have major influence on the strength and depth of the occurring in- versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' These species are also much more volatile and readily vaporised from the magma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Important to note that because of the large number of shortwave absorbers, even atmospheres that are missing major oxides such as SiO or TiO can still have deep occurring inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Previous studies have shown that pure silicate atmospheres have similar total shortwave opacity (Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This is unsurprising since the majority of shortwave absorbers come from silicate outgassing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While there are additional shortwave absorbers due to the presence of volatiles, namely SiH, MgH and VO, these are relative minor in comparison to silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Note that, due to a lack of thermal data, our models do not include FeH, the opacity of which peaks at 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For atomic species Article number, page 4 of 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' : Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes 1000 1500 2000 2500 3000 3500 4000 4500 Temperature (K) 10 8 10 6 10 4 10 2 10 0 Pressure (bar) Dayside 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01 AU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01 AU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU 10 12 10 10 10 8 10 6 10 4 10 2 10 0 Volume Mixing Ratio 10 8 10 6 10 4 10 2 10 0 Pressure (bar) SiO H2O TiO SiO H2O TiO 1 5 10 30 Wavelength (micron) 0 50 100 150 200 250 Fplanet/Fstar ppm SiO SiO TiO Hydrocarbons (CH4, C2H2, HCN) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3: The figure has been updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Extra legend in the top left panel has been added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Atmospheric models a super-Earth of 2 R⊕ orbiting at Sun-like star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In all cases, dayside confined heat redistribution (f=2/3) and a surface pressure of 1 bar are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The top left panel shows the temperature-pressure profiles at orbital distances of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='06 AU, with two highlighted cases being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01 AU (Blue) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU (Pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In the top right panel, the highlighted curves indicate abundances of SiO, H2O and TiO for the case of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01 AU with an effective planetary temperature of 3174 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In the same panel, the faded curves represent the chemistry of the same species at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU (Te f f = 1771 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The bottom panel contains the corresponding synthetic emission spectra, with the flat, thinner curves representing blackbody emission (assuming computed surface temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Major absorbers for highlighted cases are indicated via shaded areas, with SiO, TiO and hydrocarbons (CH4, C2H2 and HCN) being the primary species of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Spectra are shown at a resolution of λ/∆λ = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' we also do not use pressure-caused broadening, likely leading to some underestimation of the line widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' It is possible that many of the atomic species, especially alkali metals, are a lot more dominant in shaping the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The inherent complexity of the shortwave region makes it difficult to correctly model temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Many of the mentioned opacities here are often overlooked, leading to the- oretically incorrect temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' After the chemistry, shortwave opacities are likely to be a major source of uncertainties which can greatly affect interpretations of observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Impact of metallicity and C/O ratio The process of formation for short-period rocky planets is un- known, but it is often assumed that such are heavily enriched in metals (Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Moses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In Figure 5, we use varied metallicity to explore what effect it may have on ob- servability of silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The blue and pink curves represent the original solar models showcased in the previous section, while, for each of the orbits, the overplotted curves show atmospheres with 10, 100 and 1000 times increased metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Note that the metallicity here does not control the abundances of outgassed silicates or oxygen, but only of all volatiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The main impact of it is thus increase of C, N and the C/O ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The corresponding chemistry of the close orbit cases is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For close orbits an x10 increase in metallicity has minimum effect on atmospheric chemistry or thermal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With SiO and H2O remaining as dominant oxides, the spectral features are mostly unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This slight increase in metallicity does al- low for CO to form more efficiently, very slightly boosting its opacity at 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The unweighted opacity of CO, along with a few other species that are discussed later, are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' When the metallicity is increased to x100, the C/O ratio crosses unity and the chemistry starts prioritising the formation of CO, heavily diminishing other oxides (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Atmospheres that do not outgas or retain enough oxygen are likely to suffer this Article number, page 5 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 4: Emission contribution function of a strongly irradiated super-Earth orbiting a Sun-like star at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='015 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The temperature profile in the left panel is taken from the models showcased in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Marked in dashed is the effective temperature of the planet T = 2950 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The right panel showcases the emitting region of the atmosphere as a function of wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Major contributing molecules are marked in their respective regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Lesser contributing opacities are discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Spectra are shown at a resolution of λ/∆λ = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 5: Synthetic emission spectra for an atmosphere of in- creased metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The main cases, blue and pink, represent models with solar metallicity at two different orbital distances (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='015 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='03 AU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For each orbit, atmospheres of 10, 100 and 1000 times metallicity are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Some of the contribut- ing opacities are shown for their respective wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The in- set displays the corresponding temperature temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Note that metallicity here does not control the abundance of out- gassed silicates or oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Spectra are shown at a resolution of λ/∆λ = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' effect, erasing opacities of SiO or H2O in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With no SiO, Si either remains in atomic form or bonds with H to form SiH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Though, due to abundant N and high C/O ratio, H priori- tises bonding with CN to form HCN (rightmost panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This chemistry is now reflected in the thermal structure as inver- sions become significantly weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Pushing metallicity higher, further increases the C/O ratio, resulting in a mostly CO and hydrocarbon-dominated atmosphere, even at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Because the emitting photosphere resides mostly in the isother- mal region, the spectrum becomes largely featureless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With increasing orbital distance (Pink curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 5), the trends in the chemistry and spectra remain similar, but more severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Since the abundance of oxygen from outgassing is low Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 6: Abundance unweighted opacities of CO, HCN, SiH and PS, shown at a resolution of λ/∆λ = 2000 for a temperature of 3000 K and atmospheric pressure of 10−2 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Detailed descrip- tion of all opacities used in this work can be found in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' at these temperatures, the C/O ratio at x1 metallicity is already near unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Even at x10 the C/O ratio becomes much larger than unity causing efficient formation of hydrocarbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The dominant molecules become HCN, C2H2 and CO, while H2O and any po- tential SiO are erased from the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This results in opac- ity heavily weighted towards infrared wavelengths, thus a lack of deep inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Keeping the C/O ratio constant with metallicity The balance between carbon and oxygen is a major factor in de- termining atmospheric chemistry and whether SiO is allowed to thrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While in Figures 5 and 7 we allow outgassing to control abundances of oxygen and, therefore, the C/O ratio, in Figure 8 we set a constant C/O ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The value of oxygen is now scaled with metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The blue and pink curves represent models at same orbital distances as in the previous figures with cases of 10, 100 and 1000 times increased metallicity shown for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Article number, page 6 of 14 8 8 10 10 Sio TiO H20 Sio Sio H20 10 (bar) (bar) Pressure Pressure ( 4 10 10 10 iTeff 0 10 10 2750 3000 3250 3500 3750 4000 5 10 30 7 Temperature (K) Wavelength (micron)250 8 SiO SiO 10 5 200 10 udd 10 150 1200 2000 3000 3800 Temperature (K) 100 x10 H20 L x100 50 x1000 H20 Hydrocarbons 5 10 30 Wavelength (micron)8 CO 6 HCN 10 SiH 10 PS 2 10 10 2 10 10 6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1 10 100 Wavelength (micron)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' : Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 7: Figure has been updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Moved labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Volume mixing ratios of SiO, CO and HCN (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The models shown here are for the close orbit (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='015 AU) cases of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Each panel contains the original solar metallicity (Blue) as well as 10, 100 and 1000 times increased metallicity curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 1 5 10 30 Wavelength (micron) 0 50 100 150 200 250 Fplanet/Fstar ppm SiO SO2 CO2 H H2O C/O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='46 x10 x100 x1000 1200 2000 3000 3800 Temperature (K) 10 2 10 5 10 8 Pressure (bar) x10 x100 x1000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 8: Synthetic emission spectra as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 5, but with constant C/O ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The oxygen abundance here is scaled with metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Dark blue and and pink curves represent cases with solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Respective blackbody emission is represented with thin curves of the same colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Major contributing opac- ities for solar cases are indicated with shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The in- set shows corresponding temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Note that surface temperature increases with metallicity, shifting temperature- pressure profiles to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Spectra are shown at a resolution of λ/∆λ = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The abundance of oxygen at solar value is an order of mag- nitude lower than what would be outgassed in our close orbit case (Blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This directly results in decreased abundances of all oxides, inclusive of SiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Additionally, the solar C/O ratio is high enough for CO to form and become an abundant oxy- gen carrier in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Other oxides, such as CO2, H2O or SiO follow closely behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The resulting infrared opacity is dominated by the continuum of H–, with some contribution from H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Buried beneath are opacities of SiO and CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The atmo- sphere in this case becomes less opaque, allowing radiation to penetrate deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This causes inversions to extend nearly all the way to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' To conserve energy, the surface temperature is consequently decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Increasing metallicity to 100 results in oxygen abundance similar to what a melt-vapour equilibrium would produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This is reflected in the drastic change of the tem- perature profile, which now resembles cases presented in previ- ous figures (Blue curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The high surface temperature also produces more Si, which is why we see its features begin to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Increasing metallicity further has similar effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Regard- less of metallicity in these models, a solar C/O ratio makes CO a prolific species, which absorbs much of the oxygen and reduces the possibility of SiO being a dominant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With outgassing lessened at larger orbits (Pink curve), H2O and CO are the main oxygen carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In contrast to previous cases, increasing metallicity here results in large abundances of CO2, which has several important bands in the infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The ma- jor two being at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5 and 14 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While with high metallicity, SiO reaches a volume mixing ratio of nearly 10−3, the opacity of H2O completely overshadows it, making it invisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We additionally see emergence of some lesser species, such as SO2, which does have strong opacity at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In summary, our models indicate that it is mainly the C/O ra- tio that significantly affects atmospheric chemistry, including the final abundance of SiO, with metallicity of volatiles, being much less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Therefore, lava worlds enveloped in volatiles are likely to depend heavily on the balance between carbon and oxy- gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' High C/O ratios drive oxygen atoms away from SiO, poten- tially making SiH a species of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 6, SiH has a strong feature at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='4 µm and several other bands between 1- 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Nonetheless, in hydrogen-rich worlds, regardless of the C/O ratio, H2O and H– may further hinder observability of sili- cate species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Effect of increased surface pressure and internal heating Down from millibar silicate atmospheres to volatile-shrouded sub-Neptunes, the envelopes of rocky worlds may vary orders of magnitude in surface pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Larger, thermally-thick atmo- spheres, can act as insulators, allowing the molten state of a sur- face to exist indefinitely, even if the planet is weakly irradiated (Lopez & Fortney 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Kite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' One resulting con- sequence of this may be that insulation and trapped heat near the surface allow for much greater melt temperatures, making silicates more dominant over volatiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Of course, it is also pos- sible that due to an increase of volatiles, the outgassing becomes heavily suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In previous sections we have assumed the Article number, page 7 of 14 8 8 8 10 10 10 sio CO HCN 6 6 10 10 10 Pressure (bar) (bar) (bar) Pressure ( Pressure ( 4 10 4 10 10 x10 2 2 10 10 x100 x1000 19° 10 7 4 7 4 1 4 10 1 10 10 10 10 10 10 10 10 10 10 10 Volume Mixing Ratio Volume Mixing Ratio Volume Mixing RatioA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' main 1000 1500 2000 2500 3000 3500 4000 4500 5000 Temperature (K) 10 8 10 6 10 4 10 2 10 0 Pressure (bar) Tint = 300K 3 5 10 20 4 6 7 8 9 Wavelength (micron) 130 160 190 220 250 Fplanet/Fstar ppm SiO 1 bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='Tint = 0K 10 bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='Tint = 0K 10 bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='Tint = 300K Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 9: Temperature-pressure profiles and emission spectra of varied atmospheric pressure and internal heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The top panel contains profiles of atmospheres with 1 and 10 bar at two dif- ferent orbital distances (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For the close or- bit, an additional case with Tint=300 K is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The lower panel shows the resulting emission spectrum of the close or- bit cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The chosen wavelength region is where SiO features are expected to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Spectra are shown at a resolution of λ/∆λ = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' volatiles to be capped at 1 bar, while also keeping the internal temperature of the planet at 0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While the exact dynamics of this are out of scope for this paper, in Figure 9 we show how an increase in pressure and internal heating might affect the observ- ability of silicon-bearing species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The bright cyan curves in the top panel of the figure represent atmospheres with a surface pressure increased to 10 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With- out internal temperature enabled, there is no additional heat to transport, making convection unnecessary, resulting in a simple extension of the isothermal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' However, if the atmosphere is supplied with energy from bellow, the optically and thermally thick portion becomes unstable to convection, dramatically in- creasing the temperature of the melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Because of the exponen- tial scaling of outgassing, this can often lead to an immense rise of silicate abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' If such atmospheres are well mixed, one should expect to see substantial silicate features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The bottom panel of Figure 9 shows the spectra for the close orbit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' A 10 bar atmosphere with no internal heating re- sults in a greater dominance of the volatile component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The H– continuum becomes more effective, concealing the presence of SiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With Tint=300 K, even at 10 bar of volatile content, SiO becomes the most abundant species in the atmosphere, caus- ing its features to dominate the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Note that our arbi- trary use of unusually high Tint is purely for demonstrative pur- poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Close-in rocky planets are susceptible to various heating mechanisms outside stellar irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With an insulating, opti- cally thick atmosphere, it is possible that trapping of heat allows global magma oceans to be sustained at much hotter tempera- tures than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Observations of silicates can therefore pro- vide important constraints on the properties of the melt and the interior (Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Depletion of hydrogen Atmospheres of increasingly shorter orbital period are expected to be affected by stellar wind erosion (Owen & Wu 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Lopez & Fortney 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Lopez 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For less mas- sive worlds, such as ultra-short period super-Earths, this likely results in extreme loss of volatile species and even extensive tails of escaping particles (including silicates) (Mura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Cas- tan & Menou 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Léger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' If the internal reservoir is unable to counteract depletion, the atmosphere will increase in its mean-molecular weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With depletion of hydrogen, CO, CO2, N2 or SO2 atmospheres are not unusual outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In Fig- ure 10, we investigate models that are severely depleted of hy- drogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' As before, planets at two different orbital distances are shown with the oxygen abundance being determined via out- gassing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In addition to hydrogen depletion, by varying the carbon mixing ratio, we explore the added impact of the C/O parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For the close orbit cases the major difference caused by the depletion of hydrogen is the lack of H2O in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For a C/O ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='2 (Faint TP profile and spectrum), the thermal structure is similar to the cases discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' How- ever, with no H2O the excess oxygen makes SiO the dominant constituent, followed by CO and O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Depletion of hydrogen is yet another pathway in which SiO-dominant atmospheres are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Due to the lack of H, the H– continuum is exchanged for the opacities of CO and CO2, with a major band of CO2 ap- pearing at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5 µm (faint upper spectrum in the bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' SiO still remains the only absorber at 9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' If the C/O ratio is increased to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 (Blue curves), the formation of CO becomes even more efficient, leaving SiO nearly two orders of magnitude behind in volume mixing ratio (highlighted abundance curves in the top right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' From a theoretical perspective, for C/O ra- tios close to unity, CO atmospheres are easy to attain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Due to this, observability of the 5 µm SiO feature may not be feasible in such atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For comparison we show a pure, outgassing- produced, silicate atmosphere (Cyan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While between volatile and silicate cases, the 9 µm SiO feature remains, the presence of it at 5 µm can be lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Detecting a combination of carbon oxides and SiO could indicate that the atmosphere is of a high C/O ratio with a significant outgassing component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In the less irradiated cases, with C/O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='2, the tempera- ture profile is only inverted in very the upper region, similar to what was found in the original models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' However, the chemistry here is vastly different (see Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The lack of outgassed oxygen allows for N2 to take over as the dominant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Its abundance is also closely followed by sulphur, no- tably SO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While N2 is a weak absorber, the opacity of SO2 is significant, peaking at 4 and 7-9 µm (Faded lower spectrum of 10, features are not marked).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Sulphur, hot Venus-like atmo- spheres may be possible on irradiated rocky worlds and should be taken into consideration for observations with MIRI LRS (Schaefer & Fegley 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zolotov 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Lig- gins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Other, slightly diminished, yet still visible spec- tral features include that of CO and CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' If the C/O ratio is increased to 1 (Pink curves), N2 still remains as the dominant Article number, page 8 of 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' : Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes 1000 1500 2000 2500 3000 3500 4000 Temperature (K) 10 8 10 6 10 4 10 2 10 0 Pressure (bar) Depleted H C/O = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 C/O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='2 Silicate C/O = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 C/O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='2 Silicate 10 8 10 6 10 4 10 2 10 0 Volume Mixing Ratio 10 8 10 6 10 4 10 2 10 0 Pressure (bar) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01 AU SiO CO SiO CO 1 5 10 30 Wavelength (micron) 0 50 100 150 200 250 Fplanet/Fstar ppm SiO CO CO Silicates PS PS CO VO Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 10: Figure has been updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Moved labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Atmospheric models with severely depleted hydrogen at orbital distances of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In the top panel we show TP profiles for C/O ratios of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 (Pink and Blue) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='2 (Faded) as well as an additional model of a pure silicate atmosphere (Cyan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' All, cases assume dayside heat redistribution (f=2/3) with volatile-containing atmospheres having 1 bar surface pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The total pressure in the pure silicate case is computed using the outgassing code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The top right panel contains abundances of SiO and CO for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='01 AU case (Blue TP profile), with the faint curves representing a pure silicate atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The bottom panel displays emission spectra colour-coded for their respective TP profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Marked regions denote major opacity contributions for the cases with a C/O ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Spectral features for other cases are explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Spectra are shown at a resolution of λ/∆λ = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' component, however, previously seen oxygen-rich species, such as SO2 are no longer abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Instead, PS rises as the second most abundant molecule, causing increase in shortwave opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The photosphere becomes severely inverted, resulting in large emission features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Aside from its shortwave component, PS has potentially observable IR bands at 7 and 15 µm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 6 for its full opacity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Because of the strong VO at 10 µm, this species may be interesting for observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Observability of currently known targets Despite numerous observations with low and high resolution in- struments, no definite detections of molecules have been made on any lava world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' JWST with its broad spectral coverage and high sensitivity, is expected to shed new light on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' So far, the performance of the telescope has surpassed all ex- pectations (Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The JWST Transiting Exoplanet Community Early Release Science Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' If short period rocky planets do possess atmospheres, it is likely that JWST will be able to pick these up with greater confidence than anything before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Targets such as 55 Cnc e, or K2-141 b are ex- cellent labs to test silicate outgassing, and have been chosen to be observed during JWST’s first cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' If the observations are successful, many more targets are likely to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In Figure 12, we showcase currently confirmed planets, with a radius of 1-4 R⊕, as a function of stellar magnitude and emis- sion flux at 9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Note that here the flux ratio represents the baseline emission, not affected by possible absorption of present species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The simulated error bars are for 20 hours of observing time with MIRI LRS binned to a resolution of R = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Observ- ability of a target will depend drastically not only on the con- trast between the star and the planet but also on the brightness of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' As the surface temperature defines outgassed sili- cate abundances, the equilibrium temperature of the planet is an additional factor that needs to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For less tenuous atmospheres, this condition may be somewhat relaxed, as insu- lation of heat can allow for global magma oceans to be sustained at greater temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Despite the ample number of discov- Article number, page 9 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 11: Figure has been updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Moved labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Five most abun- dant molecules in an atmosphere corresponding to a hydrogen- depleted case with C/O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='2 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' These are, in decreas- ing abundance, N2, SO2, CO, CO2 and SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The relevant spec- trum for this case is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 10 (Pink curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 12: The figure has been updated with new noise estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Currently confirmed super-Earths and sub-Neptunes plotted as a function of stellar magnitude (K) and emission flux of the planet at 9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The flux ratio here represents baseline emission modelled with dayside redistribution (f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='667), which is not affected by present absorbing species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 9 µm is the wavelength where the largest SiO feature is expected to manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' A selec- tion of favourable targets show PANDEXO simulated noise for the MIRI LRS instrument with 20 hours observation time, binned to R=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' ered worlds, most orbit stars that are too dim to be good targets for atmospheric characterisation with JWST’s MIRI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While the brightest systems, like 55 Cnc, have simulated noise of just a few ppm, at a stellar magnitude (K) of 9, it increases close to 50 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Considering such atmospheres only reach a few hundred ppm at the 9 µm range, characterisation of any present features may be extremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' That said, there are a num- ber of planets that are potentially good follow up candidates for short duration programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' One of the brightest and most studied super-Earths is 55 Cnc e, which will be observed with NIRCam and MIRI LRS instru- ments during JWST’s first cycle (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Brandeker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Whether this planet possess an atmosphere is cur- rently debated, with a general consensus leading to an existing high-mean-molecular-weight atmosphere, possibly with an out- gassed silicate component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Compositions dominated by CO, N2 or O2 are all possible, with low metallicity, H2-rich atmospheres being less probable (Demory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Angelo & Hu 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Though there have been claims of detected HCN, which would indicate abundant H2 and a high C/O ratio (Tsiaras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Recent reanalysis of Spitzer phase curve data of 55 Cnc e also suggest a high average dayside temperature of T⋆= 3771 K, which may be caused by the pres- ence of SiO (Mercier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Still, without observations of broad spectral coverage, conclusions for this planet cannot yet be drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For high equilibrium temperature, several other targets are of various radii are of notable interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Smaller rocky worlds, such as K2-141 b, TOI-1807 b, TOI-561 b are ideal for ob- serving silicates (Hedges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Dang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zieba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' TOI-561 b is reported to have unusually low den- sity, which could imply a large volatile component (Lacedelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Larger worlds include K2-100 b, TOI-849 b or TOI- 2260 b, all of which may also host volatile, H2-rich atmospheres with underlying magma oceans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Indicated via arrows are some of the highest contrast planets, including GJ-1214 b, GJ 436 b, LHS-3844 b and several others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While the equilibrium tempera- ture of these planets is generally too low to host magma oceans, an insulating atmosphere could force larger surface temperatures and thus visible contamination of silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The search for outgassed silicates is certainly not hindered by the lack of suitable targets, but rather by how small spectral features are expected to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While we do not model full spectra for any of the suggested targets, the emission features for most of these should be expected to closely mimic our presented 2 R⊕ test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Generally, planets will not be observed for more than a few orbits, making spectral noise a considerable issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With JWST’s MIRI LRS it should be possible to characterise the presence of SiO on short-period rocky worlds and sub-Neptunes, but to do so will certainly be a grand challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Importance of heat redistribution and stellar type Characterisation of lava worlds depends as greatly on the struc- ture of the atmosphere as it does on the chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While the spectral energy distribution of the host star largely determines the features of the temperature profile and the emission spectrum of the planet, the atmosphere itself is prone to physical processes that impact observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Namely, for 1-D models, the efficiency of heat redistribution determines the total given energy budget, which subsequently decides much of the occurring chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Approximations of this effect are made through a single factor f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Tenuous, silicate atmospheres are expected to be inefficient in transporting heat, confining it to the dayside of the planet (f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='667-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' More volatile, optically thick atmospheres are more efficient, to point where a significant fraction of the incoming stellar radiation can be deposited on the night side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' A maximum, full redistribution results in f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Using analytical heat re- distribution scaling theory from Koll (2022), in Figure 13 we demonstrate the possible effect that this may have on observabil- ity of silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In general, we assume that our modelled planets do not re- distribute heat efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Most of the models are set to use f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='667, however using a scaling theory derived for tidally locked worlds, we find that strongly irradiated atmospheres with volatiles can become very efficient at transporting heat, in cases Article number, page 10 of 14 10 C/O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU 6 10 Pressure (bar) N2 10 SO2 CO CO2 10 SO 3 10 10 10 10 Volume Mixing Ratio700 3000 GJ-1214 b (1335 ppm) K2-320 b (920 ppm K2-266 b (767 ppm) TOl-2406 b (840 ppm) LHS-3844 b (812 ppm) TOl-1696 b (781 ppm)l 600 GJ 436 b (710 ppm) TO1-849 b 2500 500 2000 Dayside 400 K2-100 b 300 K2-141 b 1500 TOl-2260 b eq TOl-1807 b 200 55 Cnc e TOI-431 b 1000 HD 213885 b 100 0 500 14 12 10 6 3 8 4 Magnitude of Stellar Target (K)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Zilinskas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' : Observability of silicates in volatile atmospheres of super-Earths and sub-Neptunes 1 5 10 30 Wavelength (micron) 0 50 100 150 200 250 Fplanet/Fstar ppm SiO SiO F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='363 F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='581 F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='667 1200 2000 3000 3800 Temperature (K) 10 2 10 5 10 8 Pressure (bar) F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='363 F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='581 F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='667 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 13: Temperature profiles and emission spectra of models with computed heat redistribution factor f at orbital distances of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='015 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='04 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The atmospheres with f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='667 (Both cyan curves) are arbitrarily set to represent dayside-confined re- distribution of irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Factors of f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='363 and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='581 are calculated using the formulation of heat redistribution for rocky planets in Koll (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The parameters determining heat transport efficiency are planetary equilibrium temperature, atmo- spheric surface pressure and longwave optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Spectra are shown at a resolution of λ/∆λ = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' reaching as high as f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' While not attaining global redistri- bution (f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='25), this severely impacts the total energy budget, reducing the surface temperature and thus silicate observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' It is no surprise that volatile atmospheres increase the efficiency, but the large magnitude of it for just 1 bar of an atmosphere does indicate that even a small amount of volatiles can have a severe impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' On the contrary, planets at larger orbital distances show no large increase in redistribution efficiency, keeping most of the energy confined to the dayside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Since the primary mech- anism of heat transport is the atmosphere, phase curve measure- ments can indicate its significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Detecting temperature offsets or high nightside flux could indicate that the planet has retained a substantial, volatile-rich atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' An unusual super-Earth 55 Cnc e has been found to show signs of this (Demory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Since emission of the planet probes its thermal profile, ob- servability is also greatly impacted by the spectrum of the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In Figure 14, we take one of our solar cases and compare it to models of planets placed around stars of different type, but at an equivalent irradiation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' With increasing stellar tem- perature, its spectrum is shifted towards shorter wavelengths, causing greater incident UV flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Going from a G to a typical K type star (T⋆= 4305 K) inversions weaken drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Atmo- spheres around cool M-dwarfs are likely to contain no deep in- version that strongly affect the emitting photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For charac- terisation of lava worlds through emission spectroscopy this may present a slight issue, since strong inversions are one of the key characteristics of silicate-rich atmospheres that may help iden- tify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Conclusion In preparation for JWST observations, we have used consistent outgassing equilibrium chemistry and radiative-transfer models to assess the possibility of detecting silicates in volatile atmo- spheres of super-Earths and smaller sub-Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Placing a hy- 2500 3000 3500 4000 4500 5000 Temperature (K) 10 8 10 6 10 4 10 2 10 0 Pressure (bar) Tstar = 5750 K Tstar = 3026 K Tstar = 4305 K Tstar = 6407 K Tstar = 5750 K Tstar = 3026 K Tstar = 4305 K Tstar = 6407 K Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 14: Temperature profiles computed consistently with atmo- spheric chemistry for different stellar type hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In each case the planet is placed to an equivalent irradiation distance from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' T⋆= 3026 K spectrum represents GJ 1214 and is taken from the MUSCLES database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The other three are modelled us- ing PHOENIX spectra of the denoted temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' pothetical 2 R⊕ planet at varying orbital distances around a Sun- like star, we have explored a wide variety of viable atmospheric compositions, rich in H, C and N, that also include an outgassed silicate component, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=', Si, O, Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We modelled atmospheres of up to 10 bar surface pressure, varied in metallicity and C/O ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' A major assumption made in this work is that the atmosphere is in complete equilibrium with the underlying melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Our results are intended to guide observers towards potentially detectable species that would help characterise worlds with exposed or con- cealed lava oceans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Below are our key findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' For emission spectroscopy with JWST, SiO is likely to remain the only characterisable species that could indicate strong outgassing from an underlying magma ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' How- ever, on atmospheres containing volatiles, the main opacity bands of SiO at 5 and 9 µm can be heavily suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Only the most irradiated worlds, with melt temperatures > 2500-2800 K are expected to show super-solar abundances of silicates (Assuming 1 bar of volatiles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Unlike for pure lava worlds, we do not find features of SiO2 to be of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Very high temperature melts may also result in broad shortwave features, arising mainly from TiO and several other outgassed species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Ultimately, the visibility of silicates in volatile atmospheres will largely depend on the atmospheric properties and the efficiency of its interaction with the melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Thermal inversions are prominent in atmospheres con- taminated with silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We find that numerous outgassed silicates cause deep inversions that affect the photosphere, even if the atmosphere has strong infrared absorption origi- nating from volatiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The largest contributors to shortwave opacity are: TiO, SiO, MgO, Fe and Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We also find that alkali metals, metal hydrates and, in certain cases, PS or VO can become a source of inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Because outgassing scales exponentially with the temperature of the melt, planets with no insulating atmospheres and larger orbital distances are not likely to show strong inversions originating due to silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Article number, page 11 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' main 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The presence of hydrogen can affect observability of silicates, including of SiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Our models show that added hydrogen results in formation of ample hydrocarbons and H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Chemically, H2O and SiO are direct competitors for the outgassed oxygen, however, SiO is much less prone to thermal dissociation, making it more prominent in the lower-pressure, inversion-affected regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' On strongly irradiated worlds, the continuum of H– can also heavily obscure the 5 and 9 µm SiO features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' The C/O ratio has a large effect on SiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Even in H2-rich atmospheres, formation of CO due to a C/O ratio ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='0 can result in a drastic reduction of silicate oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Chemically, CO takes priority for oxygen, affecting all other oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' This can consequently result in Si bonding with H instead of O, forming SiH, potentially making it a species of interest for observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Detecting CO at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='5 µm and SiO at 9 µm could indicate an atmosphere with no hydrogen and a high C/O ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Atmospheric pressure, insulation and redistribution of heat are major factors in deciding whether volatile atmospheres are contaminated with silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Our models show that atmo- spheres of 10 bar with internal temperature enabled become convective, resulting in a large increase of surface tempera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Exponential scaling of outgassing can lead to SiO be- coming a dominant species, even in substantial volatile at- mospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In contrary, effective heat redistribution can re- duce surface temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Using a scaling theory for tidally locked planets, we find that even small volatile atmospheres are efficient at transporting heat, lowering surface tempera- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Observations of silicates can, therefore, provide im- portant constraints on the underlying melt properties and the balance between insulation and redistribution of heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We acknowledge funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program under grant agreement No 694513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We thank Matej Malik for the insightful discussion on HELIOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' We are also grateful for the comments of the editor and the anonymous referee, which helped improve the quality of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Software used in this work: HELIOS-K (Grimm & Heng 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' HELIOS(Malik et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=', Miguel, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2022, A&A, 661, A126 Zolotov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2018, in Oxford Research Encyclopedia of Planetary Science, 146 Article number, page 13 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' main Appendix A: Opacity Data Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='1: Description of the opacities used to calculate temperature profiles and emission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Species Source Line list Line List Reference Al DACEa VALD Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) AlH HELIOS-K AlHambra Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2018c) AlO HELIOS-Kb ATP Patrascu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) Ca DACE VALD Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) CaH HELIOS-K MoLLIST Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Bernath (2020) CaO HELIOS-K VBATHY Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2016) Fe DACE VALD Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) K DACE VALD Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) Mg DACE Kurucz Kurucz (1992) MgH HELIOS-K MoLLIST GharibNezhad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Bernath (2020) MgO HELIOS-K LiTY Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2019) Na DACE VALD Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) NaH HELIOS-K Rivlin Rivlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) Si DACE VALD Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) SiH HELIOS-K Sightly Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2018b) SiO HELIOS-K SiOUVenIR Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2022) SiO2 DACE OYT3 Owens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2020) Ti DACE VALD Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) TiH HELIOS-K MoLLIST Burrows et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2020) CH3 DACE AYYJ Adam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2019) CH4 DACE YT34to10 Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2017) C2H2 DACE aCeTY Chubb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2020) C2H4 DACE MaYTY Mant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2018) CN HELIOS-K Trihybrid Syme & McKemmish (2021) H2O DACE POKAZATEL Polyansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2018) HCN HELIOS-K Harris Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2014) HS HELIOS-K GYT Gorman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2019) H2S DACE AYT2 Azzam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2016) NH3 DACE CoYuTe Coles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2019) OH DACE HITEMP Rothman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2010) S DACE VALD Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) CS ExoMold JnK Paulose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) NS ExoMol SNaSH Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2018a) SO2 ExoMol ExoAmes Underwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2016a) SO3 ExoMol UYT2 Underwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2016b) PH3 DACE SAlTY Sousa-Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2015) PS HELIOS-K POPS Prajapat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2017) VO HELIOS-K VOMYT McKemmish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2016) Scattering and Continuum H, H2, H2O, He, O2 Scattering H– Continuum (bf & ff) John (1988);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Gray (2008) H2 –H2 HELIOS-K HITRAN Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2017) H2 –He, O2 –O2 petitRADTRANS Mollière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2019, 2020) a DACE database https://dace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='ch/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' b Opacities are computed with HELIOS-K https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content='com/exoclime/HELIOS-K (Grimm & Heng 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' c We use HITEMP2010 for temperature profiles and UCL-4000 for emission spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' d For ExoMol opacities we make use of the conversion work done by Chubb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' (2021), which are only used for generating emission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' In general, we make heavy use of the DACE (Grimm & Heng 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021), ExoMol (Chubb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2021), Kurucz (Kurucz 1992), VALD3 (Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2015) and HITRAN (Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' 2017) databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} +page_content=' Article number, page 14 of 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfpQ0e/content/2301.05190v1.pdf'} diff --git a/cNFRT4oBgHgl3EQfSjci/vector_store/index.faiss 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0000000000000000000000000000000000000000..58201d8c08ae11110dcf520cda26af895e1fda38 --- /dev/null +++ b/dNAzT4oBgHgl3EQf3P7o/content/tmp_files/2301.01828v1.pdf.txt @@ -0,0 +1,1761 @@ +On Sequential Bayesian Inference for Continual Learning +Samuel Kessler +skessler@robots.ox.ac.uk +University of Oxford +Adam Cobb +adam.cobb@sri.com +SRI International +Tim G. J. Rudner +tim.rudner@cs.ox.ac.uk +University of Oxford +Stefan Zohren +zohren@robots.ox.ac.uk +University of Oxford +Stephen J. Roberts +sjrob@robots.ox.ac.uk +University of Oxford +Abstract +Sequential Bayesian inference can be used for continual learning to prevent catastrophic +forgetting of past tasks and provide an informative prior when learning new tasks. We +revisit sequential Bayesian inference and test whether having access to the true posterior is +guaranteed to prevent catastrophic forgetting in Bayesian neural networks. To do this we +perform sequential Bayesian inference using Hamiltonian Monte Carlo. We propagate the +posterior as a prior for new tasks by fitting a density estimator on Hamiltonian Monte Carlo +samples. We find that this approach fails to prevent catastrophic forgetting demonstrating the +difficulty in performing sequential Bayesian inference in neural networks. From there we study +simple analytical examples of sequential Bayesian inference and CL and highlight the issue of +model misspecification which can lead to sub-optimal continual learning performance despite +exact inference. Furthermore, we discuss how task data imbalances can cause forgetting. +From these limitations, we argue that we need probabilistic models of the continual learning +generative process rather than relying on sequential Bayesian inference over Bayesian neural +network weights. In this vein, we also propose a simple baseline called Prototypical Bayesian +Continual Learning, which is competitive with state-of-the-art Bayesian continual learning +methods on class incremental continual learning vision benchmarks. +1 +Introduction +The goal of continual learning (CL) is to find a predictor that learns to solve a sequence of new tasks without +losing the ability to solve previously learned tasks. One key challenge of CL with neural networks (NNs) is +that model parameters from previously learned tasks are “overwritten” during gradient-based learning of new +tasks, which leads to catastrophic forgetting of previously learned abilities (French, 1999). One approach to +CL hinges on using recursive applications of Bayes’ Theorem; using the weight posterior in a Bayesian neural +network (BNN) as the prior for a new task (Kirkpatrick et al., 2017). However, obtaining a full posterior +over NN weights is computationally demanding and we often need to resort to approximations, such as the +Laplace method (MacKay, 1992) or variational inference (Graves, 2011; Blundell et al., 2015) to obtain a +neural network weight posterior. +When performing Bayesian CL, sequential Bayesian inference is performed with an approximate BNN +posterior, not the true posterior (Schwarz et al., 2018; Ritter et al., 2018; Nguyen et al., 2017; Ebrahimi et al., +2019; Kessler et al., 2019; Loo et al., 2020). If we consider the performance of sequential Bayesian inference +1 + +with a variational approximation over a BNN weight posterior then we barely observe an improvement over +simply learning new tasks with stochastic gradient descent (SGD). We will develop this statement further +in Section 2.2. So if we had access to the true BNN weight posterior, would this be enough to prevent +forgetting by sequential Bayesian inference? +Our contributions in this paper are to revisit Bayesian CL. 1) Experimentally, we perform sequential Bayesian +inference using the true Bayesian NN weight posterior. We do this by using the gold standard of Bayesian +inference methods, Hamiltonian Monte Carlo (HMC) (Neal et al., 2011). We use density estimation over +HMC samples and use this approximate posterior density as a prior for the next task within the HMC +sampling process. Surprisingly our HMC method for CL yields no noticeable benefits over an approximate +inference method (VCL Nguyen et al. (2017)) despite using samples from the true posterior. 2) As a result +we consider a simple analytical example and highlight that exact inference with a misspecified model can still +cause forgetting. 3) We show mathematically that under certain assumptions task data imbalances will cause +forgetting in Bayesian NNs. 4) We propose a new probabilistic model for CL and show that by explicitly +modeling the generative process of the data, we can achieve good performance, avoiding the need to rely +on recursive Bayesian inference over NN weights to prevent forgetting. Our proposed model, Prototypical +Bayesian Continual Learning (ProtoCL), is conceptually simple, scalable, and competitive with state of the +art Bayesian CL methods in the class-incremental learning setting. +2 +Background +2.1 +The Continual Learning Problem +Continual learning (CL) is a learning setting whereby a model must learn to make predictions over a +set of tasks sequentially while maintaining performance across all previously learned tasks. In CL, the +model is sequentially shown T tasks, denoted Tt for t = 1, . . . , T. Each task, Tt, is comprised of a dataset +Dt = {(xi, yi)}Nt +i=1 which a model needs to learn to make predictions with. More generally, tasks are denoted +by distinct tuples comprised of the conditional and marginal data distributions, {pt(y|x), pt(x)}. After task +Tt the model will lose access to the training dataset but its performance will be continually evaluated on all +tasks Ti for i ≤ t. For a comprehensive review of CL scenarios see (Hsu et al., 2018; Van de Ven & Tolias, +2019). We decompose predictors as g = h ◦ f such that ˆy = g(x) we define f as an embedding function +mapping f : X → Z and h as a head mapping to outputs h : Z → Y. Some continual learning methods use a +separate head per task {hi}T +i=1, these methods are called multi-headed while those that use one head are +called single-headed. +2.2 +Bayesian Continual Learning +We consider a setting in which task data arrives sequentially at time steps, t = 1, 2, . . . , T. At the first time +step, t = 1, the model parameterized by θ receives the dataset D1 and learns the conditional distribution +p(yi|xi, θ) for all (xi, yi) ∈ D1 (i indexes a datapoint), the parameters θ have a prior distribution p(θ). The +posterior predictive distribution for a test point, x∗ +1 is: +p(y∗ +1|x∗ +1, D1) = +� +p(y∗ +1|x∗ +1, θ)p(θ|D1)dθ. +(1) +Computing this posterior predictive distribution above requires p(θ|D1). For t = 2, a CL model is required +to fit p(yi|xi, θ) for D1 ∪ D2. The posterior predictive distribution for a new test point x∗ +2 point is: +p(y∗ +2|x∗ +2, D1, D2) = +� +p(y∗ +2|x∗ +2, θ)p(θ|D1, D2)dθ. +(2) +The posterior must thus be updated to reflect this new conditional distribution. We can use repeated +application of Bayes’ rule to calculate the posterior distributions p(θ|D1, . . . , DT ) as: +p(θ|D1, . . . , DT −1, DT ) = p(DT |θ)p(θ|D1, . . . , DT −1) +p(DT |D1, . . . , DT −1) +. +(3) +2 + +1 +2 +3 +4 +5 +Tasks +0.20 +0.40 +0.60 +0.80 +1.00 +Accuracy +Task 1 (0 vs. 1) +1 +2 +3 +4 +5 +Tasks +Task 2 (2 vs. 3) +1 +2 +3 +4 +5 +Tasks +Task 3 (4 vs. 5) +1 +2 +3 +4 +5 +Tasks +Task 4 (6 vs. 7) +1 +2 +3 +4 +5 +Tasks +Task 5 (8 vs. 9) +SGD +VCL SH +VCL MH +Figure 1: +Accuracy on Split-MNIST for various CL methods with a two-layer BNN. We compare a NN +trained with SGD (single-headed) with VCL. We consider single-headed (SH) and multi-head (MH) VCL +variants. +In the CL setting we lose access to previous training datasets: however, using repeated applications of Bayes’ +rule Eq. (3), allows us to sequentially incorporate information from past tasks in the parameters θ. At t = 1, +we have access to D1 and the posterior over weights is: +log p(θ|D1) = log p(D1|θ) + log p(θ) − log p(D1). +(4) +At t = 2, we require a posterior p(θ|D1, D2) to calculate the posterior predictive distribution in Eq. (2). +However, we have lost access to D1. According to Bayes’ rule, the posterior may be written as: +log p(θ|D1, D2) = log p(D2|θ) + log p(θ|D1) − log p(D2|D1), +(5) +where we used the conditional independence of D2 and D1 given θ. We note that the likelihood is only +dependent upon the current task dataset, D2, and that the prior encodes parameter knowledge from the +previous task. Hence, we can use the posterior at t as a prior for learning a new task at t + 1. For the MNIST +dataset (LeCun et al., 1998) we know that if we were to train a BNN we would achieve a good performance by +inferring p(θ|D). Hence if we were to Split-MNIST into 5 two-way classification tasks then we should be able +to recursively recover the multi-task posterior p(θ|D) = p(θ|D1 . . . , D5). This problem is called Split-MNIST. +From Eq. (5) we require that our model with parameters θ is a sufficient statistic of D1, making the likelihood +conditionally independent of D1 given θ. This observation motivates the use of high-capacity predictors, such +as Bayesian neural networks, that are flexible enough to learn D1. +2.3 +Variational Continual Learning +Variational CL (VCL; Nguyen et al. (2017)) simplifies the Bayesian inference problem in Eq. (5) into a +sequence of approximate Bayesian updates on the distribution over random neural network weights θ. To do +so, VCL uses the variational posterior from previous tasks as a prior for new tasks. In this way, learning +to solve the first task entails finding a variational distribution q1(θ|D1) that maximizes a corresponding +variational objective. For the subsequent task, the prior is chosen to be q1(θ|D1), and the goal becomes to +learn a variational distribution q2(θ|D2) that maximizes a corresponding variational objective under this +prior. Denoting the recursive posterior inferred from multiple datasets by qt(θ|D1:t), we can express the +variational CL objective for the t-th task as: +L(θ, Dt) = DKL [qt(θ)||qt−1(θ|D1:t−1)] − Eqt[log p(Dt|θ)]. +(6) +When applying VCL to the problem of Split-MNIST Figure 1, we can see that single-headed VCL barely +performs better than SGD when remembering past tasks. Multi-headed VCL performs better, despite not +being a requirement from sequential Bayesian inference Eq. (5). So why does single-head VCL not improve +over SGD if we can recursively build up an approximate posterior using Eq. (5)? We hypothesize that it could +be due to using a variational approximation of the posterior and so we are not actually strictly performing +the Bayesian CL process described in Section 2.2. We test this hypothesis in the next section by propagating +the true BNN posterior to verify whether we can recursively obtain the true multi-task posterior and so +improve on single-head VCL and prevent catastrophic forgetting. +3 + +3 +Bayesian Continual Learning with Hamiltonian Monte Carlo +To perform inference over BNN weights we use the HMC algorithm (Neal et al., 2011). We then use these +samples and learn a density estimator that can be used as a prior for a new task1. HMC is considered the +gold standard in approximate inference and is guaranteed to asymptotically produce samples from the true +posterior2. we use posterior samples of θ from HMC and then fit a density estimator over these samples, to +use as a prior for a new task. This allows us to use a multi-modal posterior distribution over θ. In contrast, +to a diagonal Gaussian variational posterior like in VCL. More concretely, to propagate the posterior p(θ|D1) +we use a density estimator, defined ˆp(θ|D1), to fit a probability density on HMC samples as a posterior. For +the next task T2 we can use ˆp(θ|D1) as a prior for a new HMC sampling chain and so on (see Fig. 2). The +density estimator priors need to satisfy two key conditions for use within HMC sampling. Firstly, that they +are a probability density function. Secondly, that they are differentiable with respect to the input samples. +... +Figure 2: Illustration of the posterior propagation pro- +cess; priors in blue are in the top row and posterior +samples on the bottom row. This is a two step process +where we first perform HMC with an isotropic Gaus- +sian prior for T1 then perform density estimation on +the HMC samples from the posterior to obtain ˆp1(θ|D1). +This posterior can then be used as a prior for the new +task T2 and so on. +We use a toy dataset (Fig. 3) with two classes and +inputs x ∈ R2 (Pan et al., 2020). Each task is +a binary classification problem where the decision +boundary extends from left to right for each new +task. We train a two layer BNN, with hidden state +size of 10. +We use a Gaussian Mixture Models +(GMM) as a density estimator for approximating +the posterior with HMC samples. We also tried Nor- +malizing Flows which should be more flexible (Dinh +et al., 2016) however these did not work robustly +for HMC sampling3. To the best of our knowledge +we are the first to incorporate flexible priors into +the sampling methods like HMC. +Training a BNN with HMC on the same multi- +task dataset gets a test accuracy of 1.0. Thus, the +final posterior is suitable for continual learning un- +der Eq. (3) we should be able to recursively arrive +at the multi-task posterior with our recursive infer- +ence method with HMC. The results from Fig. 3 +demonstrate that using HMC with an approximate multi-modal posterior fails to prevent forgetting and is +less effective than using multi-head VCL. In fact, multi-head VCL clearly outperforms HMC indicating that +the source of the knowledge retention is not through the propagation of the posterior but through the task +specific heads. We also repeated these experiments with another toy dataset of five binary classification tasks +where we observe similar results (Fig. 7). +For HMC we ensure that we are sampling from the posterior by assessing chain convergence and effective +sample sizes (Fig. 11). The effective sample size measures the autocorrelation in the chain. The effective +sample sizes for the HMC chains for our BNNs are similar to the literature (Cobb & Jalaian, 2021). Also, +we ensure that our GMM approximate posteriors are multi-modal and so has a more complex posterior in +comparison to VCL, and that the GMM samples produce equivalent results to HMC samples for the current +task (Fig. 10). See Appendix B for details. +We are not able to perform sequential Bayesian inference in BNNs despite using HMC which is considered +the gold standard of Bayesian deep learning. HMC and density estimation with a GMM produces richer, +accurate and multi-modal posteriors. Despite this we are still not able to sequentially build up the multi-task +posterior or get much better results than an isotropic Gaussian posterior like single-head VCL. The weak +1We considered Sequential Monte Carlo, but it is unable to scale to the dimensions required for the NNs we consider (Chopin +et al., 2020). HMC on the other hand has recently been successfully scaled to BNNs (Cobb & Jalaian, 2021; Izmailov et al., +2021). +2In the NeurIPS 2021 Bayesian Deep Learning Competition, the goal was to find an approximate inference method that is as +“close” as possible to the posterior samples from HMC. +3RealNVP was very sensitive to the choice of random seed, the samples from the learned distribution did not give accurate +predictions for the current task and led to numerical instabilities when used as a prior within HMC sampling. +4 + +0 +1 +2 +x1 +0.5 +0.0 +0.5 +1.0 +x2 +Task 1 +Task 2 +Task 3 +Task 4 +Task 5 +1 +2 +3 +4 +5 +Tasks +0.00 +0.50 +1.00 +Accuracy +Task 1 +1 +2 +3 +4 +5 +Tasks +0.60 +0.80 +1.00 +Task 2 +1 +2 +3 +4 +5 +Tasks +0.60 +0.80 +1.00 +Task 3 +1 +2 +3 +4 +5 +Tasks +0.60 +0.80 +1.00 +Task 4 +1 +2 +3 +4 +5 +Tasks +0.80 +1.00 +Task 5 +HMC +MH VCL +SH VCL +SGD +MT SGD/HMC +Figure 3: On the left is the toy dataset of 5 distinct 2-way classification tasks which involve classifying circles +and squares (Pan et al., 2020). Also, continual learning binary classification test accuracies over 10 seeds. +The pink solid line is a multi-task (MT) baseline accuracy using SGD/HMC with the same model as for the +CL experiments. +point of this method is the density estimation, the GMM removes probability mass over areas of the BNN +weight space posterior which is important for the new task. This demonstrates just how difficult a task it is +to model BNN weight posteriors. In the next section, we study a different analytical example of sequential +Bayesian inference and look at how model misspecification and task data imbalances can cause forgetting in +Bayesian CL. +4 +Bayesian Continual Learning and Model Misspecification +0 +100 +200 +t +1 +0 +1 +A +0 +100 +200 +t +B +t +task 1 data +task 2 data +Figure 4: Posterior estimate of the filtering dis- +tribution Eq. (7) for two different scenarios with +two tasks or changepoint. +We now consider a simple analytical example where we +can perform the sequential Bayesian inference Eq. (3) in +closed form using conjugacy. We consider a simple setting +where data points arrive online, one after another. +Observations y1, y2, . . . , yt arrive online, each observation +is generated by a hidden variable θ1, θ2, . . . , θt ∼ p where +p is a probability density function. At time t we wish to +infer the filtering distribution p(θt|y1, y2, . . . , yt) (Doucet +et al., 2001) using sequential Bayesian inference, similarly +to the Kalman filter (Kalman, 1960). The likelihood is +p(yt|θt) = N(yt; f( · ; θt), σ2) such that yt = f( · ; θt) + ϵ +where ϵ ∼ N(0, σ2) and f( · ; θt) = θt. We consider a +Gaussian prior over the mean parameters θ such that +p(θ0) = N(θ0; 0, σ2 +0). Since the conjugate prior for the mean is also Gaussian, the prior and posterior are +N(θt−1; ˆθt−1, ˆσ2 +t−1) and N(θt; ˆθt, ˆσ2 +t ). By using sequential Bayesian inference we can have closed-form update +equations for our posterior parameters: +ˆθt = ˆσ2 +t +� +yt +σ2 + +ˆθt−1 +ˆσ2 +t−1 +� += ˆσ2 +t +� +t +� +i=1 +yi +σ2 + +ˆθ0 +ˆσ2 +0 +� +, +1 +ˆσ2 +t += 1 +σ2 + +1 +ˆσ2 +t−1 +. +(7) +From Equation (7) the posterior mean follows a Gaussian distribution where the posterior mean is a sum of +the online observation and the online prior. So the posterior mean is a weighted sum of the data. So, if the +observations are non-stationary: if there is task change (more commonly referred to as a changepoint in the +time-series literature). Then the mean parameter will encapsulate a global mean over both tasks rather than +a mean for each task Fig. 4. The model is clearly misspecified since a single parameter cannot model both +of these tasks together. Despite performing exact inference a misspecified model can forget, Fig. 4. In the +case of HMC we verified that our Bayesian neural network had perfect performance on all tasks beforehand. +In Section 3 we had a well specified model but struggled with exact sequential Bayesian inference Eq. (3). +With this 1-d online learning scenario we are performing exact inference, however we have a misspecified +model. It is important to disentangle model misspecification and exact inference, and highlight that model +misspecification is a caveat which has not been highlighted in the CL literature as far as we are aware. +Furthermore we can only ensure that our models are well specified if we have access to data from all tasks a +5 + +priori. So in the scenario of online continual learning (De Lange et al., 2021) we cannot know if our model +will perform well on all past and future tasks without making assumptions on the task distributions. +5 +Sequential Bayesian Inference and Imbalanced Task Data +Neural Networks are complex models with a broad hypothesis space and hence are a suitably well-specified +model when tackling continual learning problems (Wilson & Izmailov, 2020). However we struggle to fit the +posterior samples from HMC to perform sequential Bayesian inference Section 3. +We continue to use Bayesian filtering and assume a Bayesian NN where the posterior is Gaussian with a full +covariance. By modelling the entire covariance we enable modelling how each individual weight varies with +respect to all others. We do this by interpreting online learning in Bayesian NNs as filtering (Ciftcioglu & +Türkcan, 1995). Our treatment is similar to Aitchison (2018) who derives an optimizer by leveraging Bayesian +filtering. We consider inference in the graphical model depicted in Fig. 5. The aim is to infer the optimal +BNN weights, θ∗ +t at t given a single observation and the BNN weight prior. The previous BNN weights are +used as a prior for inferring the posterior BNN parameters. We consider the online setting where a single +data point (xt, yt) is observed at a time. +Instead of modelling the full covariance we instead consider each parameter θi as a function of all the +other parameters θ−it. We also assume that the values of the weights are close to those of the previous +timestep (Jacot et al., 2018). To obtain the update equations for BNN parameters given a new observation +and prior we make two simplifying assumptions as follows. +Assumption 5.1. For a Bayesian neural network with output f(xt; θ) and likelihood L(xt, yt; θ), the +derivative evaluated at θt is zt = ∂L(xt, yt; θ)/∂θ|θ=θt and the Hessian is H. We assume a quadratic loss +for a data point (xt, yt) of the form: +L(xt, yt; θ) = Lt(θ) = −1 +2θ⊤Hθ + z⊤ +t θ, +(8) +the result of a second-order Taylor expansion. The Hessian is assumed to be constant with respect to (xt, yt) +(but not with respect to θ). +θ∗ +t−1 +θ∗ +t−2 +θ∗ +t +θ∗ +t+1 +yt−1 +yt−2 +yt +yt+1 +xt−1 +xt−2 +xt +xt+1 +. . . +. . . +Figure 5: Graphical model for filtering. Grey and white +nodes are observed and latent variables respectively. +To construct the dynamical equation for θ, consider +the gradient for the i-th weight while all other param- +eters are set to their current estimate at the optimal +value for the θ∗ +it: +θ∗ +it = − 1 +Hii +H⊤ +−iiθ−it, +(9) +since zit = 0 at a mode. The equation above shows +us that the dynamics of the optimal weight θ∗ +it is +dependent on all the other current values of the pa- +rameters θ−it. The dynamics of θ−it will be a com- +plex stochastic process dependent on many different +variables: such as the dataset, model architecture, +learning rate schedule, etc. +Assumption 5.2. Since reasoning about the dynamics of θ−it are intractable, we assume that at the next +time-step the optimal weights are close to the previous timesteps with a discretized Ornstein-Uhlenbeck process +for the weights θ−it with reversion speed ϑ ∈ R+ and noise variance η2 +−i: +p(θ−i,t+1|θ−i,t) = N((1 − ϑ)θ−it, η2 +−i), +(10) +this implies that the dynamics for the optimal weight is defined by +p(θ∗ +i,t+1|θ∗ +i,t) = N((1 − ϑ)θ∗ +it, η2), +(11) +where η2 = η2 +−iH⊤ +−iiH−ii. +6 + +In simple terms, in Assumption 5.2 we assume a parsimonious model of the dynamics. That the next value of +θ−i,t is close to their previous value according to a Gaussian, similarly to Aitchison (2018). +Lemma 5.3. Under Assumptions 5.1 and 5.2 the dynamics and likelihood are Gaussian. Thus we are able to +infer the posterior distribution over the optimal weights using Bayesian updates and by linearizing the BNN +the update equations for the posterior of the mean and variance of the BNN for a new data point are: +µt,post = σ2 +t,post +� +µt,prior +σ2 +t,prior(η2) + yt +σ2 g(xt) +� +and +1 +σ2 +t,post += g(xt)2 +σ2 ++ +1 +σ2 +t,prior(η2), +(12) +where we drop the notation for the i-th parameter, the posterior is N(θ∗ +t ; µt,post, σ2 +t,post) and g(xt) = ∂f(xt;θ∗ +it) +∂θ∗ +it +and σ2 +t,prior is a function of η2. +See Appendix E for the derivation of Lemma 5.3. From Eq. (12) we can notice that the posterior mean +depends linearly on the prior and a data dependent term and so will behave similarly to our previous example +in Section 4. Under Assumption 5.1 and Assumption 5.2 then if there is a data imbalance between tasks +in Eq. (12), then the data dependent term will dominate the prior term if there is more data for the current +task. +In Section 3 we showed that it is very difficult with current machine learning tools to perform sequential +Bayesian inference for simple CL problems with small Bayesian NNs. When we disentangle Bayesian inference +and model misspecification we show showed that misspecified models can forget despite exact Bayesian +inference. The only way to ensure that our model is well specified is to show that the multi-task posterior +produces reasonable posterior predictive distributions p(y|x, D) = +� +p(y|x, D, θ)p(θ|D)dθ for one’s application. +Additionally, in this section we have shown that if there is a task dataset size imbalance then we can get +forgetting under certain assumptions. +6 +Related Work +There has been a recent resurgence in the field of CL (Thrun & Mitchell, 1995) given the advent of deep +learning. Methods which approximate sequential Bayesian inference Eq. (5) have been seminal in CL’s +revival and have used a diagonal Laplace approximation (Kirkpatrick et al., 2017; Schwarz et al., 2018). The +diagonal Laplace approximation have been enhanced by modelling covariances of between neural network +weights in the same layer (Ritter et al., 2018). Instead of the Laplace approximation we can use a variational +approximation for sequential Bayesian inference (Nguyen et al., 2017; Zeno et al., 2018). Using richer priors +has also been explored (Ahn et al., 2019; Farquhar et al., 2020; Kessler et al., 2019; Mehta et al., 2021; Kumar +et al., 2021; Loo et al., 2020). Gaussian processes have also been applied to CL problems leveraging inducing +points to retain previous task functions (Titsias et al., 2019; Kapoor et al., 2021). +Bayesian methods which regularize weights have not matched up to the performance of experience replay +based CL methods (Buzzega et al., 2020) in terms of accuracy on CL image classification benchmarks. +Instead of regularizing high dimension weight spaces, regularizing task functions is a more direct approach +to combat forgetting (Benjamin et al., 2018). In particular, one can leverage the duality between the +Laplace approximation and Gaussian Processes to develop a functional regularization approach to Bayesian +CL (Swaroop et al., 2019) or using function-space variational inference (Rudner et al., 2022a;b). +7 +Prototypical Bayesian Continual Learning +We have shown that sequential Bayes over NN parameters is very difficult (Section 3), and is only suitable for +situations where the multi-task posterior is suitable for all tasks. We now show that a more fruitful approach +is to model the full data-generating process of the CL problem and we propose a simple and scalable approach +for doing so. In particular, we represent classes by prototypes (Snell et al., 2017; Rebuffi et al., 2017) to +prevent catastrophic forgetting. We refer to this framework as Prototypical Bayesian Continual Learning, or +ProtoCL for short. This approach can be viewed as a probabilistic variant of iCarl (Rebuffi et al., 2017), +which creates embedding functions for different classes which are simply class means and predictions are +7 + +made by nearest neighbors. ProtoCL also bears similarities to the few-shot learning model Probabilistic +Clustering for Online Classification (Harrison et al., 2019), developed for few-shot image classification. +Model. ProtoCL models the generative CL process. We consider classes j ∈ {1, . . . , J}, generated from a +categorical distribution with a Dirichlet prior: +yi,t ∼ Cat(p1:J), +p1:J ∼ Dir(αt). +(13) +Images are embedded into a embedding space by an encoder, z = f(x; w) with parameters w. The per class +embeddings are Gaussian whose mean has a prior which is also Gaussian: +zit|yit ∼ N(¯zyt, Σϵ), +¯zyt ∼ N(µyt, Λ−1 +yt ). +(14) +enc +enc +Embedding +Space +Embedding +Space +Figure 6: Overview of ProtoCL. +See Fig. 6 for an overview of the model. To alleviate +forgetting in CL, ProtoCL uses a coreset of past task +data to continue to embed past classes distinctly +as prototypes. The posterior distribution over class +probabilities {pj}J +j=1 and class embeddings {¯zyj}J +j=1 +is denoted in short hand as p(θ) with parameters +ηt = {αt, µ1:J,t, Λ−1 +1:J,t}. ProtoCL models each class +prototype but does not use task specific NN parame- +ters or modules like multi-head VCL. By modeling a +probabilistic model over an embedding space this al- +lows us to use powerful embedding functions f( · ; w) +without having to parameterize them probabilisti- +cally and so this approach will be more scalable than +VCL, for instance. +Inference. As the Dirichlet prior is conjugate with +the Categorical distribution and likewise the Gaus- +sian over prototypes with a Gaussian prior over the +prototype mean, we can calculate posteriors in closed +form and update the parameters ηt as new data is +observed without using gradient based updates. We +optimize the model by maximizing the posterior predictive distribution and use a softmax over class probabil- +ities to perform predictions. We perform gradient-based learning of the NN embedding function f( · ; w) and +update the parameters, ηt at each iteration of gradient descent as well, see Algorithm 1. +Sequential updates. We can obtain our parameter updates for the Dirichlet posterior by Categorical- +Dirichlet conjugacy: +αt+1,j = αt,j + +Nt +� +i=1 +I(yi +t = j), +(15) +where Nt are the number of points seen during the update at time step t. Also, due to Gaussian-Gaussian +conjugacy the posterior for the Gaussian prototypes is governed by: +Λyt+1 = Λyt + NyΣ−1 +ϵ +(16) +Λyt+1µyt+1 = NyΣ−1 +ϵ +¯zyt + Λytµyt, ∀yt ∈ Ct, +(17) +where Ny are the number of samples of class y and ¯zyt = (1/Ny) �Ny +i=1 zyi, see Appendix D.2 for the detailed +derivation. +Objective. We optimize the posterior predictive distribution of the prototypes and classes: +p(z, y) = +� +p(z, y|θt; ηt)p(θt; ηt)dθt = p(y) +Nt +� +i=1 +N(zit|yit; µyt,t, Σϵ + Λ−1 +yt,t). +(18) +8 + +Algorithm 1 ProtoCL continual learning +1: Input: task datasets T1:T , initialize embedding function: f( · ; w), coreset: M = ∅. +2: for T1 to TT do +3: +for each batch in Ti ∪ M do +4: +Optimize f(·; w) by maximizing the posterior predictive p(z, y) Eq. (18) +5: +Obtain posterior over θ by updating η, Eqs. (15) to (17). +6: +end for +7: +Add random subset from Ti to M. +8: end for +Where the p(y) = αy/ �J +j=1 αj, see Appendix D.3 for the detailed derivation. This objective can then be +optimized using gradient based optimization for learning the prototype embedding function z = f(x; w). +Predictions. To make a prediction for a test point x∗ the class with the maximum (log)-posterior predictive +is chosen, where the posterior predictive is: +p(y∗ = j|x∗, x1:t, y1:t) = p(y∗ = j|z∗, θt) = +p(y∗ = j, z∗|θt) +� +i p(y = i, z∗|θt), +(19) +see Appendix D.4 for further details. +Preventing forgetting. As we wish to retain the class prototypes. We make use of coresets: experience +from previous tasks. At the end of learning a task Tt, we retain subset Mt ⊂ Dt and augment each new task +dataset to ensure that posterior parameters ηt and prototypes are able to retain previous task information. +Class incremental learning. In this CL setting we do not tell the CL agent which task it is currently +training and evaluating with a task identifier τ. So we cannot use the task identifier to select a specific head +to use for classifying a test point, or use the task identifier to condition the model in another way (such +that p(y|x, τ) where τ is the task identifier). Also we require the CL agents to classify each class in the CL +benchmarks, for example {0, . . . , 9} for Split-MNIST and Split CIFAR10 and not just {0, 1} for each task as +commonly performed (these are called task-incremental and domain-incremental learning). We believe that +these are more realistic and the most important settings to evaluate on. +Implementation. For Split-MNIST and Split-FMNIST the baselines and ProtoCL all use two layer NNs +with hidden state size of 200. For Split-CIFAR10 and Split-CIFAR100, the baselines and ProtoCL use a four +layer convolution neural network with two fully connected layers of size 512 similarly to Pan et al. (2020). +For ProtoCL and all baselines which rely on replay we fix the size of the coreset to 200 points per task. For +all ProtoCL models we allow the prior Dirichlet parameters to be learned and set their initial value to 0.7 +found by a random search over MNIST with ProtoCL. An important hyperparameter for ProtoCL is the +embedding dimension of the Gaussian prototypes for Split-MNIST and Split-FMNIST this was set to 128 +while for the larger vision datasets this was set to 32 found using grid-search. +Table 1: Mean accuracies across all tasks over CL vision benchmarks for class incremental learning (Van de +Ven & Tolias, 2019). All results are averages and standard errors over 10 seeds. ∗Uses the predictive entropy +to make a decision about which head for class incremental learning. +Method +Coreset +Split-MNIST +Split-FMNIST +VCL (Nguyen et al., 2017) + +33.01 ± 0.08 +32.77 ± 1.25 ++ coreset + +52.98 ± 18.56 +61.12 ± 16.96 +HIBNN∗ (Kessler et al., 2019) + +85.50 ± 3.20 +43.70 ± 20.21 +FROMP (Pan et al., 2020) + +84.40 ± 0.00 +68.54 ± 0.00 +S-FSVI (Rudner et al., 2022b) + +92.94 ± 0.17 +80.55 ± 0.41 +ProtoCL (ours) + +93.73 ± 1.05 +82.73 ± 1.70 +9 + +Table 2: Mean accuracies across all tasks over CL vision benchmarks for class incremental learning (Van de +Ven & Tolias, 2019). All results are averages and standard errors over 10 seeds. ∗Uses the predictive entropy +to make a decision about which head for class incremental learning. Training times have been benchmarked +using an Nvidia RTX3090 GPU. +Method +Training time (sec) (↓) +Split CIFAR-10 (acc) (↑) +FROMP (Pan et al., 2020) +1425 ± 28 +48.92 ± 10.86 +S-FSVI (Rudner et al., 2022b) +44434 ± 91 +50.85 ± 3.87 +ProtoCL (ours) +384 ± 6 +55.81 ± 2.10 +Split CIFAR-100 (acc) +S-FSVI (Rudner et al., 2022b) +37355 ± 1135 +20.04 ± 2.37 +ProtoCL (ours) +1425 ± 28 +23.96 ± 1.34 +Results. +ProtoCL produces good results on CL benchmarks on par or better than S-FSVI (Rudner +et al., 2022b) which is state-of-the-art on the smaller CL benchmarks while being a lot more efficient +to train and without requiring expensive variational inference. ProtoCL can flexibly scale to larger CL +vision benchmarks producing better results than S-FSVI. Code to reproduce all experiments can be found +here anonymous.4open.science/r/bayes_cl_exploration. All our experiments are in the more realistic class +incremental learning setting, which is a harder setting than those reported in most CL papers, so the results +in Table 1 are lower for certain baselines than in the respective papers. We use 200 data points per task, +see Figure 12 for a sensitivity analysis of the performance over the Split-MNIST benchmark as as function of +core size for ProtoCL. +The stated aim of ProtoCL is not provide a novel state-of-the-art method for CL, but rather to propose a +simple baseline which takes an alternative route than weight-space sequential Bayesian inference. We can +achieve strong results that mitigate forgetting, namely by modeling the generative CL process and using +sequential Bayesian inference over a few parameters in the class prototype embedding space. We argue +that modeling the generative CL process is a fruitful direction for further research rather than attempting +sequential Bayesian inference over the weights of a BNN. +8 +Discussion & Conclusion +In this paper we have revisited the use of sequential Bayesian inference for CL. We can use sequential Bayes to +recursively build up the multi-task posterior Eq. (5). Previous methods have relied on approximate inference +and see little benefit over SGD. We test the hypothesis of whether this poor performance is due to the +approximate inference scheme by using HMC in two simple CL problems. HMC asymptotically samples from +the true posterior and we use a density estimator over HMC samples to use as a prior for a new task within +the HMC sampling process. This density is multi-modal and accurate with respect to the current task but is +not able to improve over using an approximate posterior. This demonstrates just how challenging it is to +work with BNN weight posteriors. The source of error comes from the density estimation step. We then look +at an analytical example of sequential Bayesian inference where we perform exact inference however due to +model misspecification, we observe forgetting. The only way to ensure a well specified model is to assess the +multi-task performance over all tasks a priori. This might not be possible in online CL settings. We then +model an analytical example over Bayesian NNs and under certain assumptions show that if there is task +data imbalances then this will cause forgetting. Because of these results, we argue against performing weight +space sequential Bayesian inference and instead model the generative CL problem. We introduce a simple +baseline called ProtoCL. ProtoCL doesn’t require complex variational optimization and achieves competitive +results to state-of-the-art in the realistic setting of class incremental learning. +This conclusion should not be a surprise since the latest Bayesian CL papers have all relied multi-head +architectures or inducing points/coresets to prevent forgetting, rather than better weight-space inference +schemes. Our observations are in line with recent theory from (Knoblauch et al., 2020) which states that +10 + +optimal CL requires perfect memory. Although the results were shown with deterministic NNs the same +results follow for BNN with a single set of parameters. Future research directions include enabling coresets of +task data to efficiently and accurately approximate the posterior of a BNN to remember previous tasks. +9 +Acknowledgments +We would like to thank Sebastian Farquhar, Laurence Aitchison, Jeremias Knoblauch and Chris Holmes for +discussions. Thanks to Phil Ball for help with writing the paper. SK acknowledges funding from the Oxford- +Man Institute of Quantitative Finance. TGJR acknowledges funding from the Rhodes Trust, Qualcomm, +and the Engineering and Physical Sciences Research Council (EPSRC). This material is based upon work +supported by the United States Air Force and DARPA under Contract No. FA8750-20-C-0002. Any opinions, +findings and conclusions or recommendations expressed in this material are those of the author(s) and do not +necessarily reflect the views of the United States Air Force and DARPA. +References +Hongjoon Ahn, Sungmin Cha, Donggyu Lee, and Taesup Moon. Uncertainty-based continual learning with +adaptive regularization. Advances in neural information processing systems, 32, 2019. +Laurence Aitchison. Bayesian filtering unifies adaptive and non-adaptive neural network optimization methods. +arXiv preprint arXiv:1807.07540, 2018. +Ari S Benjamin, David Rolnick, and Konrad Kording. 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Task agnostic continual learning using online +variational bayes. arXiv preprint arXiv:1803.10123, 2018. +13 + +2 +1 +0 +1 +2 +x1 +2 +1 +0 +1 +2 +x2 +Task 1 +Task 2 +Task 3 +Task 4 +Task 5 +1 +2 +3 +4 +5 +Tasks +0.00 +0.50 +1.00 +Accuracy +Task 1 +1 +2 +3 +4 +5 +Tasks +0.00 +0.50 +1.00 +Task 2 +1 +2 +3 +4 +5 +Tasks +0.60 +0.80 +1.00 +Task 3 +1 +2 +3 +4 +5 +Tasks +0.40 +0.60 +0.80 +1.00 +Task 4 +1 +2 +3 +4 +5 +Tasks +0.80 +0.90 +1.00 +Task 5 +HMC +MH VCL +SH VCL +SGD +MT SGD/HMC +Figure 7: Continual learning binary classification accuracies from the toy Gaussian dataset similar to (Henning +et al., 2021) using 10 random seeds. The pink solid line is a multi-task (MT) baseline test accuracy using +SGD/HMC. +Supplementary Material +A +The Toy Gaussians Dataset +See Fig. 7 for a visualization of the toy Gaussians dataset which we use as a simple CL problem. This is used +for evaluating our method for propagating the true posterior by using HMC for posterior inference and then +using a density estimator on HMC samples as a prior for a new task. We construct 5, 2-way classification +problems for CL. Each 2-way task involves classifying adjacent circles and squares Fig. 7. With a 2 layer +network with 10 neurons we get a test accuracy of 1.0 for the multi-task learning of all 5 tasks together. +Hence according to Eq. (3) a BNN with the same size should be able to learn all 5 binary classification tasks +continually by sequentially building up the posterior. +B +HMC implementation details +We set the prior for T1, to p1(θ) = N(0, τ −1I) with τ = 10. We burn-in the HMC chain for 1000 steps and +sample for 10000 more steps and run 20 different chains to obtain samples from our posterior, which we then +pass to our density estimator. We use a step size of 0.001 and trajectory length of L = 20, see Appendix C +for further implementation details of the density estimation procedure. For the GMM we optimize for the +number of components by using a holdout set of HMC samples. +C +Density Estimation Diagnostics +We provide plots to show that the HMC chains indeed sample from the posterior have converged in Figure 9 +and Figure 11. We run 20 HMC sampling chains and randomly select one chain to plot for each seed (of 10). +We run HMC over 10 seeds and aggregate the results Figure 3 and Figure 7. The posteriors p(θ|D1), . . . are +approximated with a GMM and used as a prior for the second task and so forth. +We provide empirical evidence to show that the density estimators have fit to HMC samples of the posterior +in Figure 8 and Figure 10. Where we show the number of components of the GMM density estimator which we +use as a prior for a new task are all multi-modal posteriors. We show the BNN accuracy when sampling BNN +weights from our GMM all recover the accuracy of the converged HMC samples. The effective sample size +(ESS) of the 20 chains which is a measure of how correlated the samples are (higher is better). The reported +ESS values for our experiments are in line with previous work which uses HMC for BNN inference (Cobb & +Jalaian, 2021). +14 + +1 +2 +3 +4 +5 +Task +0 +10 +20 +30 +40 +50 +ESS +1 +2 +3 +4 +5 +Task +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Acc. sampled from + GMM posterior +1 +2 +3 +4 +5 +Task +0 +5 +10 +15 +20 +25 +# GMM components + in posterior +Figure 8: Diagnostics from using a GMM prior fit to samples of the posterior generated from HMC, all results +are for 10 random seeds. Left, effective sample sizes (ESS) of the resulting HMC chains of the posterior, all +are greater than those reported in other works using HMC for BNNs (Cobb & Jalaian, 2021). Middle, the +accuracy of the BNN when using samples from the GMM density estimator instead from the samples from +HMC. Right, The optimal number of components of each GMM posterior fitted with a holdout set of HMC +samples by maximizing the likelihood. +0.00 +0.25 +0.50 +0.75 +1.00 +Accuracy +Task 1 +Task 2 +Task 3 +Task 4 +Task 5 +0.0 +0.2 +0.4 +0.6 +nll +Iteration number +Figure 9: Convergence plots from a one randomly sampled HMC chain (of 20) for each task over 10 different +runs (seeds) for 5 tasks from the toy Gaussians dataset similar to Henning et al. (2021) (visualized in Fig. 7). +We use a GMM density estimator as the prior conditioned on previous task data. +1 +2 +3 +4 +5 +Task +0 +10 +20 +30 +40 +50 +ESS +1 +2 +3 +4 +5 +Task +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Task Acc. +1 +2 +3 +4 +5 +Task +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Acc. sampled from + GMM posterior +1 +2 +3 +4 +5 +Task +0 +5 +10 +15 +20 +25 +# GMM components + in posterior +Figure 10: Diagnostics from using a GMM to fit samples of the posterior HMC samples, all results are for 10 +random seeds on the toy dataset from Pan et al. (2020) (and visualized in Fig. 3). Left, effective sample +sizes (ESS) of the resulting HMC chains of the posterior, all are greater than those reported in other works +using HMC for BNNs (Cobb & Jalaian, 2021). Middle left. the current task accuracy from HMC sampling. +Middle right, the accuracy of the BNN when using samples from the GMM density estimator instead of +the converged HMC samples. Right, The optimal number of components of each GMM posterior fitted with +a holdout set of HMC samples by maximizing the likelihood. +15 + +0.5 +1.0 +Accuracy +Task 1 +Task 2 +Task 3 +Task 4 +Task 5 +0.5 +1.0 +NLL +HMC Iteration +Figure 11: Convergence plots from a randomly sampled HMC chain (of 20) for each task over 10 different +seeds for 5 tasks from the toy dataset from (Pan et al., 2020) (see Fig. 3 for a visualization of the data). We +use a GMM density estimator as a prior. +D +Prototypical Bayesian Continual Learning +ProtoCL models the generative process of CL where new tasks are comprised of new classes j ∈ {1, . . . , J} of +a total of J and can be modeled by using a categorical distribution with a Dirichlet prior: +yi,t ∼ Cat(p1:J), +p1:J ∼ Dir(αt). +(D.1) +We learn a joint embedding space for our data with a NN, z = f(x; w) with parameters w. The embedding +space for each class is Gaussian whose mean has a prior which is also Gaussian: +zit|yit ∼ N(¯zyt, Σϵ), +¯zyt ∼ N(µyt, Λ−1 +yt ). +(D.2) +By ensuring that we have an embedding per class and using a memory of past data, we ensure that the +embedding does not drift. The posterior parameters are ηt = {αt, µ1:J,t, Λ−1 +1:J,t}. +D.1 +Inference +As the Dirichlet prior is conjugate with the Categorical distribution and so is the Gaussian distribution with +a Gaussian prior over the mean of the embedding, then we can calculate posteriors in closed form and update +our parameters as we see new data online without using gradient based updates. We perform gradient based +learning of the NN embedding function f( · ; w) with parameters w. We optimize the model by maximizing +the log-predictive posterior of the data and use the softmax over class probabilities to perform predictions. +The posterior over class probabilities {pj}J +j=1 and class embeddings {¯zyj}J +j=1 is denoted as p(θ) for short +hand and has parameters are ηt = {αt, µ1:J,t, Λ−1 +1:J,t} are updated in closed form at each iteration of gradient +descent. +D.2 +Sequential updates +We can obtain our posterior: +p(θt|Dt) ∝ p(Dt|θt)p(θt) +(D.3) += +Nt +� +i=1 +p(zi +t|yi +t; ¯zyt, Σϵ,yt)p(yi +t|p1:J)p(pi:J; αt)p(¯zyt; µyt,t, Λ−1 +yt,t) +(D.4) += N(µt+1, Σt+1)Dir(αt+1), +(D.5) +16 + +where Nt is the number of data points seen during update t. Concentrating on the Categorical-Dirichlet +conjugacy: +Dir(αt+1) ∝ p(p1:J; αt) +Nt +� +i=1 +p(yi +t; pi:J) +(D.6) +∝ +J +� +j=1 +pαj−1 +j +Nt +� +i=1 +J +� +j=1 +pI(yi +t=j) +j +(D.7) += +J +� +j=1 +p +αj−1+�Nt +i=1 I(yi +t=j) +j +. +(D.8) +Thus: +αt+1,j = αt,j + +Nt +� +i=1 +I(yi +t = j). +(D.9) +Also, due to Gaussian-Gaussian conjugacy, then the posterior for the Gaussian prototype of the embedding +for each class is: +N(µt+1, Λt+1) ∝ +Nt +� +i=1 +N(zi +t|yi +t; ¯zyt, Σϵ)N(¯zyt; µyt,t, Λ−1 +yt ) +(D.10) += +� +yt∈{1,...,J} +N(zyt|yt; ¯zyt, +1 +Nyt +Σϵ)N(¯zyt; µyt+1, Λ−1 +yt ) +(D.11) += +� +yt∈{1,...,J} +N(¯zyt; µt+1, Λ−1 +yt+1), +(D.12) +where Nyt is the number of points of class yt from the set of all classes C = {1, . . . , J}. The update equations +for the mean and variance of the posterior are: +Λyt+1 = Λyt + NytΣ−1 +ϵ , +∀yt ∈ Ct +(D.13) +Λyt+1µyt+1 = NytΣ−1 +ϵ +¯zyt + Λytµyt, +∀yt ∈ Ct. +(D.14) +D.3 +ProtoCL Objective +The posterior predictive distribution we want to optimize is: +p(z, y) = +� +p(z, y|θ; η)p(θ; η)dθ, +(D.15) +where p(θ) denotes the distributions over class probabilities {pj}J +j=1 and mean embeddings {¯zyj}J +j=1, +p(z, y) = +� +Nt +� +i=1 +p(zit|yit; ¯zyt, Σϵ)p(yit|p1:J)p(p1:J; αt)p(¯zyt; µyt,t, Λ−1 +yt,t)dp1:Jd¯zyt +(D.16) += +� +Nt +� +i=1 +p(zit|yit; zyt, Σϵ)p(¯zyt; µyt,t, Λ−1 +yt,t)d¯zyt +� +Nt +� +i=1 +p(yit|p1:J)p(p1:J; αt)dp1:J +� +�� +� +� +i p(yi)=p(y) +(D.17) += p(y) +Nt +� +i=1 +Z−1 +i +� +N(¯zyit; c, C)d¯zyt +(D.18) += p(y) +Nt +� +i=1 +N(zit|yit; µyt,t, Σϵ + Λ−1 +yt,t). +(D.19) +17 + +Figure 12: Split-MNIST average test accuracy over +all 5 tasks for different memory sizes. Accuracies are +over 10 seeds. +0 +250 500 750 1000 +1500 +2000 +mem. size +40 +50 +60 +70 +80 +90 +Accuracy +Where in Eq. (D.18) we use §8.1.8 in (Petersen et al., 2008). The term p(y) is: +p(y) = +� +p(y|p1:J)p(p1:J; αt)dp1:J +(D.20) += +� +py +Γ(�J +j=1 αj) +�J +j=1 Γ(αj) +J +� +j=1 +pαj−1 +j +dp1:J +(D.21) += +Γ(�J +j=1 αj) +�J +j=1 Γ(αj) +� +J +� +j=1 +pI(y=j)+αj−1 +j +dp1:J +(D.22) += +Γ(�J +j=1 αj) +�J +j=1 Γ(αj) +�J +j=1 Γ(I(y = j) + αj) +Γ(1 + �J +j=1 αj) +(D.23) +=  +Γ(�J +j=1 αj) +�J +j=1 Γ(αj) +�J +j=1 Γ(I(y = j) + αj) +�J +j=1 αj +Γ(�J +j=1 αj) +(D.24) += +�J +j=1,j̸=y Γ(αj) +�J +j=1 Γ(αj) +Γ(1 + αy) +�J +j=1 αj +(D.25) += +�J +j=1,j̸=y Γ(αj) +�J +j=1 Γ(αj) +αyΓ(αy) +�J +j=1 αj +(D.26) +(D.27) +where we use the identity Γ(n + 1) = nΓ(n). +D.4 +Predictions +To make a prediction for a test point x∗: +p(y∗ = j|x∗, x1:t, y1:t) = p(y∗ = j|z∗, θt) +(D.28) += +p(z∗|y∗ = j, θt)p(y∗ = j|θt) +� +i p(z∗|y∗ = i, ηt)p(y∗ = i|θt) +(D.29) += +p(y∗ = j, z∗|θt) +� +i p(y = i, z∗|θt), +(D.30) +where θt are sufficient statistics for (x1:t, y1:t). +18 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Dirichlet p +class prob: 1 +class prob: 2 +class prob: 3 +class prob: 4 +class prob: 5 +1 +2 +3 +4 +5 +Task +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Dirichlet p +class prob: 6 +1 +2 +3 +4 +5 +Task +class prob: 7 +1 +2 +3 +4 +5 +Task +class prob: 8 +1 +2 +3 +4 +5 +Task +class prob: 9 +1 +2 +3 +4 +5 +Task +class prob: 10 +Figure 13: The evolution of the Dirichlet parameters αt for each class in Split-MNIST tasks for ProtoCL. All +αj are shown over 10 seeds with ±1 standard error. By the end of training all classes are roughly equally +likely, as we have trained on equal amounts of all classes. +Preventing forgetting. As we wish to retain the task specific prototypes, at the end of learning a task Tt +we take a small subset of the data as a memory to ensure that posterior parameters and prototypes do not +drift, see Algorithm 1. +D.5 +Experimental Setup +The prototype variance, Σϵ is set to a diagonal matrix with the variances of each prototype set to 0.05. The +prototype prior precisions, Λyt are also diagonals and initialized randomly and exponentiated to ensure a +positive semi-definite covariance for the sequential updates. The parameters αj ∀j are set to 0.78 which was +found by random search over the validation set on MNIST. We also allow αj to be learned in the gradient +update step in addition to the sequential update step (lines 4 and 5 Algorithm 1), see Fig. 13 to see the +evolution of the αj or all classes j over the course of learning Split-MNIST. +For the Split-MNIST and Split-FMNIST benchmarks we use a NN with 2 layers of size 200 and trained for +50 epochs with an Adam optimizer. We perform a grid-search over learning rates, dropout rates and weight +decay coefficients. The embedding dimension is set to 128. For the Split-CIFAR10 and Split-CIFAR100 +benchmarks, we use the same network as Pan et al. (2020) which consists of 4 convolution layers and 2 linear +layers. We train the networks for 80 epochs for each task with the Adam optimizer with a learning rate of +1e − 3. The embedding dimension is set to 32. All experiments are run a single GPU NVIDIA RTX 3090. +E +Sequential Bayesian Estimation as Bayesian Neural Network optimization +We shall consider inference in the graphical model depicted in Fig. 14. The aim is to infer the optimal BNN +weights, θ∗ +t at time t given observations and the previous BNN weights. We assume a Gaussian posterior +over weights with full covariance hence we model interactions between all weights. We shall consider the +online setting where we see one data point (xt, yt) at a time and we will make no assumption as to whether +the data comes from the same task or different tasks over the course of learning. +We set up the problem of sequential Bayesian inference as a filtering problem and we leverage the work +of Aitchison (2018) which casts NN optimization as Bayesian sequential inference. We make the reasonable +assumption that the distribution over weights is a Gaussian with full covariance. Since reasoning about +the full covariance matrix of a BNN is intractable we instead consider the i-th parameter and reason about +the dynamics of the optimal estimates θ∗ +it as a function of all the other parameters θ−it. Each weight is +functionally dependent on all others. If we had access to the full covariance of the parameters then we could +reason about the unknown optimal weights θ∗ +it given the values of all the other weights θ−it. However, since +19 + +θ∗ +t−1 +θ∗ +t−2 +θ∗ +t +θ∗ +t+1 +yt−1 +yt−2 +yt +yt+1 +xt−1 +xt−2 +xt +xt+1 +. . . +. . . +Figure 14: Graphical model of under which we perform inference in Section 5. Grey nodes are observed and +white are latent variables. +we do not have access to the full covariance another approach is to reason about the dynamics of θ∗ +it given the +dynamics of θ−it and assume that the values of the weights are close to those of the previous time-step (Jacot +et al., 2018) and so we cast the problem as a dynamical system. +Consider a quadratic loss of the form: +L(xt, yt; θ) = Lt(θ) = −1 +2θ⊤Hθ + z⊤ +t θ, +(E.31) +which we can arrive by simple Taylor expansion where H is the Hessian which is assumed to constant across +data points but not across the parameters θ. If the BNN output takes the form f(xt; θ), then the derivative +evaluated at θt is zt = ∂L(xt,yt;θ) +∂θ +|θ=θt. To construct the dynamical equations for our weights, consider the +gradient for a single datapoint: +∂Lt(θ) +∂θ += −Hθ + zt. +(E.32) +If we consider the gradient for the i-th weight while all other parameters are set to their current estimate: +∂L(θi, θ−i) +∂θi += −Hiiθit − H⊤ +−iiθ−it + zti. +(E.33) +When the gradient is set to zero we recover the optimal value for θit, denoted as θ∗ +it: +θ∗ +it = − 1 +Hii +H⊤ +−iiθ−it. +(E.34) +since zti = 0 at the modes. The equation above shows us that the dynamics of the optimal weight θ∗ +it is +dependent on all the other current values of the parameters θ−it. That is, the dynamics of θ∗ +it will be governed +by the dynamics of the weights θ−it. The dynamics of θ−it will be a complex stochastic process dependent on +many different variables. Since reasoning about the dynamics is intractable we instead assume a discretized +Ornstein-Uhlenbeck process for the weights θ−it with reversion speed ϑ ∈ R+ and noise variance η2 +−i: +p(θ−i,t+1|θ−i,t) = N((1 − ϑ)θ−it, η2 +−i), +(E.35) +this implies that the dynamics for the optimal weight is defined by +p(θ∗ +i,t+1|θ∗ +i,t) = N((1 − ϑ)θ∗ +it, η2), +(E.36) +where η2 = η2 +−iH⊤ +−iiH−ii. This same assumption is made in Aitchison (2018). This assumes a parsimonious +model of the dynamics. Together with our likelihood: +p(yt|xt; θ∗ +t ) = N(yt; f(xt; θ∗ +t ), σ2) +(E.37) +20 + +where f( · , θ) is a neural network prediction with weights θ, we can now define a linear dynamical system for +the optimal weight θ∗ +i by linearizing the the Bayesian NN (Jacot et al., 2018) and by using the transition +dynamics in Eq. (E.36). Thus we are able to infer the posterior distribution over the optimal weights using +Kalman filter like updates (Kalman, 1960). As the dynamics and likelihood are Gaussian, then the prior and +posterior are also Gaussian, for ease of notation we drop the index i such that θ∗ +it = θ∗ +t : +p(θ∗ +t |(x, y)t−1, . . . , (x, y)1) = N(µt,prior, σ2 +t,prior) +(E.38) +p(θ∗ +t |(x, y)t, . . . , (x, y)1) = N(µt,post, σ2 +t,post) +(E.39) +By using the transition dynamics and the prior we can obtain closed form updates: +p(θ∗ +t |(x, y)t−1, . . . , (x, y)1) = +� +p(θ∗ +t |θ∗ +t−1)p(θ∗ +t−1|(x, y)t−1, . . . , (x, y)1)dθ∗ +t−1 +(E.40) +N(θ∗ +t ; µt,prior, σ2 +t,prior) = +� +N(θ∗ +t ; (1 − ϑ)θ∗ +t−1, η2)N(θ∗ +t−1; µt−1,post, σ2 +t−1,post)dθ∗ +t−1. +(E.41) +Integrating out θ∗ +t−1 we can get updates for the prior for the next timestep as follows: +µt,prior = (1 − ϑ)µt−1,post +(E.42) +σ2 +t,prior = η2 + (1 − ϑ)−2σ2 +t−1,post. +(E.43) +The updates for obtaining our posterior parameters: µt,post and σ2 +t,post, comes from applying Bayes’ theorem: +log N(θ∗ +t ; µt,post, σ2 +t,post) ∝ log N(yt; f(xt; θ∗ +t ), σ2) + log N(θ∗ +t ; µt,prior, σ2 +t,prior), +(E.44) +by linearizing our Bayesian NN such that f(xt, θ0) ≈ f(xt, θ0) + ∂f(xt;θ∗ +t ) +∂θ∗ +t +(θ∗ +t − θ0) and by substituting +into Eq. (E.44) we obtain our update equation for the posterior of the mean of our BNN parameters: +− +1 +2σ2 +t,post +(θ∗ +t − µt,post)2 = − 1 +2σ2 (y − g(xt)θ∗ +t )2 − +1 +2σ2 +t,prior +(θ∗ +t − µt,prior)2 +(E.45) +µt,post = σ2 +t,post +�µt,prior +σ2 +t,prior ++ y +σ2 g(xt) +� +, +(E.46) +where g(xt) = ∂f(xt;θ∗ +t ) +∂θ∗ +t +, and the update equation for the variance of the Gaussian posterior is: +1 +σ2 +t,post += g(xt)2 +σ2 ++ +1 +σ2 +t,prior +. +(E.47) +From our update equations Eq. (E.46) and Eq. (E.47) we can notice that the posterior mean depends linearly +on the prior and an additional data dependent term. These equations are similar to the filtering example +in Section 4. So under certain assumptions above, a BNN should behave similarly. If there exists a task data +imbalance then the data term will dominate the prior term in Eq. (E.46) and could lead to forgetting of +previous tasks. +21 + diff --git a/dNAzT4oBgHgl3EQf3P7o/content/tmp_files/load_file.txt b/dNAzT4oBgHgl3EQf3P7o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a72fc5c29a02af06fde817de02acfd3f1a50bca --- /dev/null +++ b/dNAzT4oBgHgl3EQf3P7o/content/tmp_files/load_file.txt @@ -0,0 +1,1040 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf,len=1039 +page_content='On Sequential Bayesian Inference for Continual Learning Samuel Kessler skessler@robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='uk University of Oxford Adam Cobb adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='cobb@sri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='com SRI International Tim G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Rudner tim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='rudner@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='uk University of Oxford Stefan Zohren zohren@robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='uk University of Oxford Stephen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Roberts sjrob@robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='uk University of Oxford Abstract Sequential Bayesian inference can be used for continual learning to prevent catastrophic forgetting of past tasks and provide an informative prior when learning new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We revisit sequential Bayesian inference and test whether having access to the true posterior is guaranteed to prevent catastrophic forgetting in Bayesian neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' To do this we perform sequential Bayesian inference using Hamiltonian Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We propagate the posterior as a prior for new tasks by fitting a density estimator on Hamiltonian Monte Carlo samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We find that this approach fails to prevent catastrophic forgetting demonstrating the difficulty in performing sequential Bayesian inference in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' From there we study simple analytical examples of sequential Bayesian inference and CL and highlight the issue of model misspecification which can lead to sub-optimal continual learning performance despite exact inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Furthermore, we discuss how task data imbalances can cause forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' From these limitations, we argue that we need probabilistic models of the continual learning generative process rather than relying on sequential Bayesian inference over Bayesian neural network weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In this vein, we also propose a simple baseline called Prototypical Bayesian Continual Learning, which is competitive with state-of-the-art Bayesian continual learning methods on class incremental continual learning vision benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 1 Introduction The goal of continual learning (CL) is to find a predictor that learns to solve a sequence of new tasks without losing the ability to solve previously learned tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' One key challenge of CL with neural networks (NNs) is that model parameters from previously learned tasks are “overwritten” during gradient-based learning of new tasks, which leads to catastrophic forgetting of previously learned abilities (French, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' One approach to CL hinges on using recursive applications of Bayes’ Theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' using the weight posterior in a Bayesian neural network (BNN) as the prior for a new task (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' However, obtaining a full posterior over NN weights is computationally demanding and we often need to resort to approximations, such as the Laplace method (MacKay, 1992) or variational inference (Graves, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2015) to obtain a neural network weight posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' When performing Bayesian CL, sequential Bayesian inference is performed with an approximate BNN posterior, not the true posterior (Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Ritter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Ebrahimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Kessler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' If we consider the performance of sequential Bayesian inference 1 with a variational approximation over a BNN weight posterior then we barely observe an improvement over simply learning new tasks with stochastic gradient descent (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We will develop this statement further in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' So if we had access to the true BNN weight posterior, would this be enough to prevent forgetting by sequential Bayesian inference?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Our contributions in this paper are to revisit Bayesian CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 1) Experimentally, we perform sequential Bayesian inference using the true Bayesian NN weight posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We do this by using the gold standard of Bayesian inference methods, Hamiltonian Monte Carlo (HMC) (Neal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We use density estimation over HMC samples and use this approximate posterior density as a prior for the next task within the HMC sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Surprisingly our HMC method for CL yields no noticeable benefits over an approximate inference method (VCL Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (2017)) despite using samples from the true posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 2) As a result we consider a simple analytical example and highlight that exact inference with a misspecified model can still cause forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 3) We show mathematically that under certain assumptions task data imbalances will cause forgetting in Bayesian NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 4) We propose a new probabilistic model for CL and show that by explicitly modeling the generative process of the data, we can achieve good performance, avoiding the need to rely on recursive Bayesian inference over NN weights to prevent forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Our proposed model, Prototypical Bayesian Continual Learning (ProtoCL), is conceptually simple, scalable, and competitive with state of the art Bayesian CL methods in the class-incremental learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='1 The Continual Learning Problem Continual learning (CL) is a learning setting whereby a model must learn to make predictions over a set of tasks sequentially while maintaining performance across all previously learned tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In CL, the model is sequentially shown T tasks, denoted Tt for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Each task, Tt, is comprised of a dataset Dt = {(xi, yi)}Nt i=1 which a model needs to learn to make predictions with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' More generally, tasks are denoted by distinct tuples comprised of the conditional and marginal data distributions, {pt(y|x), pt(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' After task Tt the model will lose access to the training dataset but its performance will be continually evaluated on all tasks Ti for i ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For a comprehensive review of CL scenarios see (Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Van de Ven & Tolias, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We decompose predictors as g = h ◦ f such that ˆy = g(x) we define f as an embedding function mapping f : X → Z and h as a head mapping to outputs h : Z → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Some continual learning methods use a separate head per task {hi}T i=1, these methods are called multi-headed while those that use one head are called single-headed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 Bayesian Continual Learning We consider a setting in which task data arrives sequentially at time steps, t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' At the first time step, t = 1, the model parameterized by θ receives the dataset D1 and learns the conditional distribution p(yi|xi, θ) for all (xi, yi) ∈ D1 (i indexes a datapoint), the parameters θ have a prior distribution p(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The posterior predictive distribution for a test point, x∗ 1 is: p(y∗ 1|x∗ 1, D1) = � p(y∗ 1|x∗ 1, θ)p(θ|D1)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (1) Computing this posterior predictive distribution above requires p(θ|D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For t = 2, a CL model is required to fit p(yi|xi, θ) for D1 ∪ D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The posterior predictive distribution for a new test point x∗ 2 point is: p(y∗ 2|x∗ 2, D1, D2) = � p(y∗ 2|x∗ 2, θ)p(θ|D1, D2)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (2) The posterior must thus be updated to reflect this new conditional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We can use repeated application of Bayes’ rule to calculate the posterior distributions p(θ|D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , DT ) as: p(θ|D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , DT −1, DT ) = p(DT |θ)p(θ|D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , DT −1) p(DT |D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , DT −1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (3) 2 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Accuracy Task 1 (0 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 1) 1 2 3 4 5 Tasks Task 2 (2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 3) 1 2 3 4 5 Tasks Task 3 (4 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 5) 1 2 3 4 5 Tasks Task 4 (6 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 7) 1 2 3 4 5 Tasks Task 5 (8 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 9) SGD VCL SH VCL MH Figure 1: Accuracy on Split-MNIST for various CL methods with a two-layer BNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We compare a NN trained with SGD (single-headed) with VCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We consider single-headed (SH) and multi-head (MH) VCL variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In the CL setting we lose access to previous training datasets: however, using repeated applications of Bayes’ rule Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (3), allows us to sequentially incorporate information from past tasks in the parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' At t = 1, we have access to D1 and the posterior over weights is: log p(θ|D1) = log p(D1|θ) + log p(θ) − log p(D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (4) At t = 2, we require a posterior p(θ|D1, D2) to calculate the posterior predictive distribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' However, we have lost access to D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' According to Bayes’ rule, the posterior may be written as: log p(θ|D1, D2) = log p(D2|θ) + log p(θ|D1) − log p(D2|D1), (5) where we used the conditional independence of D2 and D1 given θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We note that the likelihood is only dependent upon the current task dataset, D2, and that the prior encodes parameter knowledge from the previous task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Hence, we can use the posterior at t as a prior for learning a new task at t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For the MNIST dataset (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 1998) we know that if we were to train a BNN we would achieve a good performance by inferring p(θ|D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Hence if we were to Split-MNIST into 5 two-way classification tasks then we should be able to recursively recover the multi-task posterior p(θ|D) = p(θ|D1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , D5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This problem is called Split-MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (5) we require that our model with parameters θ is a sufficient statistic of D1, making the likelihood conditionally independent of D1 given θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This observation motivates the use of high-capacity predictors, such as Bayesian neural networks, that are flexible enough to learn D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='3 Variational Continual Learning Variational CL (VCL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (2017)) simplifies the Bayesian inference problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (5) into a sequence of approximate Bayesian updates on the distribution over random neural network weights θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' To do so, VCL uses the variational posterior from previous tasks as a prior for new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In this way, learning to solve the first task entails finding a variational distribution q1(θ|D1) that maximizes a corresponding variational objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For the subsequent task, the prior is chosen to be q1(θ|D1), and the goal becomes to learn a variational distribution q2(θ|D2) that maximizes a corresponding variational objective under this prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Denoting the recursive posterior inferred from multiple datasets by qt(θ|D1:t), we can express the variational CL objective for the t-th task as: L(θ, Dt) = DKL [qt(θ)||qt−1(θ|D1:t−1)] − Eqt[log p(Dt|θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (6) When applying VCL to the problem of Split-MNIST Figure 1, we can see that single-headed VCL barely performs better than SGD when remembering past tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Multi-headed VCL performs better, despite not being a requirement from sequential Bayesian inference Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' So why does single-head VCL not improve over SGD if we can recursively build up an approximate posterior using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (5)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We hypothesize that it could be due to using a variational approximation of the posterior and so we are not actually strictly performing the Bayesian CL process described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We test this hypothesis in the next section by propagating the true BNN posterior to verify whether we can recursively obtain the true multi-task posterior and so improve on single-head VCL and prevent catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 3 3 Bayesian Continual Learning with Hamiltonian Monte Carlo To perform inference over BNN weights we use the HMC algorithm (Neal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We then use these samples and learn a density estimator that can be used as a prior for a new task1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' HMC is considered the gold standard in approximate inference and is guaranteed to asymptotically produce samples from the true posterior2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' we use posterior samples of θ from HMC and then fit a density estimator over these samples, to use as a prior for a new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This allows us to use a multi-modal posterior distribution over θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In contrast, to a diagonal Gaussian variational posterior like in VCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' More concretely, to propagate the posterior p(θ|D1) we use a density estimator, defined ˆp(θ|D1), to fit a probability density on HMC samples as a posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For the next task T2 we can use ˆp(θ|D1) as a prior for a new HMC sampling chain and so on (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The density estimator priors need to satisfy two key conditions for use within HMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Firstly, that they are a probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Secondly, that they are differentiable with respect to the input samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Figure 2: Illustration of the posterior propagation pro- cess;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' priors in blue are in the top row and posterior samples on the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This is a two step process where we first perform HMC with an isotropic Gaus- sian prior for T1 then perform density estimation on the HMC samples from the posterior to obtain ˆp1(θ|D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This posterior can then be used as a prior for the new task T2 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We use a toy dataset (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 3) with two classes and inputs x ∈ R2 (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Each task is a binary classification problem where the decision boundary extends from left to right for each new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We train a two layer BNN, with hidden state size of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We use a Gaussian Mixture Models (GMM) as a density estimator for approximating the posterior with HMC samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We also tried Nor- malizing Flows which should be more flexible (Dinh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2016) however these did not work robustly for HMC sampling3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' To the best of our knowledge we are the first to incorporate flexible priors into the sampling methods like HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Training a BNN with HMC on the same multi- task dataset gets a test accuracy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Thus, the final posterior is suitable for continual learning un- der Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (3) we should be able to recursively arrive at the multi-task posterior with our recursive infer- ence method with HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The results from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 3 demonstrate that using HMC with an approximate multi-modal posterior fails to prevent forgetting and is less effective than using multi-head VCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In fact, multi-head VCL clearly outperforms HMC indicating that the source of the knowledge retention is not through the propagation of the posterior but through the task specific heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We also repeated these experiments with another toy dataset of five binary classification tasks where we observe similar results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For HMC we ensure that we are sampling from the posterior by assessing chain convergence and effective sample sizes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The effective sample size measures the autocorrelation in the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The effective sample sizes for the HMC chains for our BNNs are similar to the literature (Cobb & Jalaian, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Also, we ensure that our GMM approximate posteriors are multi-modal and so has a more complex posterior in comparison to VCL, and that the GMM samples produce equivalent results to HMC samples for the current task (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' See Appendix B for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We are not able to perform sequential Bayesian inference in BNNs despite using HMC which is considered the gold standard of Bayesian deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' HMC and density estimation with a GMM produces richer, accurate and multi-modal posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Despite this we are still not able to sequentially build up the multi-task posterior or get much better results than an isotropic Gaussian posterior like single-head VCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The weak 1We considered Sequential Monte Carlo, but it is unable to scale to the dimensions required for the NNs we consider (Chopin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' HMC on the other hand has recently been successfully scaled to BNNs (Cobb & Jalaian, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Izmailov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 2In the NeurIPS 2021 Bayesian Deep Learning Competition, the goal was to find an approximate inference method that is as “close” as possible to the posterior samples from HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 3RealNVP was very sensitive to the choice of random seed, the samples from the learned distribution did not give accurate predictions for the current task and led to numerical instabilities when used as a prior within HMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 4 0 1 2 x1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 x2 Task 1 Task 2 Task 3 Task 4 Task 5 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Accuracy Task 1 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Task 2 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Task 3 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Task 4 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Task 5 HMC MH VCL SH VCL SGD MT SGD/HMC Figure 3: On the left is the toy dataset of 5 distinct 2-way classification tasks which involve classifying circles and squares (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Also, continual learning binary classification test accuracies over 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The pink solid line is a multi-task (MT) baseline accuracy using SGD/HMC with the same model as for the CL experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' point of this method is the density estimation, the GMM removes probability mass over areas of the BNN weight space posterior which is important for the new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This demonstrates just how difficult a task it is to model BNN weight posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In the next section, we study a different analytical example of sequential Bayesian inference and look at how model misspecification and task data imbalances can cause forgetting in Bayesian CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 4 Bayesian Continual Learning and Model Misspecification 0 100 200 t 1 0 1 A 0 100 200 t B t task 1 data task 2 data Figure 4: Posterior estimate of the filtering dis- tribution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (7) for two different scenarios with two tasks or changepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We now consider a simple analytical example where we can perform the sequential Bayesian inference Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (3) in closed form using conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We consider a simple setting where data points arrive online, one after another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Observations y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , yt arrive online, each observation is generated by a hidden variable θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , θt ∼ p where p is a probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' At time t we wish to infer the filtering distribution p(θt|y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , yt) (Doucet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2001) using sequential Bayesian inference, similarly to the Kalman filter (Kalman, 1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The likelihood is p(yt|θt) = N(yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' f( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θt), σ2) such that yt = f( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θt) + ϵ where ϵ ∼ N(0, σ2) and f( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θt) = θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We consider a Gaussian prior over the mean parameters θ such that p(θ0) = N(θ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 0, σ2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Since the conjugate prior for the mean is also Gaussian, the prior and posterior are N(θt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ˆθt−1, ˆσ2 t−1) and N(θt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ˆθt, ˆσ2 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' By using sequential Bayesian inference we can have closed-form update equations for our posterior parameters: ˆθt = ˆσ2 t � yt σ2 + ˆθt−1 ˆσ2 t−1 � = ˆσ2 t � t � i=1 yi σ2 + ˆθ0 ˆσ2 0 � , 1 ˆσ2 t = 1 σ2 + 1 ˆσ2 t−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (7) From Equation (7) the posterior mean follows a Gaussian distribution where the posterior mean is a sum of the online observation and the online prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' So the posterior mean is a weighted sum of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' So, if the observations are non-stationary: if there is task change (more commonly referred to as a changepoint in the time-series literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Then the mean parameter will encapsulate a global mean over both tasks rather than a mean for each task Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The model is clearly misspecified since a single parameter cannot model both of these tasks together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Despite performing exact inference a misspecified model can forget, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In the case of HMC we verified that our Bayesian neural network had perfect performance on all tasks beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In Section 3 we had a well specified model but struggled with exact sequential Bayesian inference Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' With this 1-d online learning scenario we are performing exact inference, however we have a misspecified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' It is important to disentangle model misspecification and exact inference, and highlight that model misspecification is a caveat which has not been highlighted in the CL literature as far as we are aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Furthermore we can only ensure that our models are well specified if we have access to data from all tasks a 5 priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' So in the scenario of online continual learning (De Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2021) we cannot know if our model will perform well on all past and future tasks without making assumptions on the task distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 5 Sequential Bayesian Inference and Imbalanced Task Data Neural Networks are complex models with a broad hypothesis space and hence are a suitably well-specified model when tackling continual learning problems (Wilson & Izmailov, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' However we struggle to fit the posterior samples from HMC to perform sequential Bayesian inference Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We continue to use Bayesian filtering and assume a Bayesian NN where the posterior is Gaussian with a full covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' By modelling the entire covariance we enable modelling how each individual weight varies with respect to all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We do this by interpreting online learning in Bayesian NNs as filtering (Ciftcioglu & Türkcan, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Our treatment is similar to Aitchison (2018) who derives an optimizer by leveraging Bayesian filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We consider inference in the graphical model depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The aim is to infer the optimal BNN weights, θ∗ t at t given a single observation and the BNN weight prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The previous BNN weights are used as a prior for inferring the posterior BNN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We consider the online setting where a single data point (xt, yt) is observed at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Instead of modelling the full covariance we instead consider each parameter θi as a function of all the other parameters θ−it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We also assume that the values of the weights are close to those of the previous timestep (Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' To obtain the update equations for BNN parameters given a new observation and prior we make two simplifying assumptions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For a Bayesian neural network with output f(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ) and likelihood L(xt, yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ), the derivative evaluated at θt is zt = ∂L(xt, yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ)/∂θ|θ=θt and the Hessian is H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We assume a quadratic loss for a data point (xt, yt) of the form: L(xt, yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ) = Lt(θ) = −1 2θ⊤Hθ + z⊤ t θ, (8) the result of a second-order Taylor expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The Hessian is assumed to be constant with respect to (xt, yt) (but not with respect to θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ∗ t−1 θ∗ t−2 θ∗ t θ∗ t+1 yt−1 yt−2 yt yt+1 xt−1 xt−2 xt xt+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Figure 5: Graphical model for filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Grey and white nodes are observed and latent variables respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' To construct the dynamical equation for θ, consider the gradient for the i-th weight while all other param- eters are set to their current estimate at the optimal value for the θ∗ it: θ∗ it = − 1 Hii H⊤ −iiθ−it, (9) since zit = 0 at a mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The equation above shows us that the dynamics of the optimal weight θ∗ it is dependent on all the other current values of the pa- rameters θ−it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The dynamics of θ−it will be a com- plex stochastic process dependent on many different variables: such as the dataset, model architecture, learning rate schedule, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Since reasoning about the dynamics of θ−it are intractable, we assume that at the next time-step the optimal weights are close to the previous timesteps with a discretized Ornstein-Uhlenbeck process for the weights θ−it with reversion speed ϑ ∈ R+ and noise variance η2 −i: p(θ−i,t+1|θ−i,t) = N((1 − ϑ)θ−it, η2 −i), (10) this implies that the dynamics for the optimal weight is defined by p(θ∗ i,t+1|θ∗ i,t) = N((1 − ϑ)θ∗ it, η2), (11) where η2 = η2 −iH⊤ −iiH−ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 6 In simple terms, in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 we assume a parsimonious model of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' That the next value of θ−i,t is close to their previous value according to a Gaussian, similarly to Aitchison (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Under Assumptions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 the dynamics and likelihood are Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Thus we are able to infer the posterior distribution over the optimal weights using Bayesian updates and by linearizing the BNN the update equations for the posterior of the mean and variance of the BNN for a new data point are: µt,post = σ2 t,post � µt,prior σ2 t,prior(η2) + yt σ2 g(xt) � and 1 σ2 t,post = g(xt)2 σ2 + 1 σ2 t,prior(η2), (12) where we drop the notation for the i-th parameter, the posterior is N(θ∗ t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µt,post, σ2 t,post) and g(xt) = ∂f(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='θ∗ it) ∂θ∗ it and σ2 t,prior is a function of η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' See Appendix E for the derivation of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (12) we can notice that the posterior mean depends linearly on the prior and a data dependent term and so will behave similarly to our previous example in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='1 and Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 then if there is a data imbalance between tasks in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (12), then the data dependent term will dominate the prior term if there is more data for the current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In Section 3 we showed that it is very difficult with current machine learning tools to perform sequential Bayesian inference for simple CL problems with small Bayesian NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' When we disentangle Bayesian inference and model misspecification we show showed that misspecified models can forget despite exact Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The only way to ensure that our model is well specified is to show that the multi-task posterior produces reasonable posterior predictive distributions p(y|x, D) = � p(y|x, D, θ)p(θ|D)dθ for one’s application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Additionally, in this section we have shown that if there is a task dataset size imbalance then we can get forgetting under certain assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 6 Related Work There has been a recent resurgence in the field of CL (Thrun & Mitchell, 1995) given the advent of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Methods which approximate sequential Bayesian inference Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (5) have been seminal in CL’s revival and have used a diagonal Laplace approximation (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The diagonal Laplace approximation have been enhanced by modelling covariances of between neural network weights in the same layer (Ritter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Instead of the Laplace approximation we can use a variational approximation for sequential Bayesian inference (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Zeno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Using richer priors has also been explored (Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Farquhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Kessler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Gaussian processes have also been applied to CL problems leveraging inducing points to retain previous task functions (Titsias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Kapoor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Bayesian methods which regularize weights have not matched up to the performance of experience replay based CL methods (Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020) in terms of accuracy on CL image classification benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Instead of regularizing high dimension weight spaces, regularizing task functions is a more direct approach to combat forgetting (Benjamin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In particular, one can leverage the duality between the Laplace approximation and Gaussian Processes to develop a functional regularization approach to Bayesian CL (Swaroop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2019) or using function-space variational inference (Rudner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 7 Prototypical Bayesian Continual Learning We have shown that sequential Bayes over NN parameters is very difficult (Section 3), and is only suitable for situations where the multi-task posterior is suitable for all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We now show that a more fruitful approach is to model the full data-generating process of the CL problem and we propose a simple and scalable approach for doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In particular, we represent classes by prototypes (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Rebuffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2017) to prevent catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We refer to this framework as Prototypical Bayesian Continual Learning, or ProtoCL for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This approach can be viewed as a probabilistic variant of iCarl (Rebuffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2017), which creates embedding functions for different classes which are simply class means and predictions are 7 made by nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ProtoCL also bears similarities to the few-shot learning model Probabilistic Clustering for Online Classification (Harrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2019), developed for few-shot image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ProtoCL models the generative CL process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We consider classes j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , J}, generated from a categorical distribution with a Dirichlet prior: yi,t ∼ Cat(p1:J), p1:J ∼ Dir(αt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (13) Images are embedded into a embedding space by an encoder, z = f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' w) with parameters w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The per class embeddings are Gaussian whose mean has a prior which is also Gaussian: zit|yit ∼ N(¯zyt, Σϵ), ¯zyt ∼ N(µyt, Λ−1 yt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (14) enc enc Embedding Space Embedding Space Figure 6: Overview of ProtoCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 6 for an overview of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' To alleviate forgetting in CL, ProtoCL uses a coreset of past task data to continue to embed past classes distinctly as prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The posterior distribution over class probabilities {pj}J j=1 and class embeddings {¯zyj}J j=1 is denoted in short hand as p(θ) with parameters ηt = {αt, µ1:J,t, Λ−1 1:J,t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ProtoCL models each class prototype but does not use task specific NN parame- ters or modules like multi-head VCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' By modeling a probabilistic model over an embedding space this al- lows us to use powerful embedding functions f( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' w) without having to parameterize them probabilisti- cally and so this approach will be more scalable than VCL, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' As the Dirichlet prior is conjugate with the Categorical distribution and likewise the Gaus- sian over prototypes with a Gaussian prior over the prototype mean, we can calculate posteriors in closed form and update the parameters ηt as new data is observed without using gradient based updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We optimize the model by maximizing the posterior predictive distribution and use a softmax over class probabil- ities to perform predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We perform gradient-based learning of the NN embedding function f( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' w) and update the parameters, ηt at each iteration of gradient descent as well, see Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Sequential updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We can obtain our parameter updates for the Dirichlet posterior by Categorical- Dirichlet conjugacy: αt+1,j = αt,j + Nt � i=1 I(yi t = j), (15) where Nt are the number of points seen during the update at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Also, due to Gaussian-Gaussian conjugacy the posterior for the Gaussian prototypes is governed by: Λyt+1 = Λyt + NyΣ−1 ϵ (16) Λyt+1µyt+1 = NyΣ−1 ϵ ¯zyt + Λytµyt, ∀yt ∈ Ct, (17) where Ny are the number of samples of class y and ¯zyt = (1/Ny) �Ny i=1 zyi, see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 for the detailed derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We optimize the posterior predictive distribution of the prototypes and classes: p(z, y) = � p(z, y|θt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ηt)p(θt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ηt)dθt = p(y) Nt � i=1 N(zit|yit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µyt,t, Σϵ + Λ−1 yt,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (18) 8 Algorithm 1 ProtoCL continual learning 1: Input: task datasets T1:T , initialize embedding function: f( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' w), coreset: M = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 2: for T1 to TT do 3: for each batch in Ti ∪ M do 4: Optimize f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' w) by maximizing the posterior predictive p(z, y) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (18) 5: Obtain posterior over θ by updating η, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (15) to (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 6: end for 7: Add random subset from Ti to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 8: end for Where the p(y) = αy/ �J j=1 αj, see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='3 for the detailed derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This objective can then be optimized using gradient based optimization for learning the prototype embedding function z = f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' To make a prediction for a test point x∗ the class with the maximum (log)-posterior predictive is chosen, where the posterior predictive is: p(y∗ = j|x∗, x1:t, y1:t) = p(y∗ = j|z∗, θt) = p(y∗ = j, z∗|θt) � i p(y = i, z∗|θt), (19) see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4 for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Preventing forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' As we wish to retain the class prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We make use of coresets: experience from previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' At the end of learning a task Tt, we retain subset Mt ⊂ Dt and augment each new task dataset to ensure that posterior parameters ηt and prototypes are able to retain previous task information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Class incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' In this CL setting we do not tell the CL agent which task it is currently training and evaluating with a task identifier τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' So we cannot use the task identifier to select a specific head to use for classifying a test point, or use the task identifier to condition the model in another way (such that p(y|x, τ) where τ is the task identifier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Also we require the CL agents to classify each class in the CL benchmarks, for example {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , 9} for Split-MNIST and Split CIFAR10 and not just {0, 1} for each task as commonly performed (these are called task-incremental and domain-incremental learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We believe that these are more realistic and the most important settings to evaluate on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For Split-MNIST and Split-FMNIST the baselines and ProtoCL all use two layer NNs with hidden state size of 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For Split-CIFAR10 and Split-CIFAR100, the baselines and ProtoCL use a four layer convolution neural network with two fully connected layers of size 512 similarly to Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For ProtoCL and all baselines which rely on replay we fix the size of the coreset to 200 points per task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For all ProtoCL models we allow the prior Dirichlet parameters to be learned and set their initial value to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='7 found by a random search over MNIST with ProtoCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' An important hyperparameter for ProtoCL is the embedding dimension of the Gaussian prototypes for Split-MNIST and Split-FMNIST this was set to 128 while for the larger vision datasets this was set to 32 found using grid-search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Table 1: Mean accuracies across all tasks over CL vision benchmarks for class incremental learning (Van de Ven & Tolias, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' All results are averages and standard errors over 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ∗Uses the predictive entropy to make a decision about which head for class incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Method Coreset Split-MNIST Split-FMNIST VCL (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2017) \x17 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='08 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='77 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='25 + coreset \x13 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='98 ± 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='56 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='12 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='96 HIBNN∗ (Kessler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2019) \x17 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='50 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='20 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='70 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='21 FROMP (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020) \x13 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 S-FSVI (Rudner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2022b) \x13 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='17 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='41 ProtoCL (ours) \x13 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='73 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='05 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='73 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='70 9 Table 2: Mean accuracies across all tasks over CL vision benchmarks for class incremental learning (Van de Ven & Tolias, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' All results are averages and standard errors over 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ∗Uses the predictive entropy to make a decision about which head for class incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Training times have been benchmarked using an Nvidia RTX3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Method Training time (sec) (↓) Split CIFAR-10 (acc) (↑) FROMP (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020) 1425 ± 28 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='92 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='86 S-FSVI (Rudner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2022b) 44434 ± 91 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='85 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='87 ProtoCL (ours) 384 ± 6 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='81 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='10 Split CIFAR-100 (acc) S-FSVI (Rudner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2022b) 37355 ± 1135 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='04 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='37 ProtoCL (ours) 1425 ± 28 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='96 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='34 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ProtoCL produces good results on CL benchmarks on par or better than S-FSVI (Rudner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2022b) which is state-of-the-art on the smaller CL benchmarks while being a lot more efficient to train and without requiring expensive variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ProtoCL can flexibly scale to larger CL vision benchmarks producing better results than S-FSVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Code to reproduce all experiments can be found here anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='science/r/bayes_cl_exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' All our experiments are in the more realistic class incremental learning setting, which is a harder setting than those reported in most CL papers, so the results in Table 1 are lower for certain baselines than in the respective papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We use 200 data points per task, see Figure 12 for a sensitivity analysis of the performance over the Split-MNIST benchmark as as function of core size for ProtoCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The stated aim of ProtoCL is not provide a novel state-of-the-art method for CL, but rather to propose a simple baseline which takes an alternative route than weight-space sequential Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We can achieve strong results that mitigate forgetting, namely by modeling the generative CL process and using sequential Bayesian inference over a few parameters in the class prototype embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We argue that modeling the generative CL process is a fruitful direction for further research rather than attempting sequential Bayesian inference over the weights of a BNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 8 Discussion & Conclusion In this paper we have revisited the use of sequential Bayesian inference for CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We can use sequential Bayes to recursively build up the multi-task posterior Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Previous methods have relied on approximate inference and see little benefit over SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We test the hypothesis of whether this poor performance is due to the approximate inference scheme by using HMC in two simple CL problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' HMC asymptotically samples from the true posterior and we use a density estimator over HMC samples to use as a prior for a new task within the HMC sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This density is multi-modal and accurate with respect to the current task but is not able to improve over using an approximate posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This demonstrates just how challenging it is to work with BNN weight posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The source of error comes from the density estimation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We then look at an analytical example of sequential Bayesian inference where we perform exact inference however due to model misspecification, we observe forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The only way to ensure a well specified model is to assess the multi-task performance over all tasks a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This might not be possible in online CL settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We then model an analytical example over Bayesian NNs and under certain assumptions show that if there is task data imbalances then this will cause forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Because of these results, we argue against performing weight space sequential Bayesian inference and instead model the generative CL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We introduce a simple baseline called ProtoCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ProtoCL doesn’t require complex variational optimization and achieves competitive results to state-of-the-art in the realistic setting of class incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This conclusion should not be a surprise since the latest Bayesian CL papers have all relied multi-head architectures or inducing points/coresets to prevent forgetting, rather than better weight-space inference schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Our observations are in line with recent theory from (Knoblauch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020) which states that 10 optimal CL requires perfect memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Although the results were shown with deterministic NNs the same results follow for BNN with a single set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Future research directions include enabling coresets of task data to efficiently and accurately approximate the posterior of a BNN to remember previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 9 Acknowledgments We would like to thank Sebastian Farquhar, Laurence Aitchison, Jeremias Knoblauch and Chris Holmes for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Thanks to Phil Ball for help with writing the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' SK acknowledges funding from the Oxford- Man Institute of Quantitative Finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' TGJR acknowledges funding from the Rhodes Trust, Qualcomm, and the Engineering and Physical Sciences Research Council (EPSRC).' metadata={'source': 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arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='02099, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Sebastian Thrun and Tom M Mitchell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Lifelong robot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Robotics and autonomous systems, 15(1-2): 25–46, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Michalis K Titsias, Jonathan Schwarz, Alexander G de G Matthews, Razvan Pascanu, and Yee Whye Teh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Functional regularisation for continual learning with Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='11356, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Gido M Van de Ven and Andreas S Tolias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Three scenarios for continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='07734, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Andrew G Wilson and Pavel Izmailov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Bayesian deep learning and a probabilistic perspective of generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Advances in neural information processing systems, 33:4697–4708, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Chen Zeno, Itay Golan, Elad Hoffer, and Daniel Soudry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Task agnostic continual learning using online variational bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' arXiv preprint arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='10123, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 13 2 1 0 1 2 x1 2 1 0 1 2 x2 Task 1 Task 2 Task 3 Task 4 Task 5 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Accuracy Task 1 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Task 2 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Task 3 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Task 4 1 2 3 4 5 Tasks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Task 5 HMC MH VCL SH VCL SGD MT SGD/HMC Figure 7: Continual learning binary classification accuracies from the toy Gaussian dataset similar to (Henning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2021) using 10 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The pink solid line is a multi-task (MT) baseline test accuracy using SGD/HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Supplementary Material A The Toy Gaussians Dataset See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 7 for a visualization of the toy Gaussians dataset which we use as a simple CL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This is used for evaluating our method for propagating the true posterior by using HMC for posterior inference and then using a density estimator on HMC samples as a prior for a new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We construct 5, 2-way classification problems for CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Each 2-way task involves classifying adjacent circles and squares Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' With a 2 layer network with 10 neurons we get a test accuracy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 for the multi-task learning of all 5 tasks together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Hence according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (3) a BNN with the same size should be able to learn all 5 binary classification tasks continually by sequentially building up the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' B HMC implementation details We set the prior for T1, to p1(θ) = N(0, τ −1I) with τ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We burn-in the HMC chain for 1000 steps and sample for 10000 more steps and run 20 different chains to obtain samples from our posterior, which we then pass to our density estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We use a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='001 and trajectory length of L = 20, see Appendix C for further implementation details of the density estimation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For the GMM we optimize for the number of components by using a holdout set of HMC samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' C Density Estimation Diagnostics We provide plots to show that the HMC chains indeed sample from the posterior have converged in Figure 9 and Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We run 20 HMC sampling chains and randomly select one chain to plot for each seed (of 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We run HMC over 10 seeds and aggregate the results Figure 3 and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The posteriors p(θ|D1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' are approximated with a GMM and used as a prior for the second task and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We provide empirical evidence to show that the density estimators have fit to HMC samples of the posterior in Figure 8 and Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Where we show the number of components of the GMM density estimator which we use as a prior for a new task are all multi-modal posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We show the BNN accuracy when sampling BNN weights from our GMM all recover the accuracy of the converged HMC samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The effective sample size (ESS) of the 20 chains which is a measure of how correlated the samples are (higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The reported ESS values for our experiments are in line with previous work which uses HMC for BNN inference (Cobb & Jalaian, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 14 1 2 3 4 5 Task 0 10 20 30 40 50 ESS 1 2 3 4 5 Task 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' sampled from GMM posterior 1 2 3 4 5 Task 0 5 10 15 20 25 # GMM components in posterior Figure 8: Diagnostics from using a GMM prior fit to samples of the posterior generated from HMC, all results are for 10 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Left, effective sample sizes (ESS) of the resulting HMC chains of the posterior, all are greater than those reported in other works using HMC for BNNs (Cobb & Jalaian, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Middle, the accuracy of the BNN when using samples from the GMM density estimator instead from the samples from HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Right, The optimal number of components of each GMM posterior fitted with a holdout set of HMC samples by maximizing the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='00 Accuracy Task 1 Task 2 Task 3 Task 4 Task 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='6 nll Iteration number Figure 9: Convergence plots from a one randomly sampled HMC chain (of 20) for each task over 10 different runs (seeds) for 5 tasks from the toy Gaussians dataset similar to Henning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (2021) (visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We use a GMM density estimator as the prior conditioned on previous task data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 1 2 3 4 5 Task 0 10 20 30 40 50 ESS 1 2 3 4 5 Task 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 Task Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 1 2 3 4 5 Task 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' sampled from GMM posterior 1 2 3 4 5 Task 0 5 10 15 20 25 # GMM components in posterior Figure 10: Diagnostics from using a GMM to fit samples of the posterior HMC samples, all results are for 10 random seeds on the toy dataset from Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (2020) (and visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Left, effective sample sizes (ESS) of the resulting HMC chains of the posterior, all are greater than those reported in other works using HMC for BNNs (Cobb & Jalaian, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Middle left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' the current task accuracy from HMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Middle right, the accuracy of the BNN when using samples from the GMM density estimator instead of the converged HMC samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Right, The optimal number of components of each GMM posterior fitted with a holdout set of HMC samples by maximizing the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 Accuracy Task 1 Task 2 Task 3 Task 4 Task 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 NLL HMC Iteration Figure 11: Convergence plots from a randomly sampled HMC chain (of 20) for each task over 10 different seeds for 5 tasks from the toy dataset from (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2020) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 3 for a visualization of the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We use a GMM density estimator as a prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' D Prototypical Bayesian Continual Learning ProtoCL models the generative process of CL where new tasks are comprised of new classes j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , J} of a total of J and can be modeled by using a categorical distribution with a Dirichlet prior: yi,t ∼ Cat(p1:J), p1:J ∼ Dir(αt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='1) We learn a joint embedding space for our data with a NN, z = f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' w) with parameters w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The embedding space for each class is Gaussian whose mean has a prior which is also Gaussian: zit|yit ∼ N(¯zyt, Σϵ), ¯zyt ∼ N(µyt, Λ−1 yt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2) By ensuring that we have an embedding per class and using a memory of past data, we ensure that the embedding does not drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The posterior parameters are ηt = {αt, µ1:J,t, Λ−1 1:J,t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='1 Inference As the Dirichlet prior is conjugate with the Categorical distribution and so is the Gaussian distribution with a Gaussian prior over the mean of the embedding, then we can calculate posteriors in closed form and update our parameters as we see new data online without using gradient based updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We perform gradient based learning of the NN embedding function f( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' w) with parameters w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We optimize the model by maximizing the log-predictive posterior of the data and use the softmax over class probabilities to perform predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The posterior over class probabilities {pj}J j=1 and class embeddings {¯zyj}J j=1 is denoted as p(θ) for short hand and has parameters are ηt = {αt, µ1:J,t, Λ−1 1:J,t} are updated in closed form at each iteration of gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 Sequential updates We can obtain our posterior: p(θt|Dt) ∝ p(Dt|θt)p(θt) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='3) = Nt � i=1 p(zi t|yi t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ¯zyt, Σϵ,yt)p(yi t|p1:J)p(pi:J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' αt)p(¯zyt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µyt,t, Λ−1 yt,t) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4) = N(µt+1, Σt+1)Dir(αt+1), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='5) 16 where Nt is the number of data points seen during update t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Concentrating on the Categorical-Dirichlet conjugacy: Dir(αt+1) ∝ p(p1:J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' αt) Nt � i=1 p(yi t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' pi:J) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='6) ∝ J � j=1 pαj−1 j Nt � i=1 J � j=1 pI(yi t=j) j (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='7) = J � j=1 p αj−1+�Nt i=1 I(yi t=j) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='8) Thus: αt+1,j = αt,j + Nt � i=1 I(yi t = j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='9) Also, due to Gaussian-Gaussian conjugacy, then the posterior for the Gaussian prototype of the embedding for each class is: N(µt+1, Λt+1) ∝ Nt � i=1 N(zi t|yi t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ¯zyt, Σϵ)N(¯zyt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µyt,t, Λ−1 yt ) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='10) = � yt∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=',J} N(zyt|yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ¯zyt, 1 Nyt Σϵ)N(¯zyt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µyt+1, Λ−1 yt ) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='11) = � yt∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=',J} N(¯zyt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µt+1, Λ−1 yt+1), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='12) where Nyt is the number of points of class yt from the set of all classes C = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The update equations for the mean and variance of the posterior are: Λyt+1 = Λyt + NytΣ−1 ϵ , ∀yt ∈ Ct (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='13) Λyt+1µyt+1 = NytΣ−1 ϵ ¯zyt + Λytµyt, ∀yt ∈ Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='14) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='3 ProtoCL Objective The posterior predictive distribution we want to optimize is: p(z, y) = � p(z, y|θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' η)p(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' η)dθ, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='15) where p(θ) denotes the distributions over class probabilities {pj}J j=1 and mean embeddings {¯zyj}J j=1, p(z, y) = � Nt � i=1 p(zit|yit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' ¯zyt, Σϵ)p(yit|p1:J)p(p1:J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' αt)p(¯zyt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µyt,t, Λ−1 yt,t)dp1:Jd¯zyt (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='16) = � Nt � i=1 p(zit|yit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' zyt, Σϵ)p(¯zyt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µyt,t, Λ−1 yt,t)d¯zyt � Nt � i=1 p(yit|p1:J)p(p1:J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' αt)dp1:J � �� � � i p(yi)=p(y) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='17) = p(y) Nt � i=1 Z−1 i � N(¯zyit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' c, C)d¯zyt (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='18) = p(y) Nt � i=1 N(zit|yit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µyt,t, Σϵ + Λ−1 yt,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='19) 17 Figure 12: Split-MNIST average test accuracy over all 5 tasks for different memory sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Accuracies are over 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 0 250 500 750 1000 1500 2000 mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' size 40 50 60 70 80 90 Accuracy Where in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='18) we use §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='8 in (Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The term p(y) is: p(y) = � p(y|p1:J)p(p1:J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' αt)dp1:J (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='20) = � py Γ(�J j=1 αj) �J j=1 Γ(αj) J � j=1 pαj−1 j dp1:J (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='21) = Γ(�J j=1 αj) �J j=1 Γ(αj) � J � j=1 pI(y=j)+αj−1 j dp1:J (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='22) = Γ(�J j=1 αj) �J j=1 Γ(αj) �J j=1 Γ(I(y = j) + αj) Γ(1 + �J j=1 αj) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='23) = \x18\x18\x18\x18\x18\x18 Γ(�J j=1 αj) �J j=1 Γ(αj) �J j=1 Γ(I(y = j) + αj) �J j=1 αj\x18\x18\x18\x18\x18\x18 Γ(�J j=1 αj) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='24) = �J j=1,j̸=y Γ(αj) �J j=1 Γ(αj) Γ(1 + αy) �J j=1 αj (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='25) = �J j=1,j̸=y Γ(αj) �J j=1 Γ(αj) αyΓ(αy) �J j=1 αj (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='26) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='27) where we use the identity Γ(n + 1) = nΓ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4 Predictions To make a prediction for a test point x∗: p(y∗ = j|x∗, x1:t, y1:t) = p(y∗ = j|z∗, θt) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='28) = p(z∗|y∗ = j, θt)p(y∗ = j|θt) � i p(z∗|y∗ = i, ηt)p(y∗ = i|θt) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='29) = p(y∗ = j, z∗|θt) � i p(y = i, z∗|θt), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='30) where θt are sufficient statistics for (x1:t, y1:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='5 Dirichlet p class prob: 1 class prob: 2 class prob: 3 class prob: 4 class prob: 5 1 2 3 4 5 Task 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='5 Dirichlet p class prob: 6 1 2 3 4 5 Task class prob: 7 1 2 3 4 5 Task class prob: 8 1 2 3 4 5 Task class prob: 9 1 2 3 4 5 Task class prob: 10 Figure 13: The evolution of the Dirichlet parameters αt for each class in Split-MNIST tasks for ProtoCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' All αj are shown over 10 seeds with ±1 standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' By the end of training all classes are roughly equally likely, as we have trained on equal amounts of all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Preventing forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' As we wish to retain the task specific prototypes, at the end of learning a task Tt we take a small subset of the data as a memory to ensure that posterior parameters and prototypes do not drift, see Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='5 Experimental Setup The prototype variance, Σϵ is set to a diagonal matrix with the variances of each prototype set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The prototype prior precisions, Λyt are also diagonals and initialized randomly and exponentiated to ensure a positive semi-definite covariance for the sequential updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The parameters αj ∀j are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='78 which was found by random search over the validation set on MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We also allow αj to be learned in the gradient update step in addition to the sequential update step (lines 4 and 5 Algorithm 1), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 13 to see the evolution of the αj or all classes j over the course of learning Split-MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For the Split-MNIST and Split-FMNIST benchmarks we use a NN with 2 layers of size 200 and trained for 50 epochs with an Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We perform a grid-search over learning rates, dropout rates and weight decay coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The embedding dimension is set to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' For the Split-CIFAR10 and Split-CIFAR100 benchmarks, we use the same network as Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (2020) which consists of 4 convolution layers and 2 linear layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We train the networks for 80 epochs for each task with the Adam optimizer with a learning rate of 1e − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The embedding dimension is set to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' All experiments are run a single GPU NVIDIA RTX 3090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' E Sequential Bayesian Estimation as Bayesian Neural Network optimization We shall consider inference in the graphical model depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The aim is to infer the optimal BNN weights, θ∗ t at time t given observations and the previous BNN weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We assume a Gaussian posterior over weights with full covariance hence we model interactions between all weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We shall consider the online setting where we see one data point (xt, yt) at a time and we will make no assumption as to whether the data comes from the same task or different tasks over the course of learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We set up the problem of sequential Bayesian inference as a filtering problem and we leverage the work of Aitchison (2018) which casts NN optimization as Bayesian sequential inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' We make the reasonable assumption that the distribution over weights is a Gaussian with full covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Since reasoning about the full covariance matrix of a BNN is intractable we instead consider the i-th parameter and reason about the dynamics of the optimal estimates θ∗ it as a function of all the other parameters θ−it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Each weight is functionally dependent on all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' If we had access to the full covariance of the parameters then we could reason about the unknown optimal weights θ∗ it given the values of all the other weights θ−it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' However, since 19 θ∗ t−1 θ∗ t−2 θ∗ t θ∗ t+1 yt−1 yt−2 yt yt+1 xt−1 xt−2 xt xt+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Figure 14: Graphical model of under which we perform inference in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Grey nodes are observed and white are latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' we do not have access to the full covariance another approach is to reason about the dynamics of θ∗ it given the dynamics of θ−it and assume that the values of the weights are close to those of the previous time-step (Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018) and so we cast the problem as a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Consider a quadratic loss of the form: L(xt, yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ) = Lt(θ) = −1 2θ⊤Hθ + z⊤ t θ, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='31) which we can arrive by simple Taylor expansion where H is the Hessian which is assumed to constant across data points but not across the parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' If the BNN output takes the form f(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ), then the derivative evaluated at θt is zt = ∂L(xt,yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='θ) ∂θ |θ=θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' To construct the dynamical equations for our weights, consider the gradient for a single datapoint: ∂Lt(θ) ∂θ = −Hθ + zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='32) If we consider the gradient for the i-th weight while all other parameters are set to their current estimate: ∂L(θi, θ−i) ∂θi = −Hiiθit − H⊤ −iiθ−it + zti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='33) When the gradient is set to zero we recover the optimal value for θit, denoted as θ∗ it: θ∗ it = − 1 Hii H⊤ −iiθ−it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='34) since zti = 0 at the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The equation above shows us that the dynamics of the optimal weight θ∗ it is dependent on all the other current values of the parameters θ−it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' That is, the dynamics of θ∗ it will be governed by the dynamics of the weights θ−it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' The dynamics of θ−it will be a complex stochastic process dependent on many different variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Since reasoning about the dynamics is intractable we instead assume a discretized Ornstein-Uhlenbeck process for the weights θ−it with reversion speed ϑ ∈ R+ and noise variance η2 −i: p(θ−i,t+1|θ−i,t) = N((1 − ϑ)θ−it, η2 −i), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='35) this implies that the dynamics for the optimal weight is defined by p(θ∗ i,t+1|θ∗ i,t) = N((1 − ϑ)θ∗ it, η2), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='36) where η2 = η2 −iH⊤ −iiH−ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This same assumption is made in Aitchison (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' This assumes a parsimonious model of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Together with our likelihood: p(yt|xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ∗ t ) = N(yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' f(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ∗ t ), σ2) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='37) 20 where f( · , θ) is a neural network prediction with weights θ, we can now define a linear dynamical system for the optimal weight θ∗ i by linearizing the the Bayesian NN (Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=', 2018) and by using the transition dynamics in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' Thus we are able to infer the posterior distribution over the optimal weights using Kalman filter like updates (Kalman, 1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' As the dynamics and likelihood are Gaussian, then the prior and posterior are also Gaussian, for ease of notation we drop the index i such that θ∗ it = θ∗ t : p(θ∗ t |(x, y)t−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , (x, y)1) = N(µt,prior, σ2 t,prior) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='38) p(θ∗ t |(x, y)t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , (x, y)1) = N(µt,post, σ2 t,post) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='39) By using the transition dynamics and the prior we can obtain closed form updates: p(θ∗ t |(x, y)t−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , (x, y)1) = � p(θ∗ t |θ∗ t−1)p(θ∗ t−1|(x, y)t−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' , (x, y)1)dθ∗ t−1 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='40) N(θ∗ t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µt,prior, σ2 t,prior) = � N(θ∗ t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (1 − ϑ)θ∗ t−1, η2)N(θ∗ t−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µt−1,post, σ2 t−1,post)dθ∗ t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='41) Integrating out θ∗ t−1 we can get updates for the prior for the next timestep as follows: µt,prior = (1 − ϑ)µt−1,post (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='42) σ2 t,prior = η2 + (1 − ϑ)−2σ2 t−1,post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='43) The updates for obtaining our posterior parameters: µt,post and σ2 t,post, comes from applying Bayes’ theorem: log N(θ∗ t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µt,post, σ2 t,post) ∝ log N(yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' f(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' θ∗ t ), σ2) + log N(θ∗ t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' µt,prior, σ2 t,prior), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='44) by linearizing our Bayesian NN such that f(xt, θ0) ≈ f(xt, θ0) + ∂f(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='θ∗ t ) ∂θ∗ t (θ∗ t − θ0) and by substituting into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='44) we obtain our update equation for the posterior of the mean of our BNN parameters: − 1 2σ2 t,post (θ∗ t − µt,post)2 = − 1 2σ2 (y − g(xt)θ∗ t )2 − 1 2σ2 t,prior (θ∗ t − µt,prior)2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='45) µt,post = σ2 t,post �µt,prior σ2 t,prior + y σ2 g(xt) � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='46) where g(xt) = ∂f(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='θ∗ t ) ∂θ∗ t , and the update equation for the variance of the Gaussian posterior is: 1 σ2 t,post = g(xt)2 σ2 + 1 σ2 t,prior .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='47) From our update equations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='46) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='47) we can notice that the posterior mean depends linearly on the prior and an additional data dependent term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' These equations are similar to the filtering example in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' So under certain assumptions above, a BNN should behave similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' If there exists a task data imbalance then the data term will dominate the prior term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content='46) and could lead to forgetting of previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAzT4oBgHgl3EQf3P7o/content/2301.01828v1.pdf'} diff --git a/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf b/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f6e63edc4b1923b0f93ca0417d7e1ae114d1eaea --- /dev/null +++ b/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2377a940075686233feb9b35b7f305f22832b6d54bdfde694ed9629b6aa220e5 +size 3660377 diff --git a/f9A0T4oBgHgl3EQfHv-A/vector_store/index.pkl b/f9A0T4oBgHgl3EQfHv-A/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..de5ecb3b71391e95f86b5831642bd66a6bd85e46 --- /dev/null +++ b/f9A0T4oBgHgl3EQfHv-A/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1fa3170dacc72ec4a95771a3b9582706dd9d4b4621401a76523b777eddecfbfa +size 296954 diff --git a/f9FKT4oBgHgl3EQftC6j/content/tmp_files/2301.11885v1.pdf.txt b/f9FKT4oBgHgl3EQftC6j/content/tmp_files/2301.11885v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..227ebb61703746c1fd4c262abde5d7d4917aae44 --- /dev/null +++ b/f9FKT4oBgHgl3EQftC6j/content/tmp_files/2301.11885v1.pdf.txt @@ -0,0 +1,2395 @@ +Algorithmic Stability of Heavy-Tailed SGD with General Loss +Functions +Anant Raj∗ +Coordinated Science Laboraotry +University of Illinois Urbana-Champaign. +Inria, Ecole Normale Supérieure +PSL Research University, Paris, France. +anant.raj@inria.fr +Lingjiong Zhu∗ +Department of Mathematics +Florida State University, FL, USA. +zhu@math.fsu.edu +Mert Gürbüzbalaban +Department of Management +Science and Information Systems +Rutgers University, Piscataway, USA. +Princeton University, NJ, USA +mg1625@princeton.edu +Umut Şimşekli +Inria, CNRS, Ecole Normale Supérieure +PSL Research University, Paris, France. +umut.simsekli@inria.fr +January 30, 2023 +Abstract +Heavy-tail phenomena in stochastic gradient descent (SGD) have been reported in several +empirical studies. Experimental evidence in previous works suggests a strong interplay between +the heaviness of the tails and generalization behavior of SGD. To address this empirical +phenomena theoretically, several works have made strong topological and statistical assumptions +to link the generalization error to heavy tails. Very recently, new generalization bounds have +been proven, indicating a non-monotonic relationship between the generalization error and +heavy tails, which is more pertinent to the reported empirical observations. While these bounds +do not require additional topological assumptions given that SGD can be modeled using a +heavy-tailed stochastic differential equation (SDE), they can only apply to simple quadratic +problems. In this paper, we build on this line of research and develop generalization bounds for +a more general class of objective functions, which includes non-convex functions as well. Our +approach is based on developing Wasserstein stability bounds for heavy-tailed SDEs and their +discretizations, which we then convert to generalization bounds. Our results do not require +any nontrivial assumptions; yet, they shed more light to the empirical observations, thanks to +the generality of the loss functions. +*These authors contributed equally to this work. +1 +arXiv:2301.11885v1 [stat.ML] 27 Jan 2023 + +1 +Introduction +Many supervised learning problems can be expressed as an instance of the risk minimization +problem +min +θ∈Rd {F(θ) := Ex∼D[f(θ, x)]} , +(1) +where x ∈ X is a random data point, distributed according to an unknown probability distribution +D and taking values in the data space X, θ denotes the parameter vector of the model to be +learned and f(θ, x) is the instantaneous loss of misprediction with parameters θ corresponding +to the data point x. With different choices of the function f, we can recover many problems in +supervised learning from deep learning to logistic regression or support vector machines [26]. +As D is unknown in many scenarios, directly attacking (1) is often not possible. Assuming +we have access to a training dataset Xn = {x1, . . . , xn} ⊂ X n with n independent and identically +distributed (i.i.d.) observations, in practice, we can consider the empirical risk minimization +(ERM) problem instead, given as follows: +min +θ∈Rd +� +ˆF(θ, Xn) := 1 +n +n +� +i=1 +f(θ, xi) +� +. +One of the most popular algorithms for attacking the ERM problem is stochastic gradient +descent (SGD) that is based on the following recursion: +θk+1 = θk − η∇ ˜Fk+1(θk, Xn), +(2) +where η is the step-size (or learning-rate) and +∇ ˜Fk(θ, X) := 1 +b +� +i∈Ωk +∇f(θ, xi) +is the stochastic gradient, with Ωk ⊂ {1, . . . , n} being a random subset drawn with or without +replacement, and b := |Ωk| ≪ n being the batch-size. +Understanding the generalization properties of SGD has been a major challenge in modern +machine learning. In this context, the goal is to bound the so-called generalization error: | ˆF(θ, Xn)− +F(θ)|, either in expectation or in high probability. +While a plethora of approaches have been proposed to address this task [5, 16, 19, 21], a +promising approach among those has been based on the theoretical and empirical observations which +showed that SGD can exhibit a heavy-tailed behavior, depending on the choice of hyperparameters +(η and b), the data distribution D, and the geometry of the loss function f [11, 13]. This has +motivated the use of ‘heavy-tailed proxies’ for SGD, which –to some extent– facilitated the analysis +of SGD in terms of its generalization error. Examples of such proxies include gradient descent +with additive heavy-tailed noise: +θk+1 = θk − η∇ ˆF(θk, Xn) + ξk+1, +(3) +where (ξk)k≥1 is a sequence of heavy-tailed random vectors, potentially with unbounded higher-order +moment, i.e., E∥ξk∥p = +∞ for some p > 1 (see e.g., [20, 29, 32]). +2 + +Another popular proxy for heavy-tailed SGD is based on a continuous-time version of (3), +which is expressed by the following stochastic differential equation (SDE): +dθt = −∇ ˆF(θt, Xn)dt + σdLα +t , +(4) +where σ > 0 is a scale parameter, Lα +t is a d-dimensional α-stable Lévy process, which has heavy- +tailed increments and will be formally defined in the next section1, and α ∈ (0, 2] denotes the +‘tail-exponent’ such that as α gets smaller the process Lα +t becomes heavier-tailed. +Within this mathematical framework, Şimşekli et al. [27] proved an upper-bound (which was +then improved in [14]) for the worst-case generalization error over the trajectories of (4). The +bound informally reads as follows: with probability at least 1 − δ, it holds that +sup +θ∈Θ +��� ˆF(θ, Xn) − F(θ) +��� ≲ +� +α + I(Θ, Xn) + log(1/δ) +n +, +where Θ denotes the trajectory of (4), i.e., +Θ := +� +θ ∈ Rd : ∃t ∈ [0, 1], θ = θt +� +, +with θt being the solution of (4), and I(Θ, Xn) denotes a form of ‘mutual information’ between +the trajectory Θ and the data sample Xn (cf. [31]). This result suggests that the generalization +error is essentially determined by two terms: (i) the tail exponent α, as the tails get heavier the +generalization error will be lower, (ii) the statistical dependency between the trajectory and the +data sample, the lower the dependency the better the generalization performance. +While these results illuminated an interesting connection between heavy-tails and generalization, +they unfortunately rely on nontrivial topological assumptions on Θ and the mutual information +term cannot be controlled in an interpretable way in general. On the other hand, Barsbey et al. +[2] empirically illustrated that the relation between the tail exponent and the generalization +error might not be monotonic in practical applications; an observation which cannot be directly +supported by the bound in [27] and [14]. +Aiming to alleviate these issues, very recently, Raj et al. [24] considered the same problem +from the lens of algorithmic stability [4, 12]. They considered the SDE (4) and further simplified +it by choosing the loss function as a simple quadratic, i.e., f(θ, x) = (θ⊤x)2. They showed that +any parameter vector θ provided by (4) (or its Euler-Maruyama discretization with small enough +small step-size) cannot be algorithmically stable. However, when the algorithmic stability is +measured by a surrogate loss function instead (reminiscent of [29]), the parameter vector θ becomes +algorithmically stable, which immediately implies generalization. Their bound further illustrated +that the relation between α and the generalization error might not be monotonic, which is in line +with the observations provided in [2]. +While the bounds in [24] do not require additional topological assumptions and do not contain +a mutual information term as opposed to [14, 27], their analysis technique heavily relies on the +fact that f is a quadratic, hence cannot be directly extended beyond quadratic loss functions. +In this paper, we aim at filling this gap and prove algorithmic stability bounds the SDE (4) +(and its Euler-Maruyama discretization) with general loss functions, which can be even non-convex. +Our contributions are as follows: +1This type of SDEs have also received some attention in terms of limits of deterministic gradient descent with +dynamical regularization [18]. +3 + +• We first focus on the continuous-time setting and prove Wasserstein stability bounds for two +SDEs of the form of (4) with different drift functions. Our results cover both the finite-time case, +i.e., t < ∞ and the stationary case, i.e., t → ∞. We build upon recently introduced stochastic +analysis tools for uniform-in-time Wasserstein error bounds for Euler-Maruyama discretization [6] +to obtain a novel Wasserstein stability bound for two α-stable Lévy-driven SDEs. Our analysis +relies on an additional pseudo-Lipschitz like condition for the underlying process and the dataset +(Assumption 3) and careful adaption of the tools in [6] to our context (Lemma 18 and Theorem 12 +in the Appendix) as well as additional analysis (Lemma 7) that allows us to characterize the +dependence of our bounds on the tail-index α. Our derived bounds would be interesting on their +own to a much broader scope. +• By following [22], we translate the derived Wasserstein stability bounds to algorithmic stability +bounds. Similar to [24], our approach necessitates surrogate loss functions to measure algorithmic +stability. Our results reveal that the relation between heaviness of the tail α and the generalization +error might not be monotonic, indicating that the conclusions of [24] extends to the general case. +• By combining our results with [6], we extend our bounds to the Euler-Maruyama discretization +of (4) (that is of the form of (3)) and show that for small enough step-sizes the discrete-time +process achieves almost identical stability bounds. +Contrary to [14, 18, 27], our bounds do not rely on any topological regularity assumptions and +they further do not contain a mutual information term. Moreover, our results shed more light +to the non-monotonic relation between heavy tails and the generalization error, as empirically +observed in [2, 24], since they are applicable to non-convex losses, as opposed to [24]. We also note +that our generalization bounds and Wasserstein bounds are independent of time. Such a result was +previously shown in [9] in the context of Brownian-motion driven SDEs and their discretizations, +our work uses different techniques considering Levy-driven SDEs and studies the link between the +generalization and the coefficient of heavy tail. +2 +Notations and Technical Background +Gradients and Hessians. For any twice continuously differentiable function f : Rd → R, we +denote by ∇f and ∇2f the gradient and the Hessian of f. First-order and second-order directional +derivatives of f are defined as +∇vf(x) := lim +ϵ→0 +f(x + ϵv) − f(x) +ϵ +, +∇v2∇v1f(x) := lim +ϵ→0 +∇v1f(x + ϵv2) − ∇v1f(x) +ϵ +, +(5) +for any directions v, v1, v2 ∈ Rd. If f is three times continuously differentiable, then third-order +derivatives along the directions v1, v2 are given by +∇v2∇v1∇f(x) := lim +ϵ→0 +∇v1∇f(x + ϵv2) − ∇v1∇f(x) +ϵ +. +(6) +Wasserstein distance. +For p ≥ 1, the p-Wasserstein distance between two probability +measures µ and ν on Rd is defined as [28]: +Wp(µ, ν) = {inf E∥X − Y ∥p}1/p , +(7) +4 + +where the infimum is taken over all coupling of X ∼ µ and Y ∼ ν. In particular, the 1-Wasserstein +distance has the following dual representation [28]: +W1(µ, ν) = +sup +h∈Lip(1) +���� +� +Rd h(x)µ(dx) − +� +Rd h(x)ν(dx) +���� , +(8) +where Lip(1) consists of the functions h : Rd → R that are 1-Lipschitz. +Algorithmic stability. Algorithmic stability is an important notion in learning theory, which +has pave the way for several important theoretical results [4, 12]. Let us first state the notion of +algorithmic stability as defined in [12]. +Definition 1 (Hardt et al. [12], Definition 2.1). For a (surrogate) loss function ℓ : Rd × X → R, +an algorithm A : �∞ +n=1 X n → Rd is ε-uniformly stable if +sup +X∼ += ˆ +X +sup +z∈X +E +� +ℓ(A(X), z) − ℓ(A( ˆX), z) +� +≤ ε, +(9) +where the first supremum is taken over data X, ˆX ∈ X n that differ by one element, denoted by +X ∼= ˆX. +Here, we intentionally use a different notation for the loss ℓ (as opposed to f), as our theory +will require the algorithmic stability to be measured by using a surrogate loss function, which +might be different than the original loss f. +We now provide a result from [12] which relates algorithmic stability to the generalization +performance of a randomized algorithm. +Before stating the result, similar to ˆF and F, we +respectively define the empirical and population risks with respect to the loss ℓ as follows: +ˆR(w, Xn) := 1 +n +n +� +i=1 +ℓ(w, xi), +R(w) := Ex∼D[ℓ(w, x)]. +Theorem 2 (Hardt et al. [12], Theorem 2.2). Suppose that A is an ε-uniformly stable algorithm, +then the expected generalization error is bounded by +���EA,Xn +� +ˆR(A(Xn), Xn) +� +− R(A(Xn)) +��� ≤ ε. +(10) +Alpha-stable distributions. A scalar random variable X is said to follow a symmetric +α-stable distribution, denoted by X ∼ SαS(σ), if its characteristic function takes the form: +E +� +eiuX� += exp (−σα|u|α), for any u ∈ R, where σ > 0 is known as the scale parameter that +measures the spread of X around 0 and for α ∈ (0, 2] which is known as the tail-index that +determines the tail thickness of the distribution. The tail becomes heavier as α gets smaller. The +α-stable distribution SαS appears as the limiting distribution in the generalized central limit +theorems for a sum of i.i.d. random variables with infinite variance [17]. The probability density +function of a symmetric α-stable distribution, α ∈ (0, 2], does not yield closed-form expression in +general except for a few special cases; for example SαS reduces to the Cauchy and the Gaussian +distributions, respectively, when α = 1 and α = 2. When 0 < α < 2, the moments are finite +only up to the order α in the sense that E[|X|p] < ∞ if and only if p < α, which implies infinite +variance. +Finally, α-stable distribution can be extended to the high-dimensional case for random vectors. +One of the most commonly used extension is the rotationally symmetric α-stable distribution. X +5 + +follows a d-dimensional rotationally symmetric α-stable distribution if it admits the characteristic +function E +� +ei⟨u,X⟩� += e−σα∥u∥α +2 for any u ∈ Rd. For further details of α-stable distributions, we +refer to [25]. +Lévy processes. Lévy processes are stochastic processes with independent and stationary +increments. Their successive displacements can be viewed as the continuous-time analogue of +random walks. Lévy processes include the Poisson process, the Brownian motion, the Cauchy +process, and more generally stable processes; see e.g. [1, 3, 25]. Lévy processes in general admit +jumps and have heavy tails which are appealing in many applications; see e.g. [7]. +In this paper, we will consider the rotationally symmetric α-stable Lévy process, denoted by +Lα +t in Rd and is defined as follows. +• Lα +0 = 0 almost surely; +• For any t0 < t1 < · · · < tN, the increments Lα +tn − Lα +tn−1 are independent; +• The difference Lα +t − Lα +s and Lα +t−s have the same distribution, with the characteristic function +exp(−(t − s)α∥u∥α +2 ) for t > s; +• Lα +t has stochastically continuous sample paths, i.e. for any δ > 0 and s ≥ 0, P(∥Lα +t −Lα +s ∥ > δ) → 0 +as t → s. +When α = 2, Lα +t = +√ +2Bt, where Bt is the standard d-dimensional Brownian motion. +3 +Main Results +In this section, we present our main theoretical results. To ease the notation, we will consider the +following SDE in lieu of (4): +dθt = −∇ ˆF(θt, Xn)dt + dLα +t , +(11) +in the rest of the paper2. +Our road map is as follows. We will first consider the continuous-time case (11), i.e., we will +set the learning algorithm as A(Xn) = θt for some t ∈ [0, +∞], where θ∞ denotes a sample from +the stationary distribution of the SDE (11). As our aim is to prove algorithmic stability bounds +for this choice of algorithm, we then consider another dataset ˆXn ∼= Xn, which differ from Xn by +one element, accordingly define the following SDE: +dˆθt = −∇ ˆF(ˆθt, ˆXn)dt + dLα +t , +(12) +such that A( ˆXn) = ˆθt. Then, we will argue that, for any time t, the laws of θt and ˆθt will be close +to each other in the 1-Wasserstein metric. +By considering a surrogate loss function ℓ, which we will assume to be L-Lipschitz, our bound +on the Wasserstein distance between Law(θt) and Law(ˆθt) (Theorem 5) will immediately provide +us a generalization bound thanks to the dual representation of the 1-Wasserstein distance (cf. [22, +Lemma 3]): +���Eθt,Xn +� +ˆR(θt, Xn) +� +− R(θt) +��� +2Note that the stationary distribution of (4) is the same as the stationary distribution of dθt += +−σ−α∇ ˆF(θt, Xn)dt + dLα +t , and so that we can easily adapt our main result (Theorem 5) to the general case +σ > 0. +6 + +≤ L +sup +Xn∼ += ˆ +Xn +W1 +� +Law(θt), Law(ˆθt) +� +, +(13) +where Law(θt) and Law(ˆθt) respectively depend on Xn and ˆXn due to the form of the SDEs. The +reason why we require a surrogate loss function is the fact that we need the Lipschitz continuity of +the loss to be able to derive the bound in (13). However, as we will detail in the next subsection, +our assumptions on the true loss f will be incompatible with the Lipschitz continuity of f. +After proving a generalization bound of the form (13), we will further investigate the behavior +of the bound with respect to the heaviness of the tail which is characterized by the tail-index +α. Finally, we will consider the discrete-time case, where we will show that almost identical +results hold for the Euler-Maruyama discretizations of (11) and (12), as long as a sufficiently small +step-size is chosen. +3.1 +Assumptions +In this section we state our main assumptions and we will assume that they hold throughout the +paper. For any w ∈ Rd, we use θw +t to denote the process θt that starts at θ0 = w. +Assumption 3. For every x ∈ X, there exist universal constants K1, K2 such that +∥∇f(θ, x) − ∇f(ˆθ, ˆx)∥ ≤ K1∥θ − ˆθ∥ + K2∥x − ˆx∥(∥θ∥ + ∥ˆθ∥ + 1). +(14) +This assumption is a pseudo-Lipschitz like condition (similar to the one in [8]) on the loss f. +Under this assumption, for two datasets Xn and ˆXn, we immediately have the following property: +���∇ ˆF(θ, Xn) − ∇ ˆF +� +ˆθ, ˆXn +���� ≤ K1∥θ − ˆθ∥ + ρ(Xn, ˆXn)K2 +� +∥θ∥ + ∥ˆθ∥ + 1 +� +, +(15) +where +ρ(Xn, ˆXn) := 1 +n +n +� +i=1 +∥xi − ˆxi∥. +(16) +We will show that the term ρ(Xn, ˆXn) will have an important role in terms of Wasserstein stability. +By following [6], we also make the following assumption. +Assumption 4. For every x ∈ X, f(·, x) is three-times continuously differentiable, and for any +θ1, θ2 ∈ Rd, there exist universal constants B, m, K, L, and M such that +∥∇f(0, x)∥ ≤ B, +⟨∇f(θ1, x) − ∇f(θ2, x), θ1 − θ2⟩ ≤ −m∥θ1 − θ2∥2 + K, +and +∥∇v∇f(θ, x)∥ ≤ L∥v∥, +∥∇v1∇v2∇f(θ, x)∥ ≤ M∥v1∥∥v2∥, +for any v, v1, v2 ∈ Rd. +The first part of this assumption is common in stochastic analysis and often referred to as +dissipativity [10, 23]. The second part of the assumption amounts to requiring the drift ∇f(x) to +have bounded third-order directional derivatives (see also [6]). This would be satisfied for instance +if f has bounded third-order derivatives on the set X. +7 + +3.2 +Continuous-Time Dynamics +Now, we are ready to state our first theorem that characterizes the 1-Wasserstein distance between +θt and ˆθt at any finite time t, which is uniform in t. As a result, we also obtain an upper-bound on +the 1-Wasserstein distance between the unique invariant distribution µ of (θt)t≥0 and the unique +invariant distribution ˆµ of (ˆθt)t≥0.3 +The full statement of the theorem is rather lengthy and is given in the Section B.1 in the +Appendix. For clarity, in the next theorem, we provide our upper-bound on the distance between +the invariant distributions, i.e., t → ∞. The finite t case is handled in the Appendix. +Theorem 5. Suppose that Assumptions 3 and 4 hold. +Denote by µ, ˆµ the unique invariant +distributions of (θt)t≥0 and (ˆθt)t≥0, respectively. Then, the following inequality holds: +W1(µ, ˆµ) ≤ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2 (2C0 + 1) , +(17) +where K1, K2 and L are defined in Assumption 3 and Assumption 4 and C0, C1 and λ are some +positive real constants. +This theorem shows that, as long as the datasets Xn and ˆXn are close to each other, i.e., +ρ(Xn, ˆXn) is small, the distance between the solutions of the SDEs (11) and (12) will be small as +well for any time t. This result can be seen as a heavy-tailed version of the results presented in +[9, 23]. +Generalization Bound. +By combining Theorem 5 and (13), we can now easily obtain general- +ization bound under a Lipschitz surrogate loss function. +Corollary 6. Suppose that Assumptions 3 and 4 hold. Assume that ℓ is L-Lipschitz in θ and +supx,y∈X ∥x − y∥ ≤ D for some D < ∞. Then the following inequality holds: +���Eθ∞,Xn +� +ˆR(θ∞, Xn) +� +− R(θ∞) +��� ≤ LD +� +C1λ−1eλ + 1 +� +eLK2 (2C0 + 1) +n +. +(18) +The proof of this corollary is straightforward, hence omitted. Similar to Theorem 5, we +presented Corollary 6 for the stationary case, where t → ∞; yet, we shall underline that our theory +holds for any finite time t. +Lower bounds on algorithmic stability have been discussed in [24] for Ornstein-Uhlenbeck +process with α-stable Levy noise. While comparing with the bound obtained in this work, we can +see that the obtained bound has optimal dependence on the number of samples n. +Next, we will investigate how the constants in Theorem 5 behave with respect to varying α. +Constants in Theorem 5. In Theorem 5, we provided an upper bound on W1(µ, ˆµ) which +depends on various quantities, and our next goal is to figure out how the parameters C1, λ, L, K2 +and C0 depend on the tail-index α. +First, we notice that the parameters L and K2 only depend on the loss function. Second, the +parameters C1, λ come from the 1-Wasserstein contraction in Lemma 15 in the Appendix which is +a restatement of Proposition 2.2 in [6]; that is, for any w, y ∈ Rd: +W1 (Law (θw +t ) , Law (θy +t )) ≤ C1e−λt∥w − y∥, +(19) +3Here, we know that under our assumptions by the results of [6], invariant distributions µ and ˆµ exist. +8 + +W1 +� +Law +� +ˆθw +t +� +, Law +� +ˆθy +t +�� +≤ C1e−λt∥w − y∥, +(20) +where θw +t to denote the process θt that starts at θ0 = w. Furthermore, Proposition 2.2 in [6] +follows from Theorem 1.2. in [30]. A careful look at Theorem 1.2. in [30] reveals that C1, λ are +independent of the tail-index α. +Finally, C0 depends on α and it comes from Lemma 14 in the Appendix which is a restatement +of Proposition 2.1. in [6]; that is, which says that for any w ∈ Rd: +E∥θw +t ∥ ≤ C0(1 + ∥w∥), +for any t > 0, +(21) +E∥ˆθw +t ∥ ≤ C0(1 + ∥w∥), +for any t > 0. +(22) +Notice that Proposition 2.1. in [6] does not provide an explicit formula for C0, and in the next +result, we provide a more refined estimate to spell out the dependence of C0 on the tail-index α. +Lemma 7. Suppose that Assumptions 3 and 4 hold. For any w ∈ Rd, we have +E∥θw +t ∥ ≤ C0(1 + ∥w∥), +for any t > 0, +(23) +E∥ˆθw +t ∥ ≤ C0(1 + ∥w∥), +for any t > 0, +(24) +where we can take +C0 := 3 + 2 (K + B) +m ++ 2α+1Γ +� d+α +2 +� +π−d/2σd−1 +|Γ(−α/2)|m +� √ +d +2 − α + +1 +α − 1 +� +, +(25) +where Γ(·) is the gamma function and σd−1 = 2π +d +2 /Γ(d/2) is the surface area of the unit sphere in +Rd, and K, B, m are defined in Assumption 4. +Hence, it follows from Theorem 5 and Lemma 7 that the dependence on the tail-index α is +only via the function: +g(α; d) := +2αΓ +� d+α +2 +� √ +d +|Γ(−α/2)|(2 − α) + +2αΓ +� d+α +2 +� +|Γ(−α/2)|(α − 1). +The next result formalizes how the function g(α; d) depends on the tail-index α. +Proposition 8. Let α0 := 2(c0 − 1) ∈ (0, 2), where c0 is the unique critical value in (1, 2) such +that the gamma function Γ(x) is increasing for any x > c0 and decreasing for any 1 < x < c0. +Then, the following holds. +(i) For any d ≥ d0, where d0 := max +� +2, +1 +(log 2)2(α0−1)4 +� +, the map α �→ g(α; d) is increasing in +α ∈ [α0, 2]. +(ii) For any fixed d ∈ N, the map α �→ g(α; d) is decreasing in α ∈ [1, α′ +0], where α′ +0 ≤ α0 is +defined as α′ +0 := min +� +α0, 1 + −1+√ +1+4y−1 +0 +√ +d +2 +√ +d +� +, with y0 := log(2) + 1 +2ψ(d + α +2 ) + 3−α0 +2−α0 , where ψ(·) +is the digamma function. +The proof is given in the Section B.3 in the Appendix. This result reveals an interesting +fact. Depending on the dimension of the parameter vector d, the Wasserstein stability bound +(Theorem 5) and the generalization bound (Corollary 6) exhibit different behaviors with respect to +varying α. We observe that for sufficiently large d, there exists a critical value α0 such that the +9 + +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +α +2 +4 +6 +8 +10 +ˆg(α, d) +d = 2 +d = 5 +d = 10 +d = 20 +Figure 1: Behavior of g(α; d) with respect to α. We scale g(α; d) appropriately to fit all the plots +in the same frame which we denote as ˆg(α; d). +bound is monotonically increasing for α ≥ α0. This suggests that for d large enough, increasing +the heaviness of the tails (i.e., smaller α) can be beneficial unless α is smaller than α0. +For visualization, we also provide a pictorial illustration of the function g(α; d) in Figure 1. +The figure shows the behavior of g(α; d) with respect to α for various dimensions d. The observed +non-globally monotonic behavior for large d indicates that the conclusions of [24] extend beyond +quadratic loss functions. +On the other hand, Barsbey et al. [2] and Raj et al. [24] reported several experimental results +conducted on neural networks, which illustrated the existence of a non-globally monotonic relation +between the generalization error and the heaviness of the tails in practical settings (see Figure 7 in +[2] and Figure 2 in [24]). Our result brings a stronger theoretical justification to these empirical +observations thanks to the generality of our theoretical framework. +3.3 +Infeasibility of p-Wasserstein Distance for p ≥ α +Now, that we have provided result for the 1-Wasserstein distance between the distribution of θ +and ˆθ. A natural question to ask is whether similar results could be obtained more generally in +the p-Wasserstein distance for some arbitrary p. +Not surprisingly, the following result says that in general we do not expect to control the +p-Wasserstein distance when p is larger than the tail-index α. +Proposition 9. Let d = 1, α > 1, X ⊂ R, and f(θ, x) = (θx)2. Denote µ and ν as the invariant +measures of (11) and (12), respectively. Then for any p > α, Wp(µ, ν) = +∞. +The proof is provided in the Appendix B.4. +3.4 +Discrete-Time Dynamics +Finally, we will illustrate that our theory also extends to the discretizations of the SDEs (11) and +(12). Consider the following Euler-Maruyama discretization: +θk+1 = θk − η∇ ˆF(θk, Xn) + η1/αSk+1, +(26) +10 + +where Sk are i.i.d. rotationally invariant alpha-stable random vectors with the characteristic +function: +E +� +ei⟨u,Sk⟩� += e−∥u∥α +2 , +for any u ∈ Rd. +(27) +Similarly, with input data ˆXn, we have +ˆθk+1 = ˆθk − η∇ ˆF(ˆθk, ˆXn) + η1/αSk+1. +(28) +Let µ and ˆµ denote the stationary distributions of continuous-time θt and ˆθt as t → ∞. +Moreover, let ν and ˆν denote the stationary distributions of discrete-time θk and ˆθk as k → ∞. It +is proved in [6] that the 1-Wasserstein distance of the discretization error is of order η2/α−1. More +precisely, they showed the following result. +Lemma 10 (Theorem 1.2. in Chen et al. [6]). Suppose that Assumptions 3 and 4 hold. Let m, L +be as in Assumption 4. Then, there exists some constant Q (that may depend on B, m, K, L, M +from Assumption 4) such that for every η < min{1, m/L2, 1/m}, one has +W1(µ, ν) ≤ Qη2/α−1, +(29) +W1(ˆµ, ˆν) ≤ Qη2/α−1. +(30) +By applying the above 1-Wasserstein bound for the discretization error in Lemma 10 and +Theorem 5, we obtain the following corollary, which provides the 1-Wasserstein distance of the +stationary distributions of the discrete-time (θk)k≥0 and (ˆθk)k≥0 processes. +Corollary 11. Under the assumptions in Theorem 5 and Lemma 10, we have +W1(ν, ˆν) ≤ 2Qη2/α−1 ++ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2 (2C0 + 1) . +(31) +By using the same approach as we used in Corollary 6, we can easily obtain a generalization +bound for the discrete-time as well. Note that the upper bound in Corollary 11 depends on the +tail-index α only via η2/α−1, which is increasing in α (since η < 1 as assumed in Lemma 10), and +the constant C0 which depends on α via g(α; d) function. Therefore, by Proposition 8, the upper +bound in Corollary 11 is increasing in α ∈ [α0, 2], for any d ≥ d0, where d0 and α0 are given in +Proposition 8. Moreover, the proof of Proposition 8 reveals that +∂ +∂αg(α; d) tends to −∞ as α tends +to 1 and thus there exists some α′′ +0 < α′ +0, where α′ +0 is defined in Proposition 8, such that for any +fixed d ∈ N, the upper bound in Corollary 11 is decreasing in α ∈ [1, α′′ +0]. Hence, the conclusions +that we obtained for the continuous-time processes remain valid for the discretizations as well. +4 +Conclusion +In this work, we studied the relation between the generalization behavior and the heavy tails arising +in the SGD dynamics. Previous work on the topic obtained monotonic relationship under strong +topological and statistical regularity assumptions, with the exception of the approach in [24] which +was limited to only quadratic losses. Following the literature, we considered heavy-tailed SDEs and +their discretization for modeling the heavy-tailed behavior of SGD, and showed that the relation is +non-monotonic for a general class of losses satisfying a dissipativity condition which generalizes +the results of [24] beyond quadratic losses. Our proof technique is based on a novel 1-Wasserstein +11 + +stability bound for the symmetric α-stable Lévy-driven SDEs, that model the SGD dynamics. +Furthermore, our results, when combined with the results of [22], yield directly a generalization +bound for the class of Lipschitz functions. +Future Directions: As a future research direction, we would like to obtain similar stability +bounds without making the dissipativity assumption on the objective function as being done +for Langevin Monte Carlo in [15]. We would also like to consider specific class of functions (e.g. +one-layer neural network) and study the effect of tail-index with other parameters and its effect on +the generalization. +Acknowledgment +A.R is supported by the a Marie Sklodowska-Curie Fellowship (project NN-OVEROPT 101030817). +M.G.’s research is supported in part by the grants Office of Naval Research Award Number N00014- +21-1-2244, National Science Foundation (NSF) CCF-1814888, NSF DMS-2053485. U.Ş.’s research +is supported by the French government under management of Agence Nationale de la Recherche +as part of the “Investissements d’avenir” program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA +Institute). L.Z. is grateful to the support from a Simons Foundation Collaboration Grant and the +grants NSF DMS-2053454, NSF DMS-2208303 from the National Science Foundation. +References +[1] Applebaum, D. Lévy Processes and Stochastic Calculus. Cambridge University Press, Cam- +bridge, UK, second edition, 2009. +[2] Barsbey, M., Sefidgaran, M., Erdogdu, M. A., Richard, G., and Şimşekli, U. Heavy Tails +in SGD and Compressibility of Overparametrized Neural Networks. In Advances in Neural +Information Processing Systems, volume 34, pp. 29364–29378. Curran Associates, Inc., 2021. +[3] Bertoin, J. Lévy Processes. Cambridge University Press, Cambridge, UK, 1996. +[4] Bousquet, O. and Elisseeff, A. Stability and generalization. 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In Advances in Neural Information Processing +Systems, volume 33, pp. 15383–15393, 2020. +14 + +Algorithmic stability of heavy-tailed SGD with general loss +functions +Supplementary Document +A +Background on Markov Semigroups +In this section, we introduce the concept of Markov semigroups, that will be used in the proofs of +main results in Section B. +For a continuous-time Markov process (Xw +t )t≥0 that starts at X0 = w, its Markov semigroup +Pt is defined as for any bounded measurable function f : Rd → R, +Ptf(w) = Ef(Xw +t ), +t ≥ 0. +(32) +Similarly, for a discrete-time Markov process (Y w +k )∞ +k=0 that starts at Y0 = w, its Markov semigroup +Qk is defined as for any bounded measurable function f : Rd → R, +Qkf(w) = Ef(Y w +k ), +k = 0, 1, 2, . . . . +(33) +B +Proofs of Main Results +In this section, we provide the proofs of main results in our paper. +B.1 +Proof of Theorem 5 +We first provide the theorem statement with all the details. +Theorem 12 (Restatement of Theorem 5). Assume θ0 = ˆθ0 = w. Denote by µ, ˆµ the unique +invariant distributions of (θt)t≥0 and (ˆθt)t≥0 respectively. The following two statements hold: +(i) For every N ≥ 1 and η ∈ (0, 1), we have the following statements: +(I) If N = 1, then +W1 +� +Law(θηN), Law(ˆθηN) +� +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) · +� +(1 + ∥w∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η +� +. +(34) +(II) If 2 ≤ N ≤ η−1 + 1, then +W1 +� +Law(θηN), Law(ˆθηN) +� +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C)(1 + C0(1 + ∥w∥))η1+ 1 +α + ρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η ++ eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥))η +1 +α ++ eLρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1). +(35) +(III) If N > η−1 + 1, then +W1 +� +Law(θηN), Law(ˆθηN) +� +15 + +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥))η1+ 1 +α + ρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η ++ +� +C1λ−1eλ + 1 +� +eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥))η +1 +α ++ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1). +(36) +(ii) We have +W1(µ, ˆµ) ≤ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2 (2C0 + 1) . +(37) +Proof of Theorem 5. (i) We first prove part (i). For any h ∈ Lip(1), by the semigroup property, +we have +PNηh(w) − ˆPNηh(w) = +N +� +i=1 +� +ˆP(i−1)ηP(N−i+1)ηh(w) − ˆPiηP(N−i)ηh(w) +� += +N +� +i=1 +ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w). +Therefore, we can compute that +W1 +� +Law(θηN), Law(ˆθηN) +� += +sup +h∈Lip(1) +���PNηh(w) − ˆPNηh(w) +��� +≤ +sup +h∈Lip(1) +��� ˆP(N−1)η(Pη − ˆPη)h(w) +��� + +N−1 +� +i=1 +sup +h∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) +��� . +(38) +Let us first bound the first term in (38). For any h ∈ Lip(1) and η < 1, by applying Lemma 18, +we get +���(Pη − ˆPη)h(w) +��� ≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + ∥w∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η. +Hence, we have +sup +h∈Lip(1) +��� ˆP(N−1)η(Pη − ˆPη)h(w) +��� +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + E∥ˆθw +(N−1)η∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2E∥ˆθw +(N−1)η∥ + 1)η +(39) +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥))η1+ 1 +α + ρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η, +where we applied Lemma 14 to obtain the last inequality above. +Next, let us bound the second term in (38) and hence bound the 1-Wasserstein distance +W1 +� +Law(θηN), Law(ˆθηN) +� +. +We consider three cases: (I) N = 1; (II) 2 ≤ N ≤ η−1 + 1 and (III) N > η−1 + 1. +Case (I): N = 1. One can apply (39) and obtain +W1 +� +Law(θη), Law(ˆθη) +� +16 + +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + E∥ˆθw +0 ∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2E∥ˆθw +0 ∥ + 1)η += +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + ∥w∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η. +(40) +This completes the proof of part (I). +Case (II): 2 ≤ N ≤ η−1 + 1. By Lemma 18, for any i ≥ 1, we have +���(Pη − ˆPη)P(N−i)ηh(w) +��� +≤ ∥∇P(N−i)ηh∥∞ +�� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + ∥w∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η +� +≤ eL �� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + ∥w∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η +� +, +(41) +where we used Lemma 16 and the fact that for any i ≥ 1 and 2 ≤ N ≤ η−1 +1 we have (N −i)η ≤ 1 +in the inequality (41). +By applying Lemma 14, we obtain +sup +h∈Lip(1) +��� ˜P(i−1)η(Pη − ˆPη)P(N−i)ηh(w) +��� +≤ eL �� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) +� +1 + E∥ˆθ(i−1)η∥ +� +η1+ 1 +α + ρ(Xn, ˆXn)K2 +� +2E∥ˆθ(i−1)η∥ + 1 +� +η +� +≤ eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η1+ 1 +α ++ eLρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η. +(42) +Hence, we conclude that +W1 +� +Law(θηN), Law(ˆθηN) +� +≤ +sup +h∈Lip(1) +��� ˆP(N−1)η(Pη − ˆPη)h(w) +��� + +N−1 +� +i=1 +sup +h∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) +��� +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η1+ 1 +α + ρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η ++ (N − 1)eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η1+ 1 +α ++ (N − 1)eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η1+ 1 +α + ρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η ++ eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η +1 +α ++ eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) . +(43) +This completes the proof of (I). +Case (III): N > η−1 + 1. We can compute that +sup +h∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) +��� += +sup +h∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P1P(N−i)η−1h(w) +��� +17 + +≤ +sup +g∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P1g(w) +��� +sup +h∈Lip(1) +∥∇P(N−i)η−1h∥∞. +By Lemma 15, for any h ∈ Lip(1), +|Pth(w) − Pth(y)| ≤ C1e−λt∥w − y∥, +(44) +for any t ≥ 0 and w, y ∈ Rd. This implies that for any h ∈ Lip(1), +∥∇Pth∥∞ ≤ C1e−λt. +(45) +Hence, we conclude that +sup +h∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) +��� ≤ C1e−λ((N−i)η−1) +sup +g∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P1g(w) +��� , +where i ≤ ⌊N − η−1⌋. +Moreover, by Lemma 18, we have +���PηP1g(w) − ˆPηP1g(w) +��� +≤ ∥∇P1g∥∞ +�� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + ∥w∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η +� +≤ eL �� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + ∥w∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η +� +, +(46) +which by Lemma 14 implies that +sup +g∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P1g(w) +��� +≤ eL �� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + E∥θw +(i−1)η∥)η1+ 1 +α + ρ(Xn, ˆXn)K2 +� +2E∥θw +(i−1)η∥ + 1 +� +η +� +≤ eL �� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η1+ 1 +α + ρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η +� +. +Therefore, we have +⌊N−η−1⌋ +� +i=1 +sup +h∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) +��� +≤ +⌊N−η−1⌋ +� +i=1 +C1e−λ((N−i)η−1)eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η1+ 1 +α ++ +⌊N−η−1⌋ +� +i=1 +C1e−λ((N−i)η−1)eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η +≤ C1λ−1eλeL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥))η +1 +α ++ C1λ−1eλeLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) , +where we used the fact that +⌊N−η−1⌋ +� +i=1 +e−λ((N−i)η−1) ≤ eλ +� N−1 +⌊η−1⌋−1 +e−ληrdr ≤ eλη−1 +� ∞ +0 +e−λrdr = λeλη−1. +18 + +Next, when i ≥ ⌊N − η−1⌋ + 1, by applying (42), we have +N−1 +� +i=⌊N−η−1⌋+1 +sup +h∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) +��� +≤ +N−1 +� +i=⌊N−η−1⌋+1 +eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η1+ 1 +α ++ +N−1 +� +i=⌊N−η−1⌋+1 +eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η +≤ eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η +1 +α ++ +N−1 +� +i=⌊N−η−1⌋+1 +eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) . +Therefore, we obtain +N−1 +� +i=1 +sup +h∈Lip(1) +��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) +��� +≤ +� +C1λ−1eλ + 1 +� +eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η +1 +α ++ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) . +Hence, we conclude that +W1 +� +Law(θηN), Law(ˆθηN) +� +≤ +sup +h∈Lip(1) +��� ˆP(N−1)η +� +Pη − ˆPη +� +h(w) +��� + +N−1 +� +i=1 +sup +h∈Lip(1) +��� ˆP(i−1)η +� +Pη − ˆPη +� +P(N−i)ηh(w) +��� +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η1+ 1 +α + ρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η ++ +� +C1λ−1eλ + 1 +� +eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η +1 +α ++ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) . +(47) +This completes the proof of part (III). +(ii) Now, we are ready prove part (ii). By triangle inequality, +W1 (µ, ˆµ) ≤ W1 (Law(θηN), µ) + W1 +� +Law(θηN), Law(ˆθηN) +� ++ W1 +� +Law(ˆθηN), ˆµ +� +. +It follows from Lemma 14 that by letting N → ∞, we have +W1 (µ, ˆµ) +≤ lim sup +N→∞ +W1 +� +Law(θηN), Law(ˆθηN) +� +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η1+ 1 +α + ρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η +19 + ++ +� +C1λ−1eλ + 1 +� +eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥)) η +1 +α ++ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) , +(48) +where we used part (III) from part (i). Since W1 (µ, ˆµ) is independent of η and the initial state +x ∈ Rd, we can set η = 0 and x = 0 in (48) and conclude that +W1 (µ, ˆµ) ≤ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2 (2C0 + 1) . +The proof is complete. +B.2 +Proof of Lemma 7 +Proof of Lemma 7. First of all, the infinitesimal generator of θt process is given by +Lαf(θ) = +� +−∇ ˆF(θ, Xn), ∇f(θ) +� ++ (−∆)α/2f(θ), +(49) +where (−∆)α/2 is the fractional Laplacian operator defined as a principal value integral: +(−∆)α/2f(θ) = dα · p.v. +� +Rd(f(θ + y) − f(θ)) +dy +∥y∥α+d , +(50) +where (see e.g. [30]) +dα := 2αΓ +� d+α +2 +� +π−d/2 +|Γ(−α/2)| +. +(51) +Next, let V (w) := (1 + ∥w∥2)1/2. We derive from Assumption 4 that for any dataset Xn ∈ X n, +we have the following property: +∥∇ ˆF(0, Xn)∥ ≤ B, +� +∇ ˆF(θ1, Xn) − ∇ ˆF(θ2, Xn), θ1 − θ2 +� +≤ −m∥θ1 − θ2∥2 + K, +and +���∇v∇ ˆF(θ, Xn) +��� ≤ L∥v∥, +���∇v1∇v2∇ ˆF(θ, Xn) +��� ≤ M∥v1∥∥v2∥, +for any v, v1, v2 ∈ Rd so that we can apply Proposition 2.1. in [6]. It is shown in the proof of +Proposition 2.1. in [6] that V ∈ D(Lα), i.e. the domain of the infinitesimal generator Lα and +moreover +LαV (w) ≤ −λ1V (w) + q1, +(52) +where +λ1 := 1 +2m, +q1 := m + K + B + Cd,α, +(53) +where +Cd,α := dα +√ +dσd−1 +2 − α ++ dασd−1 +α − 1 = 2αΓ +� d+α +2 +� +π−d/2√ +dσd−1 +|Γ(−α/2)|(2 − α) ++ 2αΓ +� d+α +2 +� +π−d/2σd−1 +|Γ(−α/2)|(α − 1) +, +(54) +20 + +where σd−1 := 2π +d +2 /Γ(d/2) is the surface area of the unit sphere in Rd, a positive constant that +depends only on d. +Next, let us define the extended infinitesimal generator Lα +t : +Lα +t f(t, θ) := ∂tf(t, θ) + Lαf(t, θ). +(55) +Then, it follows from (52) that +Lα +t eλ1tV (θ) = λ1eλ1tV (θ) + eλ1tLαV (θ) ≤ λ1eλ1tV (θ) + eλ1t (−λ1V (w) + q1) = q1eλ1t. +(56) +By Dynkin’s formula, +E +� +eλ1tV (θw +t ) +� += V (w) + E +�� t +0 +Lα +s eλ1sV (θw +s ) ds +� +≤ V (w) + +� t +0 +q1eλ1sds = V (w) + q1 +eλ1t − 1 +λ1 +, +which implies that +E∥θw +t ∥ ≤ E [V (θw +t )] ≤ e−λ1tV (w) + q1 +1 − e−λ1t +λ1 +≤ 1 + ∥w∥ + q1 +λ1 +, +(57) +where we used V (w) = (1 + ∥w∥2)1/2 and the inequality ∥w∥ ≤ (1 + ∥w∥2)1/2 ≤ 1 + ∥w∥. Hence, +we have +E∥θw +t ∥ ≤ C0(1 + ∥w∥), +(58) +where we take +C0 := 1 + q1 +λ1 += 1 + 2 (m + K + B + Cd,α) +m += 3 + 2 (K + B) +m ++ 2 +m +� +2αΓ +� d+α +2 +� +π−d/2√ +dσd−1 +|Γ(−α/2)|(2 − α) ++ 2αΓ +� d+α +2 +� +π−d/2σd−1 +|Γ(−α/2)|(α − 1) +� +. +Since C0 in the above bound is uniform in the dataset, similarly, we also have +E∥ˆθw +t ∥ ≤ C0(1 + ∥w∥), +(59) +which completes the proof. +B.3 +Proof of Proposition 8 +Proof of Proposition 8. Let us first prove part (i). First, we can re-write g(α; d) as +g(α; d) = +2αΓ +� d+α +2 +� +|Γ(−α/2)|(2 − α) +�√ +d + 2 − α +α − 1 +� +. +(60) +By the properties of the gamma function, we have +Γ +� +2 − α +2 +� += +� +1 − α +2 +� +Γ +� +1 − α +2 +� += +� +1 − α +2 +� −α +2 Γ(−α/2). +21 + +Therefore, we have +|Γ(−α/2)|(2 − α) = 4 +αΓ +� +2 − α +2 +� +. +(61) +Moreover, by the properties of the gamma function, +Γ +� +2 − α +2 +� += Γ +� +1 − +�α +2 − 1 +�� += +π +sin +� +π +� α +2 − 1 +�� +Γ +� α +2 − 1 +� = +π +� α +2 − 1 +� +sin +� +π +� α +2 − 1 +�� +Γ +� α +2 +�. +Hence, we conclude that +g(α; d) = 2α−2αΓ +�d + α +2 +� +Γ +�α +2 +� +· sin +� +π +� +1 − α +2 +�� +π +� +1 − α +2 +� +�√ +d + 2 − α +α − 1 +� +, +where we used sin(−x) = − sin(x) for any x ∈ R. Let us define h(x) := sin(x) +x +for any 0 ≤ x ≤ π/2. +We can compute that h′(x) = x cos(x)−sin(x) +x2 +. Let p(x) := x cos(x) − sin(x). Then p(0) = 0 and +p′(x) = −x sin(x) < 0 for any 0 < x < π/2 which implies that p(x) < 0 and thus h′(x) < 0 for any +0 < x < π/2. Hence h(x) is decreasing in x for any 0 ≤ x ≤ π/2. As a result, the map +α �→ sin +� +π +� +1 − α +2 +�� +π +� +1 − α +2 +� +is increasing in α for any 1 < α < 2. +(62) +It is well known that gamma function x �→ Γ(x) is log-convex for x > 0 and thus convex for +any x > 0. Since Γ(1) = Γ(2) = 1, there exists a unique critical value c0 ∈ (1, 2) such that the +gamma function x �→ Γ(x) is increasing for any x ≥ c0 and decreasing for any 1 ≤ x ≤ c0. +Next, for any given α0 ∈ (1, 2) such that 1 + α0 +2 ≥ c0, we have for any 2 ≥ α2 > α1 ≥ α0 and +d ≥ 2, +g(α2; d) +g(α1; d) = +2α2−2α2Γ +� +d+α2 +2 +� +Γ +� α2 +2 +� sin(π(1− α2 +2 )) +π(1− α2 +2 ) +�√ +d + 2−α2 +α2−1 +� +2α1−2α1Γ +� +d+α1 +2 +� +Γ +� α1 +2 +� sin(π(1− α1 +2 )) +π(1− α1 +2 ) +�√ +d + 2−α1 +α1−1 +� += +2α2Γ +� +d+α2 +2 +� +Γ +� +1 + α2 +2 +� sin(π(1− α2 +2 )) +π(1− α2 +2 ) +�√ +d + 2−α2 +α2−1 +� +2α1Γ +� +d+α1 +2 +� +Γ +� +1 + α1 +2 +� sin(π(1− α1 +2 )) +π(1− α1 +2 ) +�√ +d + 2−α1 +α1−1 +� +≥ 2α2−α1 +√ +d + 2−α2 +α2−1 +√ +d + 2−α1 +α1−1 +, +(63) +where we used (62) and the fact that the gamma function x �→ Γ(x) is increasing in x ≥ 1+ α0 +2 ≥ c0. +Next, let us define the function: +q(x) := 2x−α1 +√ +d + 2−x +x−1 +√ +d + 2−α1 +α1−1 +, +(64) +where 2 ≥ x ≥ α1 ≥ α0. It is clear that q(α1) = 1 and moreover, we can compute that +q′(x) = log(2)2x−α1 +√ +d + 2−x +x−1 +√ +d + 2−α1 +α1−1 +− +2x−α1 +(x − 1)2 +1 +√ +d + 2−α1 +α1−1 +22 + += +2x−α1 +√ +d + 2−α1 +α1−1 +� +log(2) +�√ +d + 2 − x +x − 1 +� +− +1 +(x − 1)2 +� +≥ +2x−α1 +√ +d + 2−α1 +α1−1 +� +log(2) +√ +d − +1 +(α0 − 1)2 +� +≥ 0, +(65) +provided that +d ≥ +1 +(log 2)2(α0 − 1)4 . +(66) +This implies that q(x) is increasing for 2 ≥ x ≥ α1 ≥ α0 provided that d ≥ 2, 1 + α0 +2 ≥ c0 and (66) +holds. Hence, we conclude that g(α2; d) ≥ g(α1; d) for any d ≥ d0 = max +� +2, +1 +(log 2)2(α0−1)4 +� +, and +2 ≥ α2 ≥ α1 ≥ α0. +Now, let us prove part (ii) of Proposition 8. We recall from (60) that +g(α; d) = +2αΓ +� d+α +2 +� +|Γ(−α/2)|(2 − α) +�√ +d + 2 − α +α − 1 +� +. +(67) +We can compute that +∂ +∂αg(α; d) = +2αΓ +� d+α +2 +� +|Γ(−α/2)|(2 − α) +−1 +(α − 1)2 + ∂ +∂α +� +2αΓ +� d+α +2 +� +Γ(−α/2)(α − 2) +� �√ +d + 2 − α +α − 1 +� +, +where we can further compute that +∂ +∂α +� +2αΓ +� d+α +2 +� +Γ(−α/2)(α − 2) +� += log(2)2αΓ +� d+α +2 +� ++ 2α−1Γ +� d+α +2 +� +ψ +� d+α +2 +� +Γ(−α/2)(α − 2) +− 2αΓ +� d+α +2 +� � +− 1 +2(α − 2)ψ(− α +2 ) + 1 +� +Γ(−α/2)(α − 2)2 +, +where ψ(·) denotes the digamma function. This implies that +∂ +∂αg(α; d) = +2αΓ +� d+α +2 +� +|Γ(−α/2)|(2 − α)p(α; d), +(68) +where +p(α; d) := +−1 +(α − 1)2 + +� +log(2) + 1 +2ψ +�d + α +2 +�� �√ +d + 2 − α +α − 1 +� ++ +�1 +2ψ +� +−α +2 +� ++ +1 +2 − α +� �√ +d + 2 − α +α − 1 +� +. +By the property of the digamma function, we have ψ(− α +2 ) = ψ(1 − α +2 ) + 2 +α and ψ(x) is increasing +in x > 0 and ψ(−1/2) < 0. Therefore, for any 1 < α ≤ α0, we have +p(α; d) = +−1 +(α − 1)2 + +� +log(2) + 1 +2ψ +�d + α +2 +�� �√ +d + 2 − α +α − 1 +� ++ +�1 +2ψ +� +1 − α +2 +� ++ 1 +α + +1 +2 − α +� �√ +d + 2 − α +α − 1 +� +23 + +≤ +−1 +(α − 1)2 + +� +log(2) + 1 +2ψ +�d + α0 +2 +�� �√ +d + +1 +α − 1 +� ++ +� +1 + +1 +2 − α0 +� �√ +d + +1 +α − 1 +� +. +It follows that p(α; d) ≤ 0 holds if +y0 +√ +d(α − 1)2 + y0(α − 1) − 1 ≤ 0, +(69) +where y0 := log(2) + 1 +2ψ(d + α +2 ) + 3−α0 +2−α0 , and it is easy to compute that (69) holds provided that +α ≤ 1 + +−1 + +� +1 + 4y−1 +0 +√ +d +2 +√ +d +. +(70) +Hence, we conclude that p(α; d) is non-positive and thus +∂ +∂αg(α; d) is non-positive (by (68)) and +therefore g(α; d) is decreasing for any α ∈ [1, α′ +0], where α′ +0 := min +� +α0, 1 + −1+√ +1+4y−1 +0 +√ +d +2 +√ +d +� +. The +proof is complete. +B.4 +Proof of Proposition 9 +Proof. Due to our choice of the loss function f, the SDEs (11) and (12) reduce to Ornstein- +Uhlenbeck processes driven by a symmetric α-stable Lévy process. Hence, we can characterize the +invariant distributions of the SDEs as follows (see e.g. [24]): +θ∞ =d µ + σξ, +and +ˆθ∞ =d ˆµ + ˆσˆξ, +(71) +for some µ, ˆµ ∈ R and σ, ˆσ ∈ R+. Here, ξ and ˆξ are SαS(1) distributed (see Section 2 for definition) +and =d denotes equality in distribution. Now recall that µ = Law(θ∞) and ν = Law(ˆθ∞), and the +p-Wasserstein metric for one-dimensional distributions is given by, +Wp +p(µ, ν) = +inf +γ∈Γ(µ,ν) E(x,y)∼γ(x,y)|x − y|p, +where Γ(µ, ν) is the set of all couplings of µ and ν. In our case, x ∈ R and y ∈ R. For any coupling +γ⋆ ∈ Γ(µ, ν), we have +� +R×R +|x − y|p dγ⋆(x, y) = +� +R×R +� +|x − y|2�p/2 +dγ⋆(x, y) += +� +R×R +(x2 + y2 − 2xy)p/2 dγ⋆(x, y) +≥ +� +R+×R− +(x2 + y2 − 2xy)p/2 dγ⋆(x, y) +≥ +� +R+×R− +(|x|p + |y|p + |2xy|p/2) dγ⋆(x, y) +≥ +� +R+×R− +|x|p dγ⋆(x, y) + +� +R+×R− +|y|p dγ⋆(x, y) += C1 +� +R+ +|x|p dµ(x) + C2 +� +R− +|y|p dν(y) = +∞, +24 + +where C1 and C2 are some finite, positive constants. The last equation comes from the properties +of the α-stable distribution. Since it holds for any γ⋆ ∈ Γ(µ, ν), we conclude that Wp +p(µ, ν) = ∞. +This completes the proof. +B.5 +Proof of Corollary 11 +Corollary 13 (Restatement of Corollary 11). Under the assumptions in Theorem 12 and Lemma 19, +we have: +(i) For any 2 ≤ N ≤ η−1 + 1, +W1 +� +Law(θηN), Law(ˆθηN) +� +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C)(1 + C0(1 + ∥w∥))η1+ 1 +α + ρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η ++ eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥))η +1 +α ++ eLρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1) + 2Q(1 + ∥w∥)η2/α−1, +(72) +and for any N > η−1 + 1, +W1 +� +Law(θηN), Law(ˆθηN) +� +≤ +� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥))η1+ 1 +α + ρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η ++ +� +C1λ−1eλ + 1 +� +eL � +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + C0(1 + ∥w∥))η +1 +α ++ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1) + 2Q(1 + ∥w∥)η2/α−1. (73) +(ii) We have +W1(µ, ˆµ) ≤ +� +C1λ−1eλ + 1 +� +eLρ(Xn, ˆXn)K2 (2C0 + 1) + 2Qη2/α−1. +(74) +Proof. Let us prove part (ii) and the proof for part (i) is similar. It follows directly from Lemma 10 +and Theorem 5 and the inequality: +W1(ν, ˆν) ≤ W1(ν, µ) + W1(ˆν, ˆµ) + W1(µ, ˆµ). +(75) +The proof is complete. +C +Technical Lemmas +In this section, we provide some technical results that are used in the proofs of main results in +Section B. First, we have the following technical result from [6]. +Lemma 14 (Proposition 2.1. in Chen et al. [6]). Under Assumption 4, (θw +t )t≥0 and (ˆθw +t )t≥0 admit +unique invariant probability measures µ and ˆµ respectively such that +sup +|f|≤V +|E[f(θw +t )] − µ(f)| ≤ c1V (w)e−c2t, +for any t > 0, +(76) +sup +|f|≤V +���E[f(ˆθw +t )] − ˆµ(f) +��� ≤ c1V (w)e−c2t, +for any t > 0, +(77) +25 + +for some constants c1, c2 > 0 where V (w) := (1 + ∥w∥2)1/2 is a Lyapunov function. In particular, +there exists a constant C0 > 0 such that +E∥θw +t ∥ ≤ C0(1 + ∥w∥), +for any t > 0, +(78) +E∥ˆθw +t ∥ ≤ C0(1 + ∥w∥), +for any t > 0. +(79) +Moreover, we recall the following technical lemma. +Lemma 15 (Proposition 2.2 in Chen et al. [6]). There exist constants C1, λ > 0 such that for any +t > 0 and w, y ∈ Rd, we have +W1 (Law (θw +t ) , Law (θy +t )) ≤ C1e−λt∥w − y∥, +(80) +W1 +� +Law +� +ˆθw +t +� +, Law +� +ˆθy +t +�� +≤ C1e−λt∥w − y∥. +(81) +Let Pt and ˆPt denote the Markov semigroups of θt and ˆθt processes respectively, that is, for +any bounded function f : Rd → R, +Ptf(x) = Ef(θw +t ), +ˆPtf(x) = Ef(ˆθw +t ). +(82) +We have the following technical lemma from [6]. +Lemma 16 (Lemma 3.1 in Chen et al. [6]). For any h ∈ Lip(1) and v, w ∈ Rd and t ∈ (0, 1], we +have +∥∇vPth(w)∥ ≤ eL∥v∥, +∥∇v ˆPth(w)∥ ≤ eL∥v∥, +(83) +where L is defined in Assumption 4. +We recall the following technical lemma from [6]. +Lemma 17 (Lemma 3.2 in Chen et al. [6]). There exist constants C > 0 such that for all w ∈ Rd, +t ≥ 0, we have +E∥θw +t − w∥ ≤ C(1 + ∥w∥) +� +t ∨ t1/α� +, +(84) +E∥ˆθw +t − w∥ ≤ C(1 + ∥w∥) +� +t ∨ t1/α� +. +(85) +Next, we state and prove the following key technical lemma. +Lemma 18. There exist constants C > 0 such that for all w ∈ Rd, η ∈ (0, 1), f : Rd → R with +∥∇f∥∞ < ∞, we have +���Pηf(w) − ˆPηf(w) +��� +≤ ∥∇f∥∞ +�� +K1 + ρ(Xn, ˆXn)K2 +� +2C(1 + ∥w∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η +� +. +(86) +Proof of Lemma 18. We can compute that +���E +� +f +� +θw +η +� +− f +� +ˆθw +η +����� += +����E +� +f +� +w + +� η +0 +∇ ˆF(θw +r , Xn)dr + Lα +η +� +− f +� +w + +� η +0 +∇ ˆF(ˆθw +r , ˆXn)dr + Lα +η +������ +26 + +≤ ∥∇f∥∞E +���� +� η +0 +∇ ˆF(θw +r , Xn)dr − +� η +0 +∇ ˆF(ˆθw +r , ˆXn)dr +���� +≤ ∥∇f∥∞E +� η +0 +���∇ ˆF(θw +r , Xn) − ∇ ˆF(ˆθw +r , ˆXn) +��� dr +≤ ∥∇f∥∞E +� η +0 +� +K1∥θw +r − ˆθw +r ∥ + ρ(Xn, ˆXn)K2 +� +∥θw +r ∥ + ∥ˆθw +r ∥ + 1 +�� +dr += ∥∇f∥∞ +� +K1 +� η +0 +E∥θw +r − ˆθw +r ∥dr + ρ(Xn, ˆXn)K2 +� η +0 +E +� +∥θw +r ∥ + ∥ˆθw +r ∥ + 1 +� +dr +� +. +By Lemma 17, we have +� η +0 +E∥θw +r − ˆθw +r ∥dr ≤ +� η +0 +E∥θw +r − w∥dr + +� η +0 +E∥ˆθw +r − w∥dr +≤ C(1 + ∥w∥) +� η +0 +r1/αdr + C(1 + ∥w∥) +� η +0 +r1/αdr +≤ 2C(1 + ∥w∥)η1+ 1 +α . +By applying Lemma 17 again, we have +� η +0 +E +� +∥θw +r ∥ + ∥ˆθw +r ∥ + 1 +� +dr ≤ +� η +0 +E +� +∥θw +r − w∥ + ∥ˆθw +r − w∥ + 2∥w∥ + 1 +� +dr +≤ +� η +0 +� +C(1 + ∥w∥)r1/α + C(1 + ∥w∥)r1/α + 2∥w∥ + 1 +� +dr +≤ 2C(1 + ∥w∥)η1+ 1 +α + (2∥w∥ + 1)η. +Hence, we conclude that +���E +� +f(θw +η ) − f(ˆθw +η ) +���� +≤ ∥∇f∥∞ +�� +K1 + ρ(Xn, ˆXn)K2 +� +(2C) (1 + ∥w∥)η1+ 1 +α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η +� +. +This completes the proof. +Lemma 19 (Restatement of Lemma 10 (Theorem 1.2. in Chen et al. [6])). Let µt and ˆµt denote the +distributions of continuous-time θt and ˆθt and µ and ˆµ denote the distributions of continuous-time +θ∞ and ˆθ∞. Moreover, let νk and ˆνk denote the distributions of discrete-time θk and ˆθk and ν and +ˆν denote the distributions of discrete-time θ∞ and ˆθ∞. Assume the dynamics start at w at time 0. +Let m, L be as in Assumption 4. +Then, there exists some constant Q (that may depend on B, m, K, L, M from Assumption 4) +such that the followings hold. +(i) For every N ≥ 2 and η < min{1, m/(8L2), 1/m}, one has +W1(µNη, νN) ≤ Q(1 + ∥w∥)η2/α−1, +(87) +W1(ˆµNη, ˆνN) ≤ Q(1 + ∥w∥)η2/α−1. +(88) +(ii) For every η < min{1, m/L2, 1/m}, one has +W1(µ, ν) ≤ Qη2/α−1, +(89) +W1(ˆµ, ˆν) ≤ Qη2/α−1. +(90) +27 + diff --git a/f9FKT4oBgHgl3EQftC6j/content/tmp_files/load_file.txt b/f9FKT4oBgHgl3EQftC6j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2654f72f2db1ccb24f35570dcbfecedd01f400d --- /dev/null +++ b/f9FKT4oBgHgl3EQftC6j/content/tmp_files/load_file.txt @@ -0,0 +1,749 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf,len=748 +page_content='Algorithmic Stability of Heavy-Tailed SGD with General Loss Functions Anant Raj∗ Coordinated Science Laboraotry University of Illinois Urbana-Champaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Inria, Ecole Normale Supérieure PSL Research University, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' anant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='raj@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='fr Lingjiong Zhu∗ Department of Mathematics Florida State University, FL, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' zhu@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='fsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='edu Mert Gürbüzbalaban Department of Management Science and Information Systems Rutgers University, Piscataway, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Princeton University, NJ, USA mg1625@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='edu Umut Şimşekli Inria, CNRS, Ecole Normale Supérieure PSL Research University, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' umut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='simsekli@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='fr January 30, 2023 Abstract Heavy-tail phenomena in stochastic gradient descent (SGD) have been reported in several empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Experimental evidence in previous works suggests a strong interplay between the heaviness of the tails and generalization behavior of SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' To address this empirical phenomena theoretically, several works have made strong topological and statistical assumptions to link the generalization error to heavy tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Very recently, new generalization bounds have been proven, indicating a non-monotonic relationship between the generalization error and heavy tails, which is more pertinent to the reported empirical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' While these bounds do not require additional topological assumptions given that SGD can be modeled using a heavy-tailed stochastic differential equation (SDE), they can only apply to simple quadratic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' In this paper, we build on this line of research and develop generalization bounds for a more general class of objective functions, which includes non-convex functions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our approach is based on developing Wasserstein stability bounds for heavy-tailed SDEs and their discretizations, which we then convert to generalization bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our results do not require any nontrivial assumptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' yet, they shed more light to the empirical observations, thanks to the generality of the loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='11885v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='ML] 27 Jan 2023 1 Introduction Many supervised learning problems can be expressed as an instance of the risk minimization problem min θ∈Rd {F(θ) := Ex∼D[f(θ, x)]} , (1) where x ∈ X is a random data point, distributed according to an unknown probability distribution D and taking values in the data space X, θ denotes the parameter vector of the model to be learned and f(θ, x) is the instantaneous loss of misprediction with parameters θ corresponding to the data point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' With different choices of the function f, we can recover many problems in supervised learning from deep learning to logistic regression or support vector machines [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' As D is unknown in many scenarios, directly attacking (1) is often not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Assuming we have access to a training dataset Xn = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' , xn} ⊂ X n with n independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=') observations, in practice, we can consider the empirical risk minimization (ERM) problem instead, given as follows: min θ∈Rd � ˆF(θ, Xn) := 1 n n � i=1 f(θ, xi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' One of the most popular algorithms for attacking the ERM problem is stochastic gradient descent (SGD) that is based on the following recursion: θk+1 = θk − η∇ ˜Fk+1(θk, Xn), (2) where η is the step-size (or learning-rate) and ∇ ˜Fk(θ, X) := 1 b � i∈Ωk ∇f(θ, xi) is the stochastic gradient, with Ωk ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' , n} being a random subset drawn with or without replacement, and b := |Ωk| ≪ n being the batch-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Understanding the generalization properties of SGD has been a major challenge in modern machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' In this context, the goal is to bound the so-called generalization error: | ˆF(θ, Xn)− F(θ)|, either in expectation or in high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' While a plethora of approaches have been proposed to address this task [5, 16, 19, 21], a promising approach among those has been based on the theoretical and empirical observations which showed that SGD can exhibit a heavy-tailed behavior, depending on the choice of hyperparameters (η and b), the data distribution D, and the geometry of the loss function f [11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This has motivated the use of ‘heavy-tailed proxies’ for SGD, which –to some extent– facilitated the analysis of SGD in terms of its generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Examples of such proxies include gradient descent with additive heavy-tailed noise: θk+1 = θk − η∇ ˆF(θk, Xn) + ξk+1, (3) where (ξk)k≥1 is a sequence of heavy-tailed random vectors, potentially with unbounded higher-order moment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', E∥ξk∥p = +∞ for some p > 1 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', [20, 29, 32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 2 Another popular proxy for heavy-tailed SGD is based on a continuous-time version of (3), which is expressed by the following stochastic differential equation (SDE): dθt = −∇ ˆF(θt, Xn)dt + σdLα t , (4) where σ > 0 is a scale parameter, Lα t is a d-dimensional α-stable Lévy process, which has heavy- tailed increments and will be formally defined in the next section1, and α ∈ (0, 2] denotes the ‘tail-exponent’ such that as α gets smaller the process Lα t becomes heavier-tailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Within this mathematical framework, Şimşekli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [27] proved an upper-bound (which was then improved in [14]) for the worst-case generalization error over the trajectories of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The bound informally reads as follows: with probability at least 1 − δ, it holds that sup θ∈Θ ��� ˆF(θ, Xn) − F(θ) ��� ≲ � α + I(Θ, Xn) + log(1/δ) n , where Θ denotes the trajectory of (4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', Θ := � θ ∈ Rd : ∃t ∈ [0, 1], θ = θt � , with θt being the solution of (4), and I(Θ, Xn) denotes a form of ‘mutual information’ between the trajectory Θ and the data sample Xn (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This result suggests that the generalization error is essentially determined by two terms: (i) the tail exponent α, as the tails get heavier the generalization error will be lower, (ii) the statistical dependency between the trajectory and the data sample, the lower the dependency the better the generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' While these results illuminated an interesting connection between heavy-tails and generalization, they unfortunately rely on nontrivial topological assumptions on Θ and the mutual information term cannot be controlled in an interpretable way in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' On the other hand, Barsbey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [2] empirically illustrated that the relation between the tail exponent and the generalization error might not be monotonic in practical applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' an observation which cannot be directly supported by the bound in [27] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Aiming to alleviate these issues, very recently, Raj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [24] considered the same problem from the lens of algorithmic stability [4, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' They considered the SDE (4) and further simplified it by choosing the loss function as a simple quadratic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', f(θ, x) = (θ⊤x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' They showed that any parameter vector θ provided by (4) (or its Euler-Maruyama discretization with small enough small step-size) cannot be algorithmically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' However, when the algorithmic stability is measured by a surrogate loss function instead (reminiscent of [29]), the parameter vector θ becomes algorithmically stable, which immediately implies generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Their bound further illustrated that the relation between α and the generalization error might not be monotonic, which is in line with the observations provided in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' While the bounds in [24] do not require additional topological assumptions and do not contain a mutual information term as opposed to [14, 27], their analysis technique heavily relies on the fact that f is a quadratic, hence cannot be directly extended beyond quadratic loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' In this paper, we aim at filling this gap and prove algorithmic stability bounds the SDE (4) (and its Euler-Maruyama discretization) with general loss functions, which can be even non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our contributions are as follows: 1This type of SDEs have also received some attention in terms of limits of deterministic gradient descent with dynamical regularization [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 3 We first focus on the continuous-time setting and prove Wasserstein stability bounds for two SDEs of the form of (4) with different drift functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our results cover both the finite-time case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', t < ∞ and the stationary case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We build upon recently introduced stochastic analysis tools for uniform-in-time Wasserstein error bounds for Euler-Maruyama discretization [6] to obtain a novel Wasserstein stability bound for two α-stable Lévy-driven SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our analysis relies on an additional pseudo-Lipschitz like condition for the underlying process and the dataset (Assumption 3) and careful adaption of the tools in [6] to our context (Lemma 18 and Theorem 12 in the Appendix) as well as additional analysis (Lemma 7) that allows us to characterize the dependence of our bounds on the tail-index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our derived bounds would be interesting on their own to a much broader scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By following [22], we translate the derived Wasserstein stability bounds to algorithmic stability bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Similar to [24], our approach necessitates surrogate loss functions to measure algorithmic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our results reveal that the relation between heaviness of the tail α and the generalization error might not be monotonic, indicating that the conclusions of [24] extends to the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By combining our results with [6], we extend our bounds to the Euler-Maruyama discretization of (4) (that is of the form of (3)) and show that for small enough step-sizes the discrete-time process achieves almost identical stability bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Contrary to [14, 18, 27], our bounds do not rely on any topological regularity assumptions and they further do not contain a mutual information term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Moreover, our results shed more light to the non-monotonic relation between heavy tails and the generalization error, as empirically observed in [2, 24], since they are applicable to non-convex losses, as opposed to [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We also note that our generalization bounds and Wasserstein bounds are independent of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Such a result was previously shown in [9] in the context of Brownian-motion driven SDEs and their discretizations, our work uses different techniques considering Levy-driven SDEs and studies the link between the generalization and the coefficient of heavy tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 2 Notations and Technical Background Gradients and Hessians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For any twice continuously differentiable function f : Rd → R, we denote by ∇f and ∇2f the gradient and the Hessian of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' First-order and second-order directional derivatives of f are defined as ∇vf(x) := lim ϵ→0 f(x + ϵv) − f(x) ϵ , ∇v2∇v1f(x) := lim ϵ→0 ∇v1f(x + ϵv2) − ∇v1f(x) ϵ , (5) for any directions v, v1, v2 ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' If f is three times continuously differentiable, then third-order derivatives along the directions v1, v2 are given by ∇v2∇v1∇f(x) := lim ϵ→0 ∇v1∇f(x + ϵv2) − ∇v1∇f(x) ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (6) Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For p ≥ 1, the p-Wasserstein distance between two probability measures µ and ν on Rd is defined as [28]: Wp(µ, ν) = {inf E∥X − Y ∥p}1/p , (7) 4 where the infimum is taken over all coupling of X ∼ µ and Y ∼ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' In particular, the 1-Wasserstein distance has the following dual representation [28]: W1(µ, ν) = sup h∈Lip(1) ���� � Rd h(x)µ(dx) − � Rd h(x)ν(dx) ���� , (8) where Lip(1) consists of the functions h : Rd → R that are 1-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Algorithmic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Algorithmic stability is an important notion in learning theory, which has pave the way for several important theoretical results [4, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let us first state the notion of algorithmic stability as defined in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Definition 1 (Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [12], Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For a (surrogate) loss function ℓ : Rd × X → R, an algorithm A : �∞ n=1 X n → Rd is ε-uniformly stable if sup X∼ = ˆ X sup z∈X E � ℓ(A(X), z) − ℓ(A( ˆX), z) � ≤ ε, (9) where the first supremum is taken over data X, ˆX ∈ X n that differ by one element, denoted by X ∼= ˆX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Here, we intentionally use a different notation for the loss ℓ (as opposed to f), as our theory will require the algorithmic stability to be measured by using a surrogate loss function, which might be different than the original loss f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We now provide a result from [12] which relates algorithmic stability to the generalization performance of a randomized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Before stating the result, similar to ˆF and F, we respectively define the empirical and population risks with respect to the loss ℓ as follows: ˆR(w, Xn) := 1 n n � i=1 ℓ(w, xi), R(w) := Ex∼D[ℓ(w, x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Theorem 2 (Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [12], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Suppose that A is an ε-uniformly stable algorithm, then the expected generalization error is bounded by ���EA,Xn � ˆR(A(Xn), Xn) � − R(A(Xn)) ��� ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (10) Alpha-stable distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' A scalar random variable X is said to follow a symmetric α-stable distribution, denoted by X ∼ SαS(σ), if its characteristic function takes the form: E � eiuX� = exp (−σα|u|α), for any u ∈ R, where σ > 0 is known as the scale parameter that measures the spread of X around 0 and for α ∈ (0, 2] which is known as the tail-index that determines the tail thickness of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The tail becomes heavier as α gets smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The α-stable distribution SαS appears as the limiting distribution in the generalized central limit theorems for a sum of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' random variables with infinite variance [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The probability density function of a symmetric α-stable distribution, α ∈ (0, 2], does not yield closed-form expression in general except for a few special cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' for example SαS reduces to the Cauchy and the Gaussian distributions, respectively, when α = 1 and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' When 0 < α < 2, the moments are finite only up to the order α in the sense that E[|X|p] < ∞ if and only if p < α, which implies infinite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Finally, α-stable distribution can be extended to the high-dimensional case for random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' One of the most commonly used extension is the rotationally symmetric α-stable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' X 5 follows a d-dimensional rotationally symmetric α-stable distribution if it admits the characteristic function E � ei⟨u,X⟩� = e−σα∥u∥α 2 for any u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For further details of α-stable distributions, we refer to [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lévy processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lévy processes are stochastic processes with independent and stationary increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Their successive displacements can be viewed as the continuous-time analogue of random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lévy processes include the Poisson process, the Brownian motion, the Cauchy process, and more generally stable processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [1, 3, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lévy processes in general admit jumps and have heavy tails which are appealing in many applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' In this paper, we will consider the rotationally symmetric α-stable Lévy process, denoted by Lα t in Rd and is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lα 0 = 0 almost surely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For any t0 < t1 < · · · < tN, the increments Lα tn − Lα tn−1 are independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The difference Lα t − Lα s and Lα t−s have the same distribution, with the characteristic function exp(−(t − s)α∥u∥α 2 ) for t > s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lα t has stochastically continuous sample paths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' for any δ > 0 and s ≥ 0, P(∥Lα t −Lα s ∥ > δ) → 0 as t → s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' When α = 2, Lα t = √ 2Bt, where Bt is the standard d-dimensional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 3 Main Results In this section, we present our main theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' To ease the notation, we will consider the following SDE in lieu of (4): dθt = −∇ ˆF(θt, Xn)dt + dLα t , (11) in the rest of the paper2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our road map is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We will first consider the continuous-time case (11), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', we will set the learning algorithm as A(Xn) = θt for some t ∈ [0, +∞], where θ∞ denotes a sample from the stationary distribution of the SDE (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' As our aim is to prove algorithmic stability bounds for this choice of algorithm, we then consider another dataset ˆXn ∼= Xn, which differ from Xn by one element, accordingly define the following SDE: dˆθt = −∇ ˆF(ˆθt, ˆXn)dt + dLα t , (12) such that A( ˆXn) = ˆθt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Then, we will argue that, for any time t, the laws of θt and ˆθt will be close to each other in the 1-Wasserstein metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By considering a surrogate loss function ℓ, which we will assume to be L-Lipschitz, our bound on the Wasserstein distance between Law(θt) and Law(ˆθt) (Theorem 5) will immediately provide us a generalization bound thanks to the dual representation of the 1-Wasserstein distance (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [22, Lemma 3]): ���Eθt,Xn � ˆR(θt, Xn) � − R(θt) ��� 2Note that the stationary distribution of (4) is the same as the stationary distribution of dθt = −σ−α∇ ˆF(θt, Xn)dt + dLα t , and so that we can easily adapt our main result (Theorem 5) to the general case σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 6 ≤ L sup Xn∼ = ˆ Xn W1 � Law(θt), Law(ˆθt) � , (13) where Law(θt) and Law(ˆθt) respectively depend on Xn and ˆXn due to the form of the SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The reason why we require a surrogate loss function is the fact that we need the Lipschitz continuity of the loss to be able to derive the bound in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' However, as we will detail in the next subsection, our assumptions on the true loss f will be incompatible with the Lipschitz continuity of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' After proving a generalization bound of the form (13), we will further investigate the behavior of the bound with respect to the heaviness of the tail which is characterized by the tail-index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Finally, we will consider the discrete-time case, where we will show that almost identical results hold for the Euler-Maruyama discretizations of (11) and (12), as long as a sufficiently small step-size is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1 Assumptions In this section we state our main assumptions and we will assume that they hold throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For any w ∈ Rd, we use θw t to denote the process θt that starts at θ0 = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For every x ∈ X, there exist universal constants K1, K2 such that ∥∇f(θ, x) − ∇f(ˆθ, ˆx)∥ ≤ K1∥θ − ˆθ∥ + K2∥x − ˆx∥(∥θ∥ + ∥ˆθ∥ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (14) This assumption is a pseudo-Lipschitz like condition (similar to the one in [8]) on the loss f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Under this assumption, for two datasets Xn and ˆXn, we immediately have the following property: ���∇ ˆF(θ, Xn) − ∇ ˆF � ˆθ, ˆXn ���� ≤ K1∥θ − ˆθ∥ + ρ(Xn, ˆXn)K2 � ∥θ∥ + ∥ˆθ∥ + 1 � , (15) where ρ(Xn, ˆXn) := 1 n n � i=1 ∥xi − ˆxi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (16) We will show that the term ρ(Xn, ˆXn) will have an important role in terms of Wasserstein stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By following [6], we also make the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For every x ∈ X, f(·, x) is three-times continuously differentiable, and for any θ1, θ2 ∈ Rd, there exist universal constants B, m, K, L, and M such that ∥∇f(0, x)∥ ≤ B, ⟨∇f(θ1, x) − ∇f(θ2, x), θ1 − θ2⟩ ≤ −m∥θ1 − θ2∥2 + K, and ∥∇v∇f(θ, x)∥ ≤ L∥v∥, ∥∇v1∇v2∇f(θ, x)∥ ≤ M∥v1∥∥v2∥, for any v, v1, v2 ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The first part of this assumption is common in stochastic analysis and often referred to as dissipativity [10, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The second part of the assumption amounts to requiring the drift ∇f(x) to have bounded third-order directional derivatives (see also [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This would be satisfied for instance if f has bounded third-order derivatives on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2 Continuous-Time Dynamics Now, we are ready to state our first theorem that characterizes the 1-Wasserstein distance between θt and ˆθt at any finite time t, which is uniform in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' As a result, we also obtain an upper-bound on the 1-Wasserstein distance between the unique invariant distribution µ of (θt)t≥0 and the unique invariant distribution ˆµ of (ˆθt)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='3 The full statement of the theorem is rather lengthy and is given in the Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For clarity, in the next theorem, we provide our upper-bound on the distance between the invariant distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The finite t case is handled in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Denote by µ, ˆµ the unique invariant distributions of (θt)t≥0 and (ˆθt)t≥0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Then, the following inequality holds: W1(µ, ˆµ) ≤ � C1λ−1eλ + 1 � eLρ(Xn, ˆXn)K2 (2C0 + 1) , (17) where K1, K2 and L are defined in Assumption 3 and Assumption 4 and C0, C1 and λ are some positive real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This theorem shows that, as long as the datasets Xn and ˆXn are close to each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', ρ(Xn, ˆXn) is small, the distance between the solutions of the SDEs (11) and (12) will be small as well for any time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This result can be seen as a heavy-tailed version of the results presented in [9, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Generalization Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By combining Theorem 5 and (13), we can now easily obtain general- ization bound under a Lipschitz surrogate loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Assume that ℓ is L-Lipschitz in θ and supx,y∈X ∥x − y∥ ≤ D for some D < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Then the following inequality holds: ���Eθ∞,Xn � ˆR(θ∞, Xn) � − R(θ∞) ��� ≤ LD � C1λ−1eλ + 1 � eLK2 (2C0 + 1) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (18) The proof of this corollary is straightforward, hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Similar to Theorem 5, we presented Corollary 6 for the stationary case, where t → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' yet, we shall underline that our theory holds for any finite time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lower bounds on algorithmic stability have been discussed in [24] for Ornstein-Uhlenbeck process with α-stable Levy noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' While comparing with the bound obtained in this work, we can see that the obtained bound has optimal dependence on the number of samples n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Next, we will investigate how the constants in Theorem 5 behave with respect to varying α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Constants in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' In Theorem 5, we provided an upper bound on W1(µ, ˆµ) which depends on various quantities, and our next goal is to figure out how the parameters C1, λ, L, K2 and C0 depend on the tail-index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' First, we notice that the parameters L and K2 only depend on the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Second, the parameters C1, λ come from the 1-Wasserstein contraction in Lemma 15 in the Appendix which is a restatement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2 in [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' that is, for any w, y ∈ Rd: W1 (Law (θw t ) , Law (θy t )) ≤ C1e−λt∥w − y∥, (19) 3Here, we know that under our assumptions by the results of [6], invariant distributions µ and ˆµ exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 8 W1 � Law � ˆθw t � , Law � ˆθy t �� ≤ C1e−λt∥w − y∥, (20) where θw t to denote the process θt that starts at θ0 = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Furthermore, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2 in [6] follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' A careful look at Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' in [30] reveals that C1, λ are independent of the tail-index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Finally, C0 depends on α and it comes from Lemma 14 in the Appendix which is a restatement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' in [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' that is, which says that for any w ∈ Rd: E∥θw t ∥ ≤ C0(1 + ∥w∥), for any t > 0, (21) E∥ˆθw t ∥ ≤ C0(1 + ∥w∥), for any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (22) Notice that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' in [6] does not provide an explicit formula for C0, and in the next result, we provide a more refined estimate to spell out the dependence of C0 on the tail-index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For any w ∈ Rd, we have E∥θw t ∥ ≤ C0(1 + ∥w∥), for any t > 0, (23) E∥ˆθw t ∥ ≤ C0(1 + ∥w∥), for any t > 0, (24) where we can take C0 := 3 + 2 (K + B) m + 2α+1Γ � d+α 2 � π−d/2σd−1 |Γ(−α/2)|m � √ d 2 − α + 1 α − 1 � , (25) where Γ(·) is the gamma function and σd−1 = 2π d 2 /Γ(d/2) is the surface area of the unit sphere in Rd, and K, B, m are defined in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence, it follows from Theorem 5 and Lemma 7 that the dependence on the tail-index α is only via the function: g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) := 2αΓ � d+α 2 � √ d |Γ(−α/2)|(2 − α) + 2αΓ � d+α 2 � |Γ(−α/2)|(α − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The next result formalizes how the function g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) depends on the tail-index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let α0 := 2(c0 − 1) ∈ (0, 2), where c0 is the unique critical value in (1, 2) such that the gamma function Γ(x) is increasing for any x > c0 and decreasing for any 1 < x < c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Then, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (i) For any d ≥ d0, where d0 := max � 2, 1 (log 2)2(α0−1)4 � , the map α �→ g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) is increasing in α ∈ [α0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (ii) For any fixed d ∈ N, the map α �→ g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) is decreasing in α ∈ [1, α′ 0], where α′ 0 ≤ α0 is defined as α′ 0 := min � α0, 1 + −1+√ 1+4y−1 0 √ d 2 √ d � , with y0 := log(2) + 1 2ψ(d + α 2 ) + 3−α0 2−α0 , where ψ(·) is the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The proof is given in the Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='3 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This result reveals an interesting fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Depending on the dimension of the parameter vector d, the Wasserstein stability bound (Theorem 5) and the generalization bound (Corollary 6) exhibit different behaviors with respect to varying α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We observe that for sufficiently large d, there exists a critical value α0 such that the 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='0 α 2 4 6 8 10 ˆg(α, d) d = 2 d = 5 d = 10 d = 20 Figure 1: Behavior of g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We scale g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) appropriately to fit all the plots in the same frame which we denote as ˆg(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' bound is monotonically increasing for α ≥ α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This suggests that for d large enough, increasing the heaviness of the tails (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=', smaller α) can be beneficial unless α is smaller than α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For visualization, we also provide a pictorial illustration of the function g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The figure shows the behavior of g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) with respect to α for various dimensions d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The observed non-globally monotonic behavior for large d indicates that the conclusions of [24] extend beyond quadratic loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' On the other hand, Barsbey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [2] and Raj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [24] reported several experimental results conducted on neural networks, which illustrated the existence of a non-globally monotonic relation between the generalization error and the heaviness of the tails in practical settings (see Figure 7 in [2] and Figure 2 in [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our result brings a stronger theoretical justification to these empirical observations thanks to the generality of our theoretical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='3 Infeasibility of p-Wasserstein Distance for p ≥ α Now, that we have provided result for the 1-Wasserstein distance between the distribution of θ and ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' A natural question to ask is whether similar results could be obtained more generally in the p-Wasserstein distance for some arbitrary p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Not surprisingly, the following result says that in general we do not expect to control the p-Wasserstein distance when p is larger than the tail-index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let d = 1, α > 1, X ⊂ R, and f(θ, x) = (θx)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Denote µ and ν as the invariant measures of (11) and (12), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Then for any p > α, Wp(µ, ν) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The proof is provided in the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='4 Discrete-Time Dynamics Finally, we will illustrate that our theory also extends to the discretizations of the SDEs (11) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Consider the following Euler-Maruyama discretization: θk+1 = θk − η∇ ˆF(θk, Xn) + η1/αSk+1, (26) 10 where Sk are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' rotationally invariant alpha-stable random vectors with the characteristic function: E � ei⟨u,Sk⟩� = e−∥u∥α 2 , for any u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (27) Similarly, with input data ˆXn, we have ˆθk+1 = ˆθk − η∇ ˆF(ˆθk, ˆXn) + η1/αSk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (28) Let µ and ˆµ denote the stationary distributions of continuous-time θt and ˆθt as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Moreover, let ν and ˆν denote the stationary distributions of discrete-time θk and ˆθk as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' It is proved in [6] that the 1-Wasserstein distance of the discretization error is of order η2/α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' More precisely, they showed the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lemma 10 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let m, L be as in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Then, there exists some constant Q (that may depend on B, m, K, L, M from Assumption 4) such that for every η < min{1, m/L2, 1/m}, one has W1(µ, ν) ≤ Qη2/α−1, (29) W1(ˆµ, ˆν) ≤ Qη2/α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (30) By applying the above 1-Wasserstein bound for the discretization error in Lemma 10 and Theorem 5, we obtain the following corollary, which provides the 1-Wasserstein distance of the stationary distributions of the discrete-time (θk)k≥0 and (ˆθk)k≥0 processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Under the assumptions in Theorem 5 and Lemma 10, we have W1(ν, ˆν) ≤ 2Qη2/α−1 + � C1λ−1eλ + 1 � eLρ(Xn, ˆXn)K2 (2C0 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (31) By using the same approach as we used in Corollary 6, we can easily obtain a generalization bound for the discrete-time as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Note that the upper bound in Corollary 11 depends on the tail-index α only via η2/α−1, which is increasing in α (since η < 1 as assumed in Lemma 10), and the constant C0 which depends on α via g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Therefore, by Proposition 8, the upper bound in Corollary 11 is increasing in α ∈ [α0, 2], for any d ≥ d0, where d0 and α0 are given in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Moreover, the proof of Proposition 8 reveals that ∂ ∂αg(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) tends to −∞ as α tends to 1 and thus there exists some α′′ 0 < α′ 0, where α′ 0 is defined in Proposition 8, such that for any fixed d ∈ N, the upper bound in Corollary 11 is decreasing in α ∈ [1, α′′ 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence, the conclusions that we obtained for the continuous-time processes remain valid for the discretizations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 4 Conclusion In this work, we studied the relation between the generalization behavior and the heavy tails arising in the SGD dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Previous work on the topic obtained monotonic relationship under strong topological and statistical regularity assumptions, with the exception of the approach in [24] which was limited to only quadratic losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Following the literature, we considered heavy-tailed SDEs and their discretization for modeling the heavy-tailed behavior of SGD, and showed that the relation is non-monotonic for a general class of losses satisfying a dissipativity condition which generalizes the results of [24] beyond quadratic losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Our proof technique is based on a novel 1-Wasserstein 11 stability bound for the symmetric α-stable Lévy-driven SDEs, that model the SGD dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Furthermore, our results, when combined with the results of [22], yield directly a generalization bound for the class of Lipschitz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Future Directions: As a future research direction, we would like to obtain similar stability bounds without making the dissipativity assumption on the objective function as being done for Langevin Monte Carlo in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We would also like to consider specific class of functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' one-layer neural network) and study the effect of tail-index with other parameters and its effect on the generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Acknowledgment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='R is supported by the a Marie Sklodowska-Curie Fellowship (project NN-OVEROPT 101030817).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='G.’s research is supported in part by the grants Office of Naval Research Award Number N00014- 21-1-2244, National Science Foundation (NSF) CCF-1814888, NSF DMS-2053485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='Ş.’s research is supported by the French government under management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' is grateful to the support from a Simons Foundation 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 14 Algorithmic stability of heavy-tailed SGD with general loss functions Supplementary Document A Background on Markov Semigroups In this section, we introduce the concept of Markov semigroups, that will be used in the proofs of main results in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For a continuous-time Markov process (Xw t )t≥0 that starts at X0 = w, its Markov semigroup Pt is defined as for any bounded measurable function f : Rd → R, Ptf(w) = Ef(Xw t ), t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (32) Similarly, for a discrete-time Markov process (Y w k )∞ k=0 that starts at Y0 = w, its Markov semigroup Qk is defined as for any bounded measurable function f : Rd → R, Qkf(w) = Ef(Y w k ), k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (33) B Proofs of Main Results In this section, we provide the proofs of main results in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1 Proof of Theorem 5 We first provide the theorem statement with all the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Theorem 12 (Restatement of Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Assume θ0 = ˆθ0 = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Denote by µ, ˆµ the unique invariant distributions of (θt)t≥0 and (ˆθt)t≥0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The following two statements hold: (i) For every N ≥ 1 and η ∈ (0, 1), we have the following statements: (I) If N = 1, then W1 � Law(θηN), Law(ˆθηN) � ≤ � K1 + ρ(Xn, ˆXn)K2 � (2C) · � (1 + ∥w∥)η1+ 1 α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (34) (II) If 2 ≤ N ≤ η−1 + 1, then W1 � Law(θηN), Law(ˆθηN) � ≤ � K1 + ρ(Xn, ˆXn)K2 � (2C)(1 + C0(1 + ∥w∥))η1+ 1 α + ρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η + eL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥))η 1 α + eLρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (35) (III) If N > η−1 + 1, then W1 � Law(θηN), Law(ˆθηN) � 15 ≤ � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥))η1+ 1 α + ρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η + � C1λ−1eλ + 1 � eL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥))η 1 α + � C1λ−1eλ + 1 � eLρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (36) (ii) We have W1(µ, ˆµ) ≤ � C1λ−1eλ + 1 � eLρ(Xn, ˆXn)K2 (2C0 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (37) Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (i) We first prove part (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For any h ∈ Lip(1), by the semigroup property, we have PNηh(w) − ˆPNηh(w) = N � i=1 � ˆP(i−1)ηP(N−i+1)ηh(w) − ˆPiηP(N−i)ηh(w) � = N � i=1 ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Therefore, we can compute that W1 � Law(θηN), Law(ˆθηN) � = sup h∈Lip(1) ���PNηh(w) − ˆPNηh(w) ��� ≤ sup h∈Lip(1) ��� ˆP(N−1)η(Pη − ˆPη)h(w) ��� + N−1 � i=1 sup h∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (38) Let us first bound the first term in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For any h ∈ Lip(1) and η < 1, by applying Lemma 18, we get ���(Pη − ˆPη)h(w) ��� ≤ � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + ∥w∥)η1+ 1 α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence, we have sup h∈Lip(1) ��� ˆP(N−1)η(Pη − ˆPη)h(w) ��� ≤ � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + E∥ˆθw (N−1)η∥)η1+ 1 α + ρ(Xn, ˆXn)K2(2E∥ˆθw (N−1)η∥ + 1)η (39) ≤ � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥))η1+ 1 α + ρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η, where we applied Lemma 14 to obtain the last inequality above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Next, let us bound the second term in (38) and hence bound the 1-Wasserstein distance W1 � Law(θηN), Law(ˆθηN) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We consider three cases: (I) N = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (II) 2 ≤ N ≤ η−1 + 1 and (III) N > η−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Case (I): N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' One can apply (39) and obtain W1 � Law(θη), Law(ˆθη) � 16 ≤ � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + E∥ˆθw 0 ∥)η1+ 1 α + ρ(Xn, ˆXn)K2(2E∥ˆθw 0 ∥ + 1)η = � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + ∥w∥)η1+ 1 α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (40) This completes the proof of part (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Case (II): 2 ≤ N ≤ η−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By Lemma 18, for any i ≥ 1, we have ���(Pη − ˆPη)P(N−i)ηh(w) ��� ≤ ∥∇P(N−i)ηh∥∞ �� K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + ∥w∥)η1+ 1 α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η � ≤ eL �� K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + ∥w∥)η1+ 1 α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η � , (41) where we used Lemma 16 and the fact that for any i ≥ 1 and 2 ≤ N ≤ η−1 +1 we have (N −i)η ≤ 1 in the inequality (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By applying Lemma 14, we obtain sup h∈Lip(1) ��� ˜P(i−1)η(Pη − ˆPη)P(N−i)ηh(w) ��� ≤ eL �� K1 + ρ(Xn, ˆXn)K2 � (2C) � 1 + E∥ˆθ(i−1)η∥ � η1+ 1 α + ρ(Xn, ˆXn)K2 � 2E∥ˆθ(i−1)η∥ + 1 � η � ≤ eL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η1+ 1 α + eLρ(Xn, ˆXn)K2(2C0(1 + ∥w∥) + 1)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (42) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' we conclude that W1 � Law(θηN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Law(ˆθηN) � ≤ sup h∈Lip(1) ��� ˆP(N−1)η(Pη − ˆPη)h(w) ��� + N−1 � i=1 sup h∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) ��� ≤ � K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η1+ 1 α + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 (2C0(1 + ∥w∥) + 1) η + (N − 1)eL � K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η1+ 1 α + (N − 1)eLρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 (2C0(1 + ∥w∥) + 1) η ≤ � K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η1+ 1 α + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 (2C0(1 + ∥w∥) + 1) η + eL � K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η 1 α + eLρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 (2C0(1 + ∥w∥) + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (43) This completes the proof of (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Case (III): N > η−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We can compute that sup h∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) ��� = sup h∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P1P(N−i)η−1h(w) ��� 17 ≤ sup g∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P1g(w) ��� sup h∈Lip(1) ∥∇P(N−i)η−1h∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By Lemma 15, for any h ∈ Lip(1), |Pth(w) − Pth(y)| ≤ C1e−λt∥w − y∥, (44) for any t ≥ 0 and w, y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This implies that for any h ∈ Lip(1), ∥∇Pth∥∞ ≤ C1e−λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (45) Hence, we conclude that sup h∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) ��� ≤ C1e−λ((N−i)η−1) sup g∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P1g(w) ��� , where i ≤ ⌊N − η−1⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' by Lemma 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' we have ���PηP1g(w) − ˆPηP1g(w) ��� ≤ ∥∇P1g∥∞ �� K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + ∥w∥)η1+ 1 α + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2(2∥w∥ + 1)η � ≤ eL �� K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + ∥w∥)η1+ 1 α + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2(2∥w∥ + 1)η � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (46) which by Lemma 14 implies that sup g∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P1g(w) ��� ≤ eL �� K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + E∥θw (i−1)η∥)η1+ 1 α + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � 2E∥θw (i−1)η∥ + 1 � η � ≤ eL �� K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η1+ 1 α + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 (2C0(1 + ∥w∥) + 1) η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Therefore, we have ⌊N−η−1⌋ � i=1 sup h∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) ��� ≤ ⌊N−η−1⌋ � i=1 C1e−λ((N−i)η−1)eL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η1+ 1 α + ⌊N−η−1⌋ � i=1 C1e−λ((N−i)η−1)eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η ≤ C1λ−1eλeL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥))η 1 α + C1λ−1eλeLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) , where we used the fact that ⌊N−η−1⌋ � i=1 e−λ((N−i)η−1) ≤ eλ � N−1 ⌊η−1⌋−1 e−ληrdr ≤ eλη−1 � ∞ 0 e−λrdr = λeλη−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 18 Next, when i ≥ ⌊N − η−1⌋ + 1, by applying (42), we have N−1 � i=⌊N−η−1⌋+1 sup h∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) ��� ≤ N−1 � i=⌊N−η−1⌋+1 eL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η1+ 1 α + N−1 � i=⌊N−η−1⌋+1 eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η ≤ eL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η 1 α + N−1 � i=⌊N−η−1⌋+1 eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Therefore, we obtain N−1 � i=1 sup h∈Lip(1) ��� ˆP(i−1)η(Pη − ˆPη)P(N−i)ηh(w) ��� ≤ � C1λ−1eλ + 1 � eL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η 1 α + � C1λ−1eλ + 1 � eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence, we conclude that W1 � Law(θηN), Law(ˆθηN) � ≤ sup h∈Lip(1) ��� ˆP(N−1)η � Pη − ˆPη � h(w) ��� + N−1 � i=1 sup h∈Lip(1) ��� ˆP(i−1)η � Pη − ˆPη � P(N−i)ηh(w) ��� ≤ � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η1+ 1 α + ρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η + � C1λ−1eλ + 1 � eL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η 1 α + � C1λ−1eλ + 1 � eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (47) This completes the proof of part (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (ii) Now, we are ready prove part (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By triangle inequality, W1 (µ, ˆµ) ≤ W1 (Law(θηN), µ) + W1 � Law(θηN), Law(ˆθηN) � + W1 � Law(ˆθηN), ˆµ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' It follows from Lemma 14 that by letting N → ∞, we have W1 (µ, ˆµ) ≤ lim sup N→∞ W1 � Law(θηN), Law(ˆθηN) � ≤ � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η1+ 1 α + ρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) η 19 + � C1λ−1eλ + 1 � eL � K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥)) η 1 α + � C1λ−1eλ + 1 � eLρ(Xn, ˆXn)K2 (2C0(1 + ∥w∥) + 1) , (48) where we used part (III) from part (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Since W1 (µ, ˆµ) is independent of η and the initial state x ∈ Rd, we can set η = 0 and x = 0 in (48) and conclude that W1 (µ, ˆµ) ≤ � C1λ−1eλ + 1 � eLρ(Xn, ˆXn)K2 (2C0 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2 Proof of Lemma 7 Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' First of all, the infinitesimal generator of θt process is given by Lαf(θ) = � −∇ ˆF(θ, Xn), ∇f(θ) � + (−∆)α/2f(θ), (49) where (−∆)α/2 is the fractional Laplacian operator defined as a principal value integral: (−∆)α/2f(θ) = dα · p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' � Rd(f(θ + y) − f(θ)) dy ∥y∥α+d , (50) where (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [30]) dα := 2αΓ � d+α 2 � π−d/2 |Γ(−α/2)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (51) Next, let V (w) := (1 + ∥w∥2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We derive from Assumption 4 that for any dataset Xn ∈ X n, we have the following property: ∥∇ ˆF(0, Xn)∥ ≤ B, � ∇ ˆF(θ1, Xn) − ∇ ˆF(θ2, Xn), θ1 − θ2 � ≤ −m∥θ1 − θ2∥2 + K, and ���∇v∇ ˆF(θ, Xn) ��� ≤ L∥v∥, ���∇v1∇v2∇ ˆF(θ, Xn) ��� ≤ M∥v1∥∥v2∥, for any v, v1, v2 ∈ Rd so that we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' It is shown in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' in [6] that V ∈ D(Lα), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' the domain of the infinitesimal generator Lα and moreover LαV (w) ≤ −λ1V (w) + q1, (52) where λ1 := 1 2m, q1 := m + K + B + Cd,α, (53) where Cd,α := dα √ dσd−1 2 − α + dασd−1 α − 1 = 2αΓ � d+α 2 � π−d/2√ dσd−1 |Γ(−α/2)|(2 − α) + 2αΓ � d+α 2 � π−d/2σd−1 |Γ(−α/2)|(α − 1) , (54) 20 where σd−1 := 2π d 2 /Γ(d/2) is the surface area of the unit sphere in Rd, a positive constant that depends only on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Next, let us define the extended infinitesimal generator Lα t : Lα t f(t, θ) := ∂tf(t, θ) + Lαf(t, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (55) Then, it follows from (52) that Lα t eλ1tV (θ) = λ1eλ1tV (θ) + eλ1tLαV (θ) ≤ λ1eλ1tV (θ) + eλ1t (−λ1V (w) + q1) = q1eλ1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (56) By Dynkin’s formula, E � eλ1tV (θw t ) � = V (w) + E �� t 0 Lα s eλ1sV (θw s ) ds � ≤ V (w) + � t 0 q1eλ1sds = V (w) + q1 eλ1t − 1 λ1 , which implies that E∥θw t ∥ ≤ E [V (θw t )] ≤ e−λ1tV (w) + q1 1 − e−λ1t λ1 ≤ 1 + ∥w∥ + q1 λ1 , (57) where we used V (w) = (1 + ∥w∥2)1/2 and the inequality ∥w∥ ≤ (1 + ∥w∥2)1/2 ≤ 1 + ∥w∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence, we have E∥θw t ∥ ≤ C0(1 + ∥w∥), (58) where we take C0 := 1 + q1 λ1 = 1 + 2 (m + K + B + Cd,α) m = 3 + 2 (K + B) m + 2 m � 2αΓ � d+α 2 � π−d/2√ dσd−1 |Γ(−α/2)|(2 − α) + 2αΓ � d+α 2 � π−d/2σd−1 |Γ(−α/2)|(α − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Since C0 in the above bound is uniform in the dataset, similarly, we also have E∥ˆθw t ∥ ≤ C0(1 + ∥w∥), (59) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='3 Proof of Proposition 8 Proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let us first prove part (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' First, we can re-write g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) as g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) = 2αΓ � d+α 2 � |Γ(−α/2)|(2 − α) �√ d + 2 − α α − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (60) By the properties of the gamma function, we have Γ � 2 − α 2 � = � 1 − α 2 � Γ � 1 − α 2 � = � 1 − α 2 � −α 2 Γ(−α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' 21 Therefore, we have |Γ(−α/2)|(2 − α) = 4 αΓ � 2 − α 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (61) Moreover, by the properties of the gamma function, Γ � 2 − α 2 � = Γ � 1 − �α 2 − 1 �� = π sin � π � α 2 − 1 �� Γ � α 2 − 1 � = π � α 2 − 1 � sin � π � α 2 − 1 �� Γ � α 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence, we conclude that g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) = 2α−2αΓ �d + α 2 � Γ �α 2 � sin � π � 1 − α 2 �� π � 1 − α 2 � �√ d + 2 − α α − 1 � , where we used sin(−x) = − sin(x) for any x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let us define h(x) := sin(x) x for any 0 ≤ x ≤ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We can compute that h′(x) = x cos(x)−sin(x) x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let p(x) := x cos(x) − sin(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Then p(0) = 0 and p′(x) = −x sin(x) < 0 for any 0 < x < π/2 which implies that p(x) < 0 and thus h′(x) < 0 for any 0 < x < π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence h(x) is decreasing in x for any 0 ≤ x ≤ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' As a result, the map α �→ sin � π � 1 − α 2 �� π � 1 − α 2 � is increasing in α for any 1 < α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (62) It is well known that gamma function x �→ Γ(x) is log-convex for x > 0 and thus convex for any x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Since Γ(1) = Γ(2) = 1, there exists a unique critical value c0 ∈ (1, 2) such that the gamma function x �→ Γ(x) is increasing for any x ≥ c0 and decreasing for any 1 ≤ x ≤ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Next, for any given α0 ∈ (1, 2) such that 1 + α0 2 ≥ c0, we have for any 2 ≥ α2 > α1 ≥ α0 and d ≥ 2, g(α2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) g(α1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) = 2α2−2α2Γ � d+α2 2 � Γ � α2 2 � sin(π(1− α2 2 )) π(1− α2 2 ) �√ d + 2−α2 α2−1 � 2α1−2α1Γ � d+α1 2 � Γ � α1 2 � sin(π(1− α1 2 )) π(1− α1 2 ) �√ d + 2−α1 α1−1 � = 2α2Γ � d+α2 2 � Γ � 1 + α2 2 � sin(π(1− α2 2 )) π(1− α2 2 ) �√ d + 2−α2 α2−1 � 2α1Γ � d+α1 2 � Γ � 1 + α1 2 � sin(π(1− α1 2 )) π(1− α1 2 ) �√ d + 2−α1 α1−1 � ≥ 2α2−α1 √ d + 2−α2 α2−1 √ d + 2−α1 α1−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (63) where we used (62) and the fact that the gamma function x �→ Γ(x) is increasing in x ≥ 1+ α0 2 ≥ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Next, let us define the function: q(x) := 2x−α1 √ d + 2−x x−1 √ d + 2−α1 α1−1 , (64) where 2 ≥ x ≥ α1 ≥ α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' It is clear that q(α1) = 1 and moreover, we can compute that q′(x) = log(2)2x−α1 √ d + 2−x x−1 √ d + 2−α1 α1−1 − 2x−α1 (x − 1)2 1 √ d + 2−α1 α1−1 22 = 2x−α1 √ d + 2−α1 α1−1 � log(2) �√ d + 2 − x x − 1 � − 1 (x − 1)2 � ≥ 2x−α1 √ d + 2−α1 α1−1 � log(2) √ d − 1 (α0 − 1)2 � ≥ 0, (65) provided that d ≥ 1 (log 2)2(α0 − 1)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (66) This implies that q(x) is increasing for 2 ≥ x ≥ α1 ≥ α0 provided that d ≥ 2, 1 + α0 2 ≥ c0 and (66) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence, we conclude that g(α2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) ≥ g(α1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) for any d ≥ d0 = max � 2, 1 (log 2)2(α0−1)4 � , and 2 ≥ α2 ≥ α1 ≥ α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Now, let us prove part (ii) of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We recall from (60) that g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) = 2αΓ � d+α 2 � |Γ(−α/2)|(2 − α) �√ d + 2 − α α − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (67) We can compute that ∂ ∂αg(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) = 2αΓ � d+α 2 � |Γ(−α/2)|(2 − α) −1 (α − 1)2 + ∂ ∂α � 2αΓ � d+α 2 � Γ(−α/2)(α − 2) � �√ d + 2 − α α − 1 � , where we can further compute that ∂ ∂α � 2αΓ � d+α 2 � Γ(−α/2)(α − 2) � = log(2)2αΓ � d+α 2 � + 2α−1Γ � d+α 2 � ψ � d+α 2 � Γ(−α/2)(α − 2) − 2αΓ � d+α 2 � � − 1 2(α − 2)ψ(− α 2 ) + 1 � Γ(−α/2)(α − 2)2 , where ψ(·) denotes the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This implies that ∂ ∂αg(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) = 2αΓ � d+α 2 � |Γ(−α/2)|(2 − α)p(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d), (68) where p(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) := −1 (α − 1)2 + � log(2) + 1 2ψ �d + α 2 �� �√ d + 2 − α α − 1 � + �1 2ψ � −α 2 � + 1 2 − α � �√ d + 2 − α α − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By the property of the digamma function, we have ψ(− α 2 ) = ψ(1 − α 2 ) + 2 α and ψ(x) is increasing in x > 0 and ψ(−1/2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Therefore, for any 1 < α ≤ α0, we have p(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) = −1 (α − 1)2 + � log(2) + 1 2ψ �d + α 2 �� �√ d + 2 − α α − 1 � + �1 2ψ � 1 − α 2 � + 1 α + 1 2 − α � �√ d + 2 − α α − 1 � 23 ≤ −1 (α − 1)2 + � log(2) + 1 2ψ �d + α0 2 �� �√ d + 1 α − 1 � + � 1 + 1 2 − α0 � �√ d + 1 α − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' It follows that p(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) ≤ 0 holds if y0 √ d(α − 1)2 + y0(α − 1) − 1 ≤ 0, (69) where y0 := log(2) + 1 2ψ(d + α 2 ) + 3−α0 2−α0 , and it is easy to compute that (69) holds provided that α ≤ 1 + −1 + � 1 + 4y−1 0 √ d 2 √ d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (70) Hence, we conclude that p(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) is non-positive and thus ∂ ∂αg(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) is non-positive (by (68)) and therefore g(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' d) is decreasing for any α ∈ [1, α′ 0], where α′ 0 := min � α0, 1 + −1+√ 1+4y−1 0 √ d 2 √ d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='4 Proof of Proposition 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Due to our choice of the loss function f, the SDEs (11) and (12) reduce to Ornstein- Uhlenbeck processes driven by a symmetric α-stable Lévy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence, we can characterize the invariant distributions of the SDEs as follows (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [24]): θ∞ =d µ + σξ, and ˆθ∞ =d ˆµ + ˆσˆξ, (71) for some µ, ˆµ ∈ R and σ, ˆσ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Here, ξ and ˆξ are SαS(1) distributed (see Section 2 for definition) and =d denotes equality in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Now recall that µ = Law(θ∞) and ν = Law(ˆθ∞), and the p-Wasserstein metric for one-dimensional distributions is given by, Wp p(µ, ν) = inf γ∈Γ(µ,ν) E(x,y)∼γ(x,y)|x − y|p, where Γ(µ, ν) is the set of all couplings of µ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' In our case, x ∈ R and y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For any coupling γ⋆ ∈ Γ(µ, ν), we have � R×R |x − y|p dγ⋆(x, y) = � R×R � |x − y|2�p/2 dγ⋆(x, y) = � R×R (x2 + y2 − 2xy)p/2 dγ⋆(x, y) ≥ � R+×R− (x2 + y2 − 2xy)p/2 dγ⋆(x, y) ≥ � R+×R− (|x|p + |y|p + |2xy|p/2) dγ⋆(x, y) ≥ � R+×R− |x|p dγ⋆(x, y) + � R+×R− |y|p dγ⋆(x, y) = C1 � R+ |x|p dµ(x) + C2 � R− |y|p dν(y) = +∞, 24 where C1 and C2 are some finite, positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' The last equation comes from the properties of the α-stable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Since it holds for any γ⋆ ∈ Γ(µ, ν), we conclude that Wp p(µ, ν) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='5 Proof of Corollary 11 Corollary 13 (Restatement of Corollary 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Under the assumptions in Theorem 12 and Lemma 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' we have: (i) For any 2 ≤ N ≤ η−1 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' W1 � Law(θηN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Law(ˆθηN) � ≤ � K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C)(1 + C0(1 + ∥w∥))η1+ 1 α + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2(2C0(1 + ∥w∥) + 1)η + eL � K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥))η 1 α + eLρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2(2C0(1 + ∥w∥) + 1) + 2Q(1 + ∥w∥)η2/α−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (72) and for any N > η−1 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' W1 � Law(θηN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Law(ˆθηN) � ≤ � K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥))η1+ 1 α + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2(2C0(1 + ∥w∥) + 1)η + � C1λ−1eλ + 1 � eL � K1 + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � (2C) (1 + C0(1 + ∥w∥))η 1 α + � C1λ−1eλ + 1 � eLρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2(2C0(1 + ∥w∥) + 1) + 2Q(1 + ∥w∥)η2/α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (73) (ii) We have W1(µ, ˆµ) ≤ � C1λ−1eλ + 1 � eLρ(Xn, ˆXn)K2 (2C0 + 1) + 2Qη2/α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (74) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let us prove part (ii) and the proof for part (i) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' It follows directly from Lemma 10 and Theorem 5 and the inequality: W1(ν, ˆν) ≤ W1(ν, µ) + W1(ˆν, ˆµ) + W1(µ, ˆµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (75) The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' C Technical Lemmas In this section, we provide some technical results that are used in the proofs of main results in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' First, we have the following technical result from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lemma 14 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Under Assumption 4, (θw t )t≥0 and (ˆθw t )t≥0 admit unique invariant probability measures µ and ˆµ respectively such that sup |f|≤V |E[f(θw t )] − µ(f)| ≤ c1V (w)e−c2t, for any t > 0, (76) sup |f|≤V ���E[f(ˆθw t )] − ˆµ(f) ��� ≤ c1V (w)e−c2t, for any t > 0, (77) 25 for some constants c1, c2 > 0 where V (w) := (1 + ∥w∥2)1/2 is a Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' In particular, there exists a constant C0 > 0 such that E∥θw t ∥ ≤ C0(1 + ∥w∥), for any t > 0, (78) E∥ˆθw t ∥ ≤ C0(1 + ∥w∥), for any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (79) Moreover, we recall the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lemma 15 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2 in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' There exist constants C1, λ > 0 such that for any t > 0 and w, y ∈ Rd, we have W1 (Law (θw t ) , Law (θy t )) ≤ C1e−λt∥w − y∥, (80) W1 � Law � ˆθw t � , Law � ˆθy t �� ≤ C1e−λt∥w − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (81) Let Pt and ˆPt denote the Markov semigroups of θt and ˆθt processes respectively, that is, for any bounded function f : Rd → R, Ptf(x) = Ef(θw t ), ˆPtf(x) = Ef(ˆθw t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (82) We have the following technical lemma from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lemma 16 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='1 in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' For any h ∈ Lip(1) and v, w ∈ Rd and t ∈ (0, 1], we have ∥∇vPth(w)∥ ≤ eL∥v∥, ∥∇v ˆPth(w)∥ ≤ eL∥v∥, (83) where L is defined in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We recall the following technical lemma from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lemma 17 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2 in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' There exist constants C > 0 such that for all w ∈ Rd, t ≥ 0, we have E∥θw t − w∥ ≤ C(1 + ∥w∥) � t ∨ t1/α� , (84) E∥ˆθw t − w∥ ≤ C(1 + ∥w∥) � t ∨ t1/α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (85) Next, we state and prove the following key technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' There exist constants C > 0 such that for all w ∈ Rd, η ∈ (0, 1), f : Rd → R with ∥∇f∥∞ < ∞, we have ���Pηf(w) − ˆPηf(w) ��� ≤ ∥∇f∥∞ �� K1 + ρ(Xn, ˆXn)K2 � 2C(1 + ∥w∥)η1+ 1 α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (86) Proof of Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' We can compute that ���E � f � θw η � − f � ˆθw η ����� = ����E � f � w + � η 0 ∇ ˆF(θw r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Xn)dr + Lα η � − f � w + � η 0 ∇ ˆF(ˆθw r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)dr + Lα η ������ 26 ≤ ∥∇f∥∞E ���� � η 0 ∇ ˆF(θw r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Xn)dr − � η 0 ∇ ˆF(ˆθw r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)dr ���� ≤ ∥∇f∥∞E � η 0 ���∇ ˆF(θw r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Xn) − ∇ ˆF(ˆθw r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn) ��� dr ≤ ∥∇f∥∞E � η 0 � K1∥θw r − ˆθw r ∥ + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � ∥θw r ∥ + ∥ˆθw r ∥ + 1 �� dr = ∥∇f∥∞ � K1 � η 0 E∥θw r − ˆθw r ∥dr + ρ(Xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' ˆXn)K2 � η 0 E � ∥θw r ∥ + ∥ˆθw r ∥ + 1 � dr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By Lemma 17, we have � η 0 E∥θw r − ˆθw r ∥dr ≤ � η 0 E∥θw r − w∥dr + � η 0 E∥ˆθw r − w∥dr ≤ C(1 + ∥w∥) � η 0 r1/αdr + C(1 + ∥w∥) � η 0 r1/αdr ≤ 2C(1 + ∥w∥)η1+ 1 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' By applying Lemma 17 again, we have � η 0 E � ∥θw r ∥ + ∥ˆθw r ∥ + 1 � dr ≤ � η 0 E � ∥θw r − w∥ + ∥ˆθw r − w∥ + 2∥w∥ + 1 � dr ≤ � η 0 � C(1 + ∥w∥)r1/α + C(1 + ∥w∥)r1/α + 2∥w∥ + 1 � dr ≤ 2C(1 + ∥w∥)η1+ 1 α + (2∥w∥ + 1)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Hence, we conclude that ���E � f(θw η ) − f(ˆθw η ) ���� ≤ ∥∇f∥∞ �� K1 + ρ(Xn, ˆXn)K2 � (2C) (1 + ∥w∥)η1+ 1 α + ρ(Xn, ˆXn)K2(2∥w∥ + 1)η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Lemma 19 (Restatement of Lemma 10 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' [6])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let µt and ˆµt denote the distributions of continuous-time θt and ˆθt and µ and ˆµ denote the distributions of continuous-time θ∞ and ˆθ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Moreover, let νk and ˆνk denote the distributions of discrete-time θk and ˆθk and ν and ˆν denote the distributions of discrete-time θ∞ and ˆθ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Assume the dynamics start at w at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Let m, L be as in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' Then, there exists some constant Q (that may depend on B, m, K, L, M from Assumption 4) such that the followings hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (i) For every N ≥ 2 and η < min{1, m/(8L2), 1/m}, one has W1(µNη, νN) ≤ Q(1 + ∥w∥)η2/α−1, (87) W1(ˆµNη, ˆνN) ≤ Q(1 + ∥w∥)η2/α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (88) (ii) For every η < min{1, m/L2, 1/m}, one has W1(µ, ν) ≤ Qη2/α−1, (89) W1(ˆµ, ˆν) ≤ Qη2/α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} +page_content=' (90) 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FKT4oBgHgl3EQftC6j/content/2301.11885v1.pdf'} diff --git a/hNE2T4oBgHgl3EQfcQei/content/2301.03894v1.pdf b/hNE2T4oBgHgl3EQfcQei/content/2301.03894v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1f258f793632141aaecc394bbc6f6fe214691ef4 --- /dev/null +++ b/hNE2T4oBgHgl3EQfcQei/content/2301.03894v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:069e440dfb508b2cbcedea7341b40cdd8e87e2411a21c486740c411baa6440db +size 899698 diff --git a/hNE2T4oBgHgl3EQfcQei/vector_store/index.pkl b/hNE2T4oBgHgl3EQfcQei/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8fa9e9d3598272e837fd6e8685e7b0f433fc6c14 --- /dev/null +++ b/hNE2T4oBgHgl3EQfcQei/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f707adcbc30c21f9a1aabdf2726a20056571a9e06dfe5b5b2328f72d4a5e9069 +size 195891 diff --git a/hdAyT4oBgHgl3EQfXvcJ/content/tmp_files/2301.00187v1.pdf.txt b/hdAyT4oBgHgl3EQfXvcJ/content/tmp_files/2301.00187v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6120c20a1c4d5c4a290028a1657a6d11fda5af59 --- /dev/null +++ b/hdAyT4oBgHgl3EQfXvcJ/content/tmp_files/2301.00187v1.pdf.txt @@ -0,0 +1,1515 @@ +Collective modes in the anisotropic collisional hot QCD medium in the presence of +finite chemical potential +Mohammad Yousuf Jamal1, ∗ +1Indian Institute of Technology Goa, Ponda, Goa, 403401, India +The collective modes that are the collective excitations in the hot QCD medium produced in +the heavy-ion collision experiments have been studied. These modes being real or imaginary and +stable or unstable affect the evolution of the medium. To go into the more profound analysis we +incorporated several aspects of the medium such as anisotropy, the presence of medium particle +collisions, and finite baryonic chemical potential. The first two facets have been studied several +times from different perspectives but the inclusion of finite chemical potential is a missing part. +Therefore, to have a close analysis, we have combined these effects. The medium particle collision +has been included using BGK- collisional kernel, the anisotropy, and finite chemical potential enter +via quarks, anti-quarks and gluons distribution functions. Our results indicate that the presence +of finite chemical potential enhances the magnitude of unstable modes which may affect the fast +thermalization of the hot QCD medium. +Moreover, it is interesting to see the consequences of +finite chemical potential along with the other aspects of the created medium from the viewpoint +of low-energy heavy-ion collision experiments that operate at low temperatures and finite baryon +density. +Keywords: +Quark-Gluon-Plasma, Collective excitations, Collective modes, particle distribu- +tions function, BGK collisional kernel, Anisotropic QCD, Gluon self-energy, finite chemical +potential. +I. +INTRODUCTION +In the relativistic heavy-ion collision (HIC) experi- +ments we obtain such an extremely high energy density +than any we know of in the present natural universe and +where the QCD predicts a new form of matter consisting +of an extended volume of interacting quarks, antiquarks, +and gluons known as quark-gluon plasma (QGP) [1, 2]. +Such a high energy density phase, however, is thought +to have existed a few microseconds after the Big Bang. +In this phase, the effective degrees of freedom are quarks, +anti-quarks, and gluons rather than the hadronic matter. +The historic discovery of the J/Ψ, a bound state of the +charmed quark and its anti-quark in the year 1974 first +time proved the existence of the heavy quarks which indi- +cated that the nucleus–nucleus collisions at high energy +are very different from a simple superposition of nucleon- +nucleon interactions. As it mimics the cosmic Big Bang, +the term mini-bang has been coined to describe these in- +teractions, in which nuclear collisions are thought to pro- +ceed in a series and cause the formation of matter. The +presumed line of events begins with an extremely high- +temperature region inhabited by the two nuclei at the +moment of collision, as a large fraction of their kinetic +energy is converted into heat. +At thermal equilibrium +it forms a high-temperature plasma medium consisting +of quarks, anti-quarks, and gluons that instantly begins +to expand and cool, passing down through the critical +temperature at which the QGP condenses into a system +of mesons, baryons, or simply hadrons. On further ex- +pansion, the system reaches its “freeze-out” density, at +∗ mohammad@iitgoa.ac.in +which the hadrons no longer interact with each other and +stream into the detectors. +There are several theoretical investigations based on +various approaches such as semi-classical transport or ki- +netic, hard-thermal loop (HTL), ASD-CFT, holographic +theories have been invoked to study the produced mat- +ter [3, 4]. There have been several signatures observed +that support the formation of the QGP in these experi- +ments such as suppression of heavy quarkonia (a colorless +and flavorless bound state of a heavy quark and its anti- +quark) yields at high transverse momentum, color screen- +ing [5–8], Landau damping [9, 10], energy loss or gain of +heavy quarks [11–15], etc. The observables that we see +at the detector are affected by the presence of the QGP +medium that in turn, is influenced by several plasma as- +pects. One of the important factors among them is the +collective excitations or collective modes produced in the +hot QGP medium. It has been explored in previous stud- +ies that these modes can be real or imaginary, stable or +unstable [16–21]. Each of them plays a significant role +in altering the observed outcomes at the detector. +In many-body systems, the spectrum of collective exci- +tations is a fundamental part as it possesses information +regarding the thermodynamic and transport properties +of that system along with the temporal evolution of a +non-equilibrium system +[22]. The investigation begins +with the analogical idea from Quantum Electro Dynam- +ics (QED) that says that the evolving medium produced +in the HIC could also contain instabilities that are sim- +ilar to Weibel or filamentation instabilities [23] present +in electromagnetic plasma (EMP) that may contribute +to accomplishing the fast thermalization +[24]. In par- +ticular, at very high-temperature, QCD resembles QED +due to the small coupling constant. The most common +arXiv:2301.00187v1 [nucl-th] 31 Dec 2022 + +2 +feature of EMP and QGP is collective behavior that sug- +gests the QGP could also have a very rich spectrum of +collective modes they are there in EMP [25, 26]. In the +present context, these modes could refer to plasma parti- +cles that could be real or imaginary and stable or unsta- +ble. The stable modes have infinite lifetimes whereas the +unstable modes decay quickly. The stable modes have a +long-range interaction with the heavy quarks traversing +through the QGP medium whereas the unstable modes +are only found in anisotropic plasma and may help in the +fast thermalization/equilibration of the system, a lesser- +known puzzle that attracts the curiosity of investigating +the unstable modes and hence, the prior is less studied +and discussed in the literature though the latter has been +discussed widely. +In several analyses the question about the formation, +existence, and effects of the collective modes have been +discussed considering various plasma aspects such as +anisotropy, medium particle collisions, etc, [27–34]. Here, +we are interested in a crucial aspect which is to under- +stand the inclusion and effects of finite baryon density in +the study of these modes. Thrusting to very high baryon +density at low temperatures, i.e., squeezing nuclei with- +out heating them takes us into another fascinating region +of the QCD phase diagram. The high baryon density gen- +erally means the surplus of quarks over antiquarks in the +hot system parameterized by the baryon chemical poten- +tial, µ. The zero chemical potential refers to the equal +densities of quarks and antiquarks which is a good ap- +proximation for the matter produced at midrapidity at +very high energy HIC such as RHIC and LHC, and ex- +ceptionally good for the early Universe but not at the +lower energies. It is essential from the viewpoint of the +experimental facilities which perform at moderate tem- +peratures along with finite baryon density such as the +Super Proton Synchrotron (SPS) at CERN, Switzerland, +and also upcoming planned experimental facilities such +as the Facility for Antiproton and Ion Research (FAIR) +in Darmstadt, Germany and the Nuclotron-Based Ion +Collider Facility (NICA) in Dubna, Russia, and at the +Japan Proton Accelerator Research Complex (J-PARC) +in T¯okai, Japan. To have a comprehensive study of the +collective modes we shall also include the momentum +anisotropy and medium particle collisions along with the +finite chemical potential. The momentum anisotropy and +the chemical potential will enter via particle distribution +functions and for medium particle collisions we shall con- +sider the Bhatnagar–Gross–Krook (BGK) collisional ker- +nel in the Boltzman transport equation. The dispersion +relations for the modes are acquired from the poles of +the (gluon) propagator. Based on their solutions it can +be identified if the modes are stable or unstable and real +or imaginary which we shall discuss in detail in the up- +coming sections. +The current manuscript is arranged as follows. In sec- +tion II, the methodology to obtain the dispersion rela- +tions of the modes will be discussed. Section +III, con- +tains a detailed discussion of the results. Finally, sec- +tion IV, is dedicated to the summary and conclusions of +the present work along with the potential future prob- +lems. Here, natural units are used throughout the text +with c = kB = ℏ = 1. We use a bold typeface to dis- +play three vectors and a regular font to indicate four +vectors. +The centre dot depicts the four-vector scalar +product with gµν = diag(1, −1, −1, −1). +II. +METHODOLOGY +The formation of collective modes in the QGP medium +can be understood as if one assumes the QGP initially +in a homogeneous and stationary state with no net lo- +cal color charges and no net currents. +Next, suppose +this state is slightly perturbed by either a random fluc- +tuation or some external field that induces local charges +or currents in the plasma that in turn generate chromo- +electric and chromo-magnetic fields. These fields interact +back with colored partons that contributed to their for- +mation. If the wavelength of the perturbation surpasses +the typical inter-particle spacing, the plasma undergoes a +collective motion known as collective modes/excitations. +The collective modes can be specified by the nature of +the solutions of their dispersion relations (ω(k), k being +the wave vector) that can be fetched from the poles of the +gluon propagator. The solutions to the dispersion equa- +tions, ω(k) could be complex-valued. For ℑ(ω(k)) = 0 +there exist only stable modes. If ℑ(ω(k)) > 0 the am- +plitude of the mode exponentially grows with time as +eℑ(ω(k))t that represents the instability. If ℑ(ω(k)) < 0 +the mode is damped and the mode amplitude exponen- +tially decays with time as e−ℑ(ω(k))t. Likewise, if we have, +−ℑ(ω(k)) ≥ |ℜ(ω(k))| the modes will be over damped. +To obtain the gluon propagator we first derive the gluon +polarization tensor that contains the medium informa- +tion such as medium anisotropy, finite chemical poten- +tial, the effect of medium particle collision, etc that we +shall discuss next. +A. +Gluon polarization tensor +The gluon polarization tensor or gluon self-energy +(Πµν), bears the information of the QCD medium as it +represents the interactions term in the effective action +of the QCD. Before we start our derivation, we spec- +ify that our working scale is the soft momentum scale +where the plasma aspects first appear, i.e., k ∼ gT ≪ T, +( g being the strong coupling constant). At this scale, +one can obtain that the strength of field fluctuations, +Aµ ∼ O(√gT), and the derivatives, ∂x ∼ O(gT). Now, +if we examine the field strength tensor, +F µν +a += ∂µAν +a − ∂νAµ +a − ig [Aµ +b , Aν +c] , +(1) +we notice that the order of the non-abelian term is O(g2) +which is smaller than the order of the first two terms i.e., +O(g3/2) in F µν and hence can be neglected. Now we end + +3 +up with the abelian part of F µν. Next, in the abelian +limit, the linearized semi-classical transport equations for +each color channel are given as [16, 28, 31, 35], +vµ∂µδf i +a(p, X) + gθivµF µν +a (X)∂(p) +ν f i(p) = Ci +a(p, X), +(2) +where the index ‘i‘ refers to the particle species (quark, +anti-quark, and gluon), ‘a‘, ‘b‘, and‘c‘ are the color index +and θi ∈ {θg, θq, θ¯q} and have the values θg = θq = 1 +and θ¯q = −1. The four vectors xµ = (t, x) = X and +vµ = (1, v) = V , are respectively, the four space-time +coordinate and the velocity of the plasma particle with +v = p/|p|. The ∂µ, ∂(p) +ν +are the partial four derivatives +corresponding to space and momentum, respectively. As +mentioned earlier the collisional term Ci +a(p, X), is the +BGK - kernel +[28, 35–37] that describes the effects of +collisions between hard particles in a hot QCD medium +given as, +Ci +a(p, X) = −ν +� +f i +a(p, X) − N i +a(X) +N ieq +f i +eq(|p|) +� +, +(3) +where, +f i +a(p, X) = f i(p) + δf i +a(p, X), +(4) +are the distribution functions of quarks, anti-quarks and +gluons, f i(p) is equilibrium part while, δf i +a(p, X) per- +turbed part of the distribution function and +N i +a(X) = +� +d3p +(2π)3 f i +a(p, X), +N i +eq = +� +d3p +(2π)3 f i +eq(|p|) = +� +d3p +(2π)3 f i(p). +(5) +is the particle number and its equilibrium value. +The +BGK - kernel [36] depicts the equilibration of the system +due to the collisions in a time proportional to ν−1. Here, +we consider the collision frequency ν to be independent of +momentum and particle species. We preferred to choose +BGK -kernel because it conserves the particle number +instantaneously as, +� +d3p +(2π)3 Ci +a(p, X) = 0. +(6) +At finite baryon or quark chemical potential µ, the +momentum distributions of gluon, quark, and anti-quark +are given as, +fg = +exp[−βEg] +� +1 − exp[−βEg] +�, +fq = +exp[−β(Eq − µ)] +� +1 + exp[−β(Eq − µ)] +�, +f¯q = +exp[−β(Eq + µ)] +� +1 + exp[−β(Eq + µ)] +�. +(7) +To include the momentum anisotropy, we track the strat- +egy given in Ref. [31]. In this method, the anisotropic dis- +tribution function was obtained from the isotropic distri- +bution function given in Eq. (7) by rescaling (stretching +and squeezing) in one direction of the momentum space +as follows, +f(p) ≡ fξ(p) = f( +� +p2 + ξ(p · ˆn)2), +(8) +where, ˆn is an unit vector (ˆn2 = 1) showing the di- +rection of momentum anisotropy and ξ the strength of +anisotropy. It refers to squeezing when ξ > 0 and stretch- +ing when −1 < ξ < 0 of the distribution function in the +ˆn direction. +Next, the gluon polarization tensor can be obtained +from the current induced due to the change in the parti- +cles distribution functions in the Fourier space as, +Πµν +ab (K) = ∂Jµ +ind a(K) +∂Abν(K) , +(9) +where the induced current is given by [28, 31, 35, 38], +Jµ +ind,a(X) = g +� +d3p +(2π)3 V µ{2Ncδf g +a(p, X) + Nf[δf q +a(p, X) +− δf ¯q +a(p, X)]}. +(10) +Rewriting Eq. (2) as, +vµ∂µδf i +a(p, X) + gθivµF µν +a (X)∂(p) +ν f i(p) = +ν +� +f i +eq(|p|) − f i(p) +� +− νδf i +a(p, X) ++νf i +eq(|p|) +N ieq +�� +d3p′ +(2π)3 δf i +a(p′, X) +� +. +(11) +On solving Eq.(11) for δf i +a(p, X) in the Fourier space we +obtained, + +4 +δf i +a(p, K) = +−igθivµF µν +a (K)∂(p) +ν f i(p) + iν(f i +eq(|p|) − f i(p)) + iνf i +eq(|p|) +�� +d3p +(2π)3 δf i +a(p′, K) +� +/Neq +ω − v · k + iν +, +(12) +where, δf i(p, K) and F µν(K), are the Fourier trans- +forms of δf i(p, X) and F µν(X), respectively. +Where, +K = kµ = (ω, k) Now taking the Fourier transform of the +induced current from Eq. (10) and employing δf i +a(p, K), +from Eq. (12) we get, +Jµ +ind,a(K) = g2 +� +d3p +(2π)3 vµ∂(p) +ν f(p)Mνα(K, V )D−1(K, v, ν)Aαa + giν{2NcSg(K, ν) + Nf(Sq(K, ν) − S ¯q(K, ν))} ++g(iν) +� dΩ +4π vµD−1(K, v, ν) +� +d3p′ +(2π)3 +� +g∂(p′) +ν +f(p′)Mνα(K, V ′)D−1(K, v′, ν)W−1(K, ν)Aαa ++iν(feq(|p′|) − f(p′))D−1(K, v′, ν) +� +W−1(K, ν) , +(13) +where, +D(K, v, ν) = ω + iν − k · v, +Mνα(K, V ) = gνα(ω − k · v) − Kνvα, +f(p) = 2Ncf g(p) + Nf +� +f q(p) + f ¯q(p) +� +, +feq(|p′|) = 2Ncf g +eq(|p′|) + Nf +� +f q +eq(|p′|) + f ¯q +eq(|p′|) +� +, +W(K, ν) = 1 − iν +� dΩ +4π D−1(K, v, ν), +Si(K, ν) = − +� +d3p +(2π)3 vµ[f i(p) − f i +eq(|p|)]D−1(K, v, ν). +(14) +Implying Eq. (9) in Eq.(13), we get +Πµν +ab (K) = δabg2 +� +d3p +(2π)3 vµ∂(p) +β f(p)Mβν(K, V ) +×D−1(K, v, ν) + δabg2(iν) +� dΩ +4π vµ +×D−1(K, v, ν) +� +d3p′ +(2π)3 ∂(p′) +β +f(p′) +Mβν(K, V ′)D−1(K, v′, ν)W−1(K, ν). +(15) +Rewriting Eq. (15), in temporal gauge for anisotropic hot +QCD medium at finite chemical potential, +Πij(K) = m2 +D +� dΩ +4π vi vl + ξ(v · ˆn)nl +(1 + ξ(v · ˆn)2)2 +� +δjl(ω − k · v) + vjkl� +D−1(K, v, ν) + (iν)m2 +D +� dΩ′ +4π (v′)i +×D−1(K, v′, ν) +� dΩ +4π +vl + ξ(v · ˆn)nl +(1 + ξ(v · ˆn)2)2 +� +δjl(ω − k · v) + vjkl� +D−1(K, v, ν)W−1(K, ν), +(16) +where the squared Debye mass is given as, +m2 +D = − g2 +2π2 +� ∞ +0 +dp p2 dfiso(p) +dp +, +(17) +In the limit µ/T ≪ 1, we have +m2 +D = 4παsT 2 +�Nc +3 + Nf +6 + µ2 +T 2 +Nf +2π2 +� +, +(18) +and the strong coupling constant [39], αs = g2/4π at +finite chemical potential is given as, +αs = +6π − +18π(153−19Nf ) ln +� +2 ln +��� +T +ΛT +�2+ +µ2 +π2T 2 +�� +(33−2Nf )2 ln +��� +T +ΛT +�2+ +µ2 +π2T 2 +� +(33 − 2Nf) ln +��� +T +ΛT +�2 ++ +µ2 +π2T 2 +� +. +(19) + +5 +where ΛT ≪ T is the QCD scale parameter. +Here, +Tc = 0.155 GeV +T=2Tc +T=4Tc +0 +20 +40 +60 +80 +100 +0 +10 +20 +30 +40 +μ (GeV) +mD[μ,T] (GeV) +FIG. 1. Variation of screening mass with respect to chemical +potential at different temperatures. +Eq. (17) is fully solved numerically and the variation +of screening mass with respect to the chemical poten- +tial at different temperatures is shown in Fig. 1. It has +been found that mD increases with the chemical poten- +tial whereas the effect of different values of temperature +is negligible. Next, we shall discuss the derivation of the +gluon propagator. +B. +Gluon Propagator +To obtain the gluon propagator, we initiate with +Maxwell’s equation in the Fourier space, +−ikνF νµ(K) = Jµ +ind(K) + Jµ +ext(K). +(20) +Here, Jµ +ext(K) is the external current. The induced cur- +rent, Jµ +ind(K) can be described in terms of self-energy, +Πµν(K) as follows, +Jµ +ind(K) = Πµν(K)Aν(K). +(21) +The Eq.( 20) can also be noted as, +[K2gµν − kµkν + Πµν(K)]Aν(K) = −Jµ +ext(K). (22) +The quantity in the square bracket is the needed gluon +propagator. +Now assuming the temporal gauge, the +above equation can be written as, +[∆−1(K)]ijEj = [(k2 − ω2)δij − kikj + Πij(K)]Ej += iωJi +ext(k), +(23) +where Ej is the physical electric field and +[∆−1(K)]ij = (k2 − ω2)δij − kikj + Πij(K), +(24) +is the inverse of the propagator. The dispersion equations +for collective modes can be obtained from the poles of the +propagator, [∆(K)]ij. +Since, it is a tensorial equation +and hence, can not be simply integrated. To solve it we +first need to construct an analytical form of Πij using the +available symmetric tensors such as, +Πij = αP ij +T + βP ij +L + γP ij +n + δP ij +kn, +(25) +where, P ij +T += δij − kikj/k2, P ij +L += kikj/k2, P ij +n += +˜ni˜nj/˜n2 and P ij +kn = ki˜nj + kj˜ni [31, 33, 40], where, +˜ni = (δij − kikj +k2 )ˆnj is a vector orthogonal to, ki ,i.e., +˜n · k = 0. The structure functions α, β, γ and δ, can be +determined by taking the appropriate projections of the +Eq.(25) as follows, +α = (P ij +T − P ij +n )Πij, +β = P ij +L Πij, +γ = (2P ij +n − P ij +T )Πij, +δ = +1 +2k2˜n2 P ij +knΠij. +(26) +The structure functions mainly depend on k, ω, ξ, +k · ˆn = cos θn, and ν. We confined our analysis in the +small anisotropy limit, ξ < 1, where all the structure +functions can be calculated analytically up to linear or- +der in ξ, given as +α = (P ij +T − P ij +n )Πij, +β = P ij +L Πij, +γ = (2P ij +n − P ij +T )Πij, +δ = +1 +2k2˜n2 P ij +knΠij. +(27) +The structure functions mainly depend on k, ω, ξ ν +and k · ˆn = cos θn. In the small anisotropy limit, ξ < 1, +all the structure functions can be calculated analytically +up to linear order in ξ, given as +α (K) = m2 +D +48k +� +24kz2 − 2kξ +� +9z4 − 13z2 + 4 +� ++ 2iνz +� +ξ +� +9z2 − 7 +� +− 12 +� +− 2ξ cos 2θn +� +k +� +15z4 − 19z2 + 4 +� ++iνz +� +13 − 15z2�� ++ +� +z2 − 1 +� � +3ξ +� +kz +� +5z2 − 3 +� ++ iν +� +1 − 5z2�� +cos 2θn + kz +� +−7ξ + 9ξz2 − 12 +� ++iν +� +ξ − 9ξz2 + 12 +� � +ln z + 1 +z − 1 +� +, +(28) + +6 +β (K) = −m2 +D +k +2(kz − iν)2 +ν ln z+1 +z−1 + 2ik +� +1 − 1 +2z ln z + 1 +z − 1 + 1 +12ξ (1 + 3 cos 2θn) +� +2 − 6z2 + +� +3z2 − 2 +� +z ln z + 1 +z − 1 +� � +, +(29) +γ (K) = −m2 +D +12k ξ +� +k +� +z2 − 1 +� +− iνz +� � +4 − 6z2 + 3 +� +z2 − 1 +� +z ln z + 1 +z − 1 +� +sin2 θn, +(30) +δ (K) = m2 +D +24k2 ξ +(kz − iν) +2k − iν ln z+1 +z−1 +� +k +� +88z − 96z3� ++ 8iν +� +6z2 − 1 +� ++ ln z + 1 +z − 1 +× +� +12k +� +4z4 − 5z2 + 1 +� +− 10iνz − 3 iν +� +4z4 − 5z2 + 1 +� +ln z + 1 +z − 1 +� � +cos θn, +(31) +where z = ω+iν +k +, and +ln z + 1 +z − 1 = ln |z + 1| +|z − 1| + i +� +arg +�z + 1 +z − 1 +� ++ 2πN +� +. +(32) +where, N- corresponds to the number of Riemannian +sheets. Now, we can rewrite Eq.(24) as, +[∆−1(K)]ij = (k2 − ω2 + α)P ij +T + (−ω2 + β)P ij +L ++γP ij +n + δP ij +kn. +(33) +Next, we know that both a tensor and its inverse lie in +a space spanned by the same basis vectors or projection +operators. Hence, [∆(K)]ij can also be expanded as its +inverse, as, +[∆(K)]ij = aP ij +L + bP ij +T + cP ij +n + dP ij +kn. +(34) +Now, using the relation [∆−1(K)]ij[∆(K)]jl = δil, we +obtained the following expressions for the coefficients a, +b, c and d, +a = ∆G(k2 − ω2 + α + γ), +b = ∆A +c = ∆G(β − ω2) − ∆A, +d = −∆Gδ +(35) +where, +∆A = (k2 − ω2 + α)−1 +∆G = [(k2 − ω2 + α + γ)(β − ω2) − k2˜n2δ2]−1. (36) +Considering the linear ξ, approximation we ignore δ2, as +it will be of order ξ2, we have +∆G = [(k2 − ω2 + α + γ)(β − ω2)]−1. +(37) +Rewriting, Eq. (34) as, +[∆(K)]ij = ∆A(P ij +T − P ij +n ) + ∆G +� +(k2 − ω2 + α + γ)P ij +L ++(β − ω2)P ij +n − δP ij +kn +� +, +(38) +This is a simplified structure of the gluon propagator +that could be easily used to obtain the poles. +C. +Modes dispersion relation +As noted earlier, the dispersion relation of the collec- +tive modes can be acquired from the poles of the gluon +propagator. The poles of Eq. (38) are given as, +∆−1 +A (K) = k2 − ω2 + α = 0, +(39) +∆−1 +G (K) = (k2 − ω2 + α + γ)(β − ω2) = 0. +(40) +∆−1 +G (K), can further splits as, +∆−1 +G (K) = ∆−1 +G1(k) ∆−1 +G2(k) = 0. +(41) +Thus, we have two more dispersion equations, +∆−1 +G1(K) = k2 − ω2 + α + γ = 0, +(42) +∆−1 +G2(K) = β − ω2 = 0. +(43) +Note that we have got three dispersion equations (39), +(42), and (43). +We call these A-, G1- and G2-mode +dispersion equations, respectively. Now, we have three +dispersion relations for the collective modes. Based on +their solutions, ωk we can identify if the modes are real +or imaginary, stable or unstable which we shall discuss +in the next section. +III. +RESULTS AND DISCUSSIONS +We shall examine the results by solving the disper- +sion equations (39), (42) and (43) numerically. +We +normalize the frequency ω and wave vector k by the +plasma frequency in the absence of chemical potential +(ωp = mD(µ = 0)/ +√ +3). To avoid the bulk of plots that +are already available in the literature [19, 20, 27, 28, 31], +we shall primarily focus on the effects of finite chemical +potential and to have a comparative study we shall also +highlight the anisotropy and medium particles collisions +effects in the context of collective modes. We have fixed +the temperature as T = 2Tc, where Tc = 0.155 GeV the + +7 +μ = 0 +μ = 5 GeV +μ = 10GeV +0.0 +0.5 +1.0 +1.5 +2.0 +0 +2 +4 +6 +8 +10 +12 +ω/ωp +ℛ[α]/ωp +k = ωp, ν = 0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10GeV +0.0 +0.5 +1.0 +1.5 +2.0 +-8 +-6 +-4 +-2 +0 +ω/ωp +ℑ[α]/ωp +k = ωp, ν = 0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10 GeV +0.0 +0.5 +1.0 +1.5 +2.0 +0 +5 +10 +ω/ωp +ℛ[β]/ωp +k = ωp, ν = 0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10 GeV +0.0 +0.5 +1.0 +1.5 +2.0 +-14 +-12 +-10 +-8 +-6 +-4 +-2 +0 +ω/ωp +ℑ[β]/ωp +k = ωp, ν = 0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10GeV +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.1 +0.2 +0.3 +0.4 +ω/ωp +ℛ[γ]/ωp +k = ωp, ν = 0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10GeV +0.0 +0.5 +1.0 +1.5 +2.0 +0.00 +0.05 +0.10 +0.15 +0.20 +ω/ωp +ℑ[γ]/ωp +k = ωp, ν = 0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10 GeV +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ω/ωp +ℛ[δ]/ωp +k = ωp, ν = 0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10 GeV +0.0 +0.5 +1.0 +1.5 +2.0 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +ω/ωp +ℑ[δ]/ωp +k = ωp, ν = 0.3ωp, ξ = 0.3, θn = π +6 +FIG. 2. Variation of real and imaginary parts of structure functions at ν = 0.3ωp, ξ = 0.3, θn = π/6 and T = 2Tc where +Tc = 0.155GeV at different µ. +direction, θn = π/6, the strength of anisotropy, ξ = 0.3, +and the finite collision frequency, ν = 0.3ωp. + +8 +μ = 0 +μ = 5 GeV +μ = 10GeV +0 +2 +4 +6 +8 +10 +0 +2 +4 +6 +8 +10 +k/ωp +ℛ[ωA]/ωp +ν=0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10 GeV +0 +2 +4 +6 +8 +10 +0 +2 +4 +6 +8 +10 +k/ωp +ℛ[ωG1]/ωp +ν=0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10 GeV +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +k/ωp +ℛ[ωG2]/ωp +ν=0.3ωp, ξ = 0.3, θn = π +6 +FIG. 3. Variation of real stable modes at ν = 0.3ωp, ξ = 0.3, θn = π/6 and T = 2Tc where Tc = 0.155GeV at different µ. +We shall start with the discussion of structure func- +tions given in Eq. +(28) to Eq. (31) as they are the +important parts of the gluon selfenergy, gluon propaga- +tor and hence, important for collective modes. We have +plotted all the structure functions, i.e., α, β, γ and δ +with respect to ω normalized by ωp at different values of +chemical potential but at a fixed collisions frequency and +momentum anisotropy as shown in Fig. 2. It can be no- +ticed that with the increase in the chemical potential, the +magnitudes of both the real and imaginary parts of the +structure functions grow. The behavior of all the struc- +ture functions are changing at ω ∼ ωp. At very high ω, +only the real parts of structure functions survive whereas, +their imaginary parts vanish. +The stable modes are long-range modes. As shown in +Fig. 3, the real stable modes are found to increase with +the k at mentioned parameters. The stable G2- mode is +highly, whereas the G1- mode is marginally suppressed +as compared to stable A- mode. The magnitude of these +modes rise with the chemical potential. +The imaginary parts are emerging only because of the +collisional effects. Here, the finite ν generates the imag- +inary modes with Im(ω(k)) < 0, i.e., the modes are +damped and their amplitudes exponentially decay with +time as e−Im(ω(k))t whereas it does not further satisfy +the over-damped condition. If we consider the collision- +less plasma, i.e., ν = 0 these modes disappear. +This +fact has already been shown in the studies by different +groups [19, 27, 31]. +These results for the collisionless +plasma remain untouched in our case as well. In Fig. 4 +the imaginary stable modes are plotted. The magnitude +of imaginary A- and G1- modes are growing with the +chemical potential and their magnitudes are not very dis- +tinguishable. An explanation for that is the difference +between the A- and G1- modes are only the structure +functions, γ whose magnitude as compared to α is very +small and hence does not carry a high impact that could +make it distinguishable. The imaginary G2- mode on the +other hand shows the opposite behavior. This is because +G2- mode depends on β whose magnitude at µ = 10 GeV +is very high. +A crucial aspect in the current study is the unsta- +ble modes that are shown in Fig. 5 for weakly squeezed +plasma,ξ = 0.3 and non-zero collisions frequency, ν = +0.3ωp at a fixed angle, θn = π/6 for different values of +chemical potential. We have encountered only two un- +stable modes, i.e., unstable A- and G1- modes whereas, +the unstable G2- mode is missing. This is because the +unstable modes are the imaginary solutions of ω, in the +dispersion equations (39), (42) and (43) and to obtain +their solutions we consider ω, to be purely imaginary, i.e., +ω = iΓ. In this substitution, we marked that β > 0 and +consequently Γ2 + β = 0 from Eq. (43) will never be sat- +isfied. A similar case was discussed in Ref. [19, 27, 28, 31] +in the absence of medium particle collision. These modes +are short-lived though the presence of chemical potential +enhances their magnitude which may further support the +fast thermalization of the created hot medium. +To comprehend the dependence of instability on +anisotropy and chemical potential and medium particle +collisions, we obtained kmax and νmax at which the un- +stable modes were completely suppressed at fixed values +of the other parameters. The results are shown in Fig. 6 +and Fig. 7. The values of kmax, at which the unstable A- +and G1- modes are completely suppressed are obtained +by substituting ω = 0, in Eq. (39) and (42), respectively. +In Fig. 6 for A- mode (left panel) and G1- mode (right +panel), it is shown that kmax grows with the chemical po- +tential and the existence of anisotropy further enhances +it at fixed θn and ν. The corresponding values of G1- +mode are suppressed as compared to A- mode. This sup- +ports our prior discussion that the unstable modes exist +up to higher values of k at a large chemical potential. +Following the similar procedure discussed above, we +obtained νmax, at the point where the unstable A- and +G1- modes are fully suppressed. In Fig. ??, the behavior +of νmax/ωp, for A- mode (left panel) and G1- mode (right +panel) with respect to µ are shown for different ξ (ξ = +0.2, 0.4 and 0.6) at fixed θn and k. Again it is found that +with the increase in chemical potential, the maximum +values of collision frequency increase, and the results are +further enhanced with the momentum anisotropy. + +9 +μ = 0 +μ = 5 GeV +μ = 10GeV +0.0 +0.5 +1.0 +1.5 +2.0 +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +k/ωp +ℑ[ωA]/ωp +ν=0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10 GeV +0.0 +0.5 +1.0 +1.5 +2.0 +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +k/ωp +ℑ[ωG1]/ωp +ν=0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10 GeV +0.0 +0.5 +1.0 +1.5 +2.0 +-0.4 +-0.3 +-0.2 +-0.1 +0.0 +k/ωp +ℑ[ωG2]/ωp +ν = 0.3ωp, ξ = 0.3, θn = π +6 +FIG. 4. Variation of imaginary stable modes at ν = 0.3ωp, ξ = 0.3, θn = π/6 and T = 2Tc where Tc = 0.155GeV at different µ. +μ = 0 +μ = 5 GeV +μ = 10GeV +0.0 +0.5 +1.0 +1.5 +2.0 +-0.04 +-0.02 +0.00 +0.02 +0.04 +k/ωp +ΓA/ωp +ν=0.3ωp, ξ = 0.3, θn = π +6 +μ = 0 +μ = 5 GeV +μ = 10 GeV +0.0 +0.5 +1.0 +1.5 +2.0 +-0.04 +-0.02 +0.00 +0.02 +0.04 +k/ωp +ΓG1/ωp +ν=0.3ωp, ξ = 0.3, θn = π +6 +FIG. 5. Variation of unstable modes at ν = 0.3ωp, ξ = 0.3, θn = π/6 and T = 2Tc where Tc = 0.155GeV at different µ. +ξ=0.2 +ξ = 0.4 +ξ = 0.6 +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μ (GeV) +kmax/ωp +ν=0.3ωp, θn = π +6 +ξ=0.2 +ξ = 0.4 +ξ = 0.6 +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μ (GeV) +kmax/ωp +ν=0.3ωp, θn = π +6 +FIG. 6. Variation of the maximum values of propagation vector for unstable A- mode (left) and G1- mode (right) at ν = 0.3ωp, +θn = π/6 and T = 2Tc where Tc = 0.155GeV at different ξ. +Moreover, it can be seen that in both the cases, i.e., +kmax and νmax, respectively in Fig. +6 and Fig. +7, the +results for G1- mode are suppressed as compared to A- +mode. It is mainly because of the behavior of the struc- +ture function, γ that tries to suppress the G1- mode. +This indicates that if we have a high baryon density, the +collision frequency among the medium particles also in- +creases, and hence, one can infer that at a large baryon + +10 +ξ=0.2 +ξ = 0.4 +ξ = 0.6 +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +μ (GeV) +νmax/ωp +k = ωp, θn = π +6 +ξ=0.2 +ξ = 0.4 +ξ = 0.6 +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +μ (GeV) +νmax/ωp +k = ωp, θn = π +6 +FIG. 7. Variation of the maximum values of collision frequency for unstable A- mode (left) and G1- mode (right) at k = ωp, +θn = π/6 and T = 2Tc where Tc = 0.155GeV at different ξ. +density, the medium will thermalize faster. +IV. +SUMMARY AND CONCLUSIONS +In this manuscript, we attempted to assimilate the fi- +nite chemical potential in the study of collective excita- +tion present in the hot QGP medium and examine its +effect in the presence of medium particle collision and +small but finite momentum anisotropy. We begin with +the derivation of gluon self-energy in terms of structure +functions where we included the finite chemical potential +at the particle distribution level along with the momen- +tum anisotropy straightforwardly either by stretching or +squeezing the distribution function in one of the direc- +tions (denoted as ˆn). We solved the Boltzman equation +after incorporating the medium particle collision via the +BGK collisional kernel to obtain the change in the distri- +bution function of each species viz., quarks, anti-quarks +and gluons. Next, using the Yang-Mills equation or clas- +sical Maxwell’s equation, we formulated the gluon propa- +gator. From the poles of this propagator, we fetched the +dispersion equation for the collective modes. +As men- +tioned in the introduction section, we classified the modes +as real or imaginary and stable or unstable based on the +solution of the dispersion equations. +In the present formalism, one can analyze the modes +at any angular direction by changing θn, (i.e.,) the an- +gle between k and ˆn at different values of anisotropic +strength in the given range, −1 < ξ < 1. Just to avoid +the bulk of plots that are available in the prior literature, +we did not display the variation of θn and ξ. Entertain- +ing the anisotropy in the analysis is important otherwise +there will be no unstable modes appearing. As shown +in Ref. [28] that the maximum possible value of collision +frequency could be 0.62 mD, i.e., 0.36 ωp, we fixed the +value of collision frequency, ν = 0.3ωp. +We first investigate the structure functions of the self- +energy and found that they intensely depend on the +chemical potential. We noticed that at θn = 0, δ dis- +appears whereas at θn = π/2, γ vanishes. +From the +poles of the propagator, we got three dispersion equations +for collective modes. Based on their solutions, we found +three real and imaginary stable modes. Whereas, only +two unstable modes were found. The imaginary stable +modes disappear in the absence of medium particle col- +lision whereas the unstable modes vanish in the absence +of anisotropy. Also, the two unstable modes overlap at +θn = π/2 as the structure function γ goes to zero. In the +collisionless isotropic medium, we obtain only three real +stable modes. It has been found that all kinds of modes +rely strongly on the chemical potential and mainly en- +hance their magnitude. +We also investigate the suppression of unstable modes +through kmax and νmax via plotting them against µ. It +has been observed that with µ the maximum values of +the wave vector and collision frequency increase which +further enhance with anisotropy. +In the near future, the current project can be extended +through the inclusion of a non-local BGK kernel. More- +over, an essential aspect that is missing in the study of +modes is the study of group velocity as these modes are +nothing but plasma waves. The non-abelian term of the +field strength tensor could also be included to make the +analysis more realistic and closer to the experimental ob- +servations. +V. +ACKNOWLEDGEMENTS +I would like to that SERB for providing NPDF (project +ID 2022/F/SKD/014). 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B +355, 1 (1991). + diff --git a/hdAyT4oBgHgl3EQfXvcJ/content/tmp_files/load_file.txt b/hdAyT4oBgHgl3EQfXvcJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e347284e2b1aa49b80363bc7f3980c4f34182058 --- /dev/null +++ b/hdAyT4oBgHgl3EQfXvcJ/content/tmp_files/load_file.txt @@ -0,0 +1,834 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf,len=833 +page_content='Collective modes in the anisotropic collisional hot QCD medium in the presence of finite chemical potential Mohammad Yousuf Jamal1, ∗ 1Indian Institute of Technology Goa, Ponda, Goa, 403401, India The collective modes that are the collective excitations in the hot QCD medium produced in the heavy-ion collision experiments have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' These modes being real or imaginary and stable or unstable affect the evolution of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' To go into the more profound analysis we incorporated several aspects of the medium such as anisotropy, the presence of medium particle collisions, and finite baryonic chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The first two facets have been studied several times from different perspectives but the inclusion of finite chemical potential is a missing part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Therefore, to have a close analysis, we have combined these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The medium particle collision has been included using BGK- collisional kernel, the anisotropy, and finite chemical potential enter via quarks, anti-quarks and gluons distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Our results indicate that the presence of finite chemical potential enhances the magnitude of unstable modes which may affect the fast thermalization of the hot QCD medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Moreover, it is interesting to see the consequences of finite chemical potential along with the other aspects of the created medium from the viewpoint of low-energy heavy-ion collision experiments that operate at low temperatures and finite baryon density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Keywords: Quark-Gluon-Plasma, Collective excitations, Collective modes, particle distribu- tions function, BGK collisional kernel, Anisotropic QCD, Gluon self-energy, finite chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' INTRODUCTION In the relativistic heavy-ion collision (HIC) experi- ments we obtain such an extremely high energy density than any we know of in the present natural universe and where the QCD predicts a new form of matter consisting of an extended volume of interacting quarks, antiquarks, and gluons known as quark-gluon plasma (QGP) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Such a high energy density phase, however, is thought to have existed a few microseconds after the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In this phase, the effective degrees of freedom are quarks, anti-quarks, and gluons rather than the hadronic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The historic discovery of the J/Ψ, a bound state of the charmed quark and its anti-quark in the year 1974 first time proved the existence of the heavy quarks which indi- cated that the nucleus–nucleus collisions at high energy are very different from a simple superposition of nucleon- nucleon interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' As it mimics the cosmic Big Bang, the term mini-bang has been coined to describe these in- teractions, in which nuclear collisions are thought to pro- ceed in a series and cause the formation of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The presumed line of events begins with an extremely high- temperature region inhabited by the two nuclei at the moment of collision, as a large fraction of their kinetic energy is converted into heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' At thermal equilibrium it forms a high-temperature plasma medium consisting of quarks, anti-quarks, and gluons that instantly begins to expand and cool, passing down through the critical temperature at which the QGP condenses into a system of mesons, baryons, or simply hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' On further ex- pansion, the system reaches its “freeze-out” density, at ∗ mohammad@iitgoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='in which the hadrons no longer interact with each other and stream into the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' There are several theoretical investigations based on various approaches such as semi-classical transport or ki- netic, hard-thermal loop (HTL), ASD-CFT, holographic theories have been invoked to study the produced mat- ter [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' There have been several signatures observed that support the formation of the QGP in these experi- ments such as suppression of heavy quarkonia (a colorless and flavorless bound state of a heavy quark and its anti- quark) yields at high transverse momentum, color screen- ing [5–8], Landau damping [9, 10], energy loss or gain of heavy quarks [11–15], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The observables that we see at the detector are affected by the presence of the QGP medium that in turn, is influenced by several plasma as- pects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' One of the important factors among them is the collective excitations or collective modes produced in the hot QGP medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' It has been explored in previous stud- ies that these modes can be real or imaginary, stable or unstable [16–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Each of them plays a significant role in altering the observed outcomes at the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In many-body systems, the spectrum of collective exci- tations is a fundamental part as it possesses information regarding the thermodynamic and transport properties of that system along with the temporal evolution of a non-equilibrium system [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The investigation begins with the analogical idea from Quantum Electro Dynam- ics (QED) that says that the evolving medium produced in the HIC could also contain instabilities that are sim- ilar to Weibel or filamentation instabilities [23] present in electromagnetic plasma (EMP) that may contribute to accomplishing the fast thermalization [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In par- ticular, at very high-temperature, QCD resembles QED due to the small coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The most common arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='00187v1 [nucl-th] 31 Dec 2022 2 feature of EMP and QGP is collective behavior that sug- gests the QGP could also have a very rich spectrum of collective modes they are there in EMP [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In the present context, these modes could refer to plasma parti- cles that could be real or imaginary and stable or unsta- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The stable modes have infinite lifetimes whereas the unstable modes decay quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The stable modes have a long-range interaction with the heavy quarks traversing through the QGP medium whereas the unstable modes are only found in anisotropic plasma and may help in the fast thermalization/equilibration of the system, a lesser- known puzzle that attracts the curiosity of investigating the unstable modes and hence, the prior is less studied and discussed in the literature though the latter has been discussed widely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In several analyses the question about the formation, existence, and effects of the collective modes have been discussed considering various plasma aspects such as anisotropy, medium particle collisions, etc, [27–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Here, we are interested in a crucial aspect which is to under- stand the inclusion and effects of finite baryon density in the study of these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Thrusting to very high baryon density at low temperatures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', squeezing nuclei with- out heating them takes us into another fascinating region of the QCD phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The high baryon density gen- erally means the surplus of quarks over antiquarks in the hot system parameterized by the baryon chemical poten- tial, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The zero chemical potential refers to the equal densities of quarks and antiquarks which is a good ap- proximation for the matter produced at midrapidity at very high energy HIC such as RHIC and LHC, and ex- ceptionally good for the early Universe but not at the lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' It is essential from the viewpoint of the experimental facilities which perform at moderate tem- peratures along with finite baryon density such as the Super Proton Synchrotron (SPS) at CERN, Switzerland, and also upcoming planned experimental facilities such as the Facility for Antiproton and Ion Research (FAIR) in Darmstadt, Germany and the Nuclotron-Based Ion Collider Facility (NICA) in Dubna, Russia, and at the Japan Proton Accelerator Research Complex (J-PARC) in T¯okai, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' To have a comprehensive study of the collective modes we shall also include the momentum anisotropy and medium particle collisions along with the finite chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The momentum anisotropy and the chemical potential will enter via particle distribution functions and for medium particle collisions we shall con- sider the Bhatnagar–Gross–Krook (BGK) collisional ker- nel in the Boltzman transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The dispersion relations for the modes are acquired from the poles of the (gluon) propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Based on their solutions it can be identified if the modes are stable or unstable and real or imaginary which we shall discuss in detail in the up- coming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The current manuscript is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In sec- tion II, the methodology to obtain the dispersion rela- tions of the modes will be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Section III, con- tains a detailed discussion of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Finally, sec- tion IV, is dedicated to the summary and conclusions of the present work along with the potential future prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Here, natural units are used throughout the text with c = kB = ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We use a bold typeface to dis- play three vectors and a regular font to indicate four vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The centre dot depicts the four-vector scalar product with gµν = diag(1, −1, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' METHODOLOGY The formation of collective modes in the QGP medium can be understood as if one assumes the QGP initially in a homogeneous and stationary state with no net lo- cal color charges and no net currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Next, suppose this state is slightly perturbed by either a random fluc- tuation or some external field that induces local charges or currents in the plasma that in turn generate chromo- electric and chromo-magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' These fields interact back with colored partons that contributed to their for- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' If the wavelength of the perturbation surpasses the typical inter-particle spacing, the plasma undergoes a collective motion known as collective modes/excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The collective modes can be specified by the nature of the solutions of their dispersion relations (ω(k), k being the wave vector) that can be fetched from the poles of the gluon propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The solutions to the dispersion equa- tions, ω(k) could be complex-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' For ℑ(ω(k)) = 0 there exist only stable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' If ℑ(ω(k)) > 0 the am- plitude of the mode exponentially grows with time as eℑ(ω(k))t that represents the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' If ℑ(ω(k)) < 0 the mode is damped and the mode amplitude exponen- tially decays with time as e−ℑ(ω(k))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Likewise, if we have, −ℑ(ω(k)) ≥ |ℜ(ω(k))| the modes will be over damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' To obtain the gluon propagator we first derive the gluon polarization tensor that contains the medium informa- tion such as medium anisotropy, finite chemical poten- tial, the effect of medium particle collision, etc that we shall discuss next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Gluon polarization tensor The gluon polarization tensor or gluon self-energy (Πµν), bears the information of the QCD medium as it represents the interactions term in the effective action of the QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Before we start our derivation, we spec- ify that our working scale is the soft momentum scale where the plasma aspects first appear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', k ∼ gT ≪ T, ( g being the strong coupling constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' At this scale, one can obtain that the strength of field fluctuations, Aµ ∼ O(√gT), and the derivatives, ∂x ∼ O(gT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Now, if we examine the field strength tensor, F µν a = ∂µAν a − ∂νAµ a − ig [Aµ b , Aν c] , (1) we notice that the order of the non-abelian term is O(g2) which is smaller than the order of the first two terms i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', O(g3/2) in F µν and hence can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Now we end 3 up with the abelian part of F µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Next, in the abelian limit, the linearized semi-classical transport equations for each color channel are given as [16, 28, 31, 35], vµ∂µδf i a(p, X) + gθivµF µν a (X)∂(p) ν f i(p) = Ci a(p, X), (2) where the index ‘i‘ refers to the particle species (quark, anti-quark, and gluon), ‘a‘, ‘b‘, and‘c‘ are the color index and θi ∈ {θg, θq, θ¯q} and have the values θg = θq = 1 and θ¯q = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The four vectors xµ = (t, x) = X and vµ = (1, v) = V , are respectively, the four space-time coordinate and the velocity of the plasma particle with v = p/|p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The ∂µ, ∂(p) ν are the partial four derivatives corresponding to space and momentum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' As mentioned earlier the collisional term Ci a(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' is the BGK - kernel [28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 35–37] that describes the effects of collisions between hard particles in a hot QCD medium given as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Ci a(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' X) = −ν � f i a(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' X) − N i a(X) N ieq f i eq(|p|) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (3) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' f i a(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' X) = f i(p) + δf i a(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (4) are the distribution functions of quarks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' anti-quarks and gluons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' f i(p) is equilibrium part while,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' δf i a(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' X) per- turbed part of the distribution function and N i a(X) = � d3p (2π)3 f i a(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' N i eq = � d3p (2π)3 f i eq(|p|) = � d3p (2π)3 f i(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (5) is the particle number and its equilibrium value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The BGK - kernel [36] depicts the equilibration of the system due to the collisions in a time proportional to ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Here, we consider the collision frequency ν to be independent of momentum and particle species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We preferred to choose BGK -kernel because it conserves the particle number instantaneously as, � d3p (2π)3 Ci a(p, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (6) At finite baryon or quark chemical potential µ, the momentum distributions of gluon, quark, and anti-quark are given as, fg = exp[−βEg] � 1 − exp[−βEg] �, fq = exp[−β(Eq − µ)] � 1 + exp[−β(Eq − µ)] �, f¯q = exp[−β(Eq + µ)] � 1 + exp[−β(Eq + µ)] �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (7) To include the momentum anisotropy, we track the strat- egy given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In this method, the anisotropic dis- tribution function was obtained from the isotropic distri- bution function given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (7) by rescaling (stretching and squeezing) in one direction of the momentum space as follows, f(p) ≡ fξ(p) = f( � p2 + ξ(p · ˆn)2), (8) where, ˆn is an unit vector (ˆn2 = 1) showing the di- rection of momentum anisotropy and ξ the strength of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' It refers to squeezing when ξ > 0 and stretch- ing when −1 < ξ < 0 of the distribution function in the ˆn direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Next, the gluon polarization tensor can be obtained from the current induced due to the change in the parti- cles distribution functions in the Fourier space as, Πµν ab (K) = ∂Jµ ind a(K) ∂Abν(K) , (9) where the induced current is given by [28, 31, 35, 38], Jµ ind,a(X) = g � d3p (2π)3 V µ{2Ncδf g a(p, X) + Nf[δf q a(p, X) − δf ¯q a(p, X)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (10) Rewriting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (2) as, vµ∂µδf i a(p, X) + gθivµF µν a (X)∂(p) ν f i(p) = ν � f i eq(|p|) − f i(p) � − νδf i a(p, X) +νf i eq(|p|) N ieq �� d3p′ (2π)3 δf i a(p′, X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (11) On solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (11) for δf i a(p, X) in the Fourier space we obtained, 4 δf i a(p, K) = −igθivµF µν a (K)∂(p) ν f i(p) + iν(f i eq(|p|) − f i(p)) + iνf i eq(|p|) �� d3p (2π)3 δf i a(p′, K) � /Neq ω − v · k + iν , (12) where, δf i(p, K) and F µν(K), are the Fourier trans- forms of δf i(p, X) and F µν(X), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Where, K = kµ = (ω, k) Now taking the Fourier transform of the induced current from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (10) and employing δf i a(p, K), from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (12) we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Jµ ind,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='a(K) = g2 � d3p (2π)3 vµ∂(p) ν f(p)Mνα(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' V )D−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν)Aαa + giν{2NcSg(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) + Nf(Sq(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) − S ¯q(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν))} +g(iν) � dΩ 4π vµD−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) � d3p′ (2π)3 � g∂(p′) ν f(p′)Mνα(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' V ′)D−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν)W−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν)Aαa +iν(feq(|p′|) − f(p′))D−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) � W−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (13) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' D(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) = ω + iν − k · v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Mνα(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' V ) = gνα(ω − k · v) − Kνvα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' f(p) = 2Ncf g(p) + Nf � f q(p) + f ¯q(p) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' feq(|p′|) = 2Ncf g eq(|p′|) + Nf � f q eq(|p′|) + f ¯q eq(|p′|) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' W(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) = 1 − iν � dΩ 4π D−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Si(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) = − � d3p (2π)3 vµ[f i(p) − f i eq(|p|)]D−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (14) Implying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (9) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (13), we get Πµν ab (K) = δabg2 � d3p (2π)3 vµ∂(p) β f(p)Mβν(K, V ) ×D−1(K, v, ν) + δabg2(iν) � dΩ 4π vµ ×D−1(K, v, ν) � d3p′ (2π)3 ∂(p′) β f(p′) Mβν(K, V ′)D−1(K, v′, ν)W−1(K, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (15) Rewriting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' in temporal gauge for anisotropic hot QCD medium at finite chemical potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Πij(K) = m2 D � dΩ 4π vi vl + ξ(v · ˆn)nl (1 + ξ(v · ˆn)2)2 � δjl(ω − k · v) + vjkl� D−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) + (iν)m2 D � dΩ′ 4π (v′)i ×D−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν) � dΩ 4π vl + ξ(v · ˆn)nl (1 + ξ(v · ˆn)2)2 � δjl(ω − k · v) + vjkl� D−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν)W−1(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (16) where the squared Debye mass is given as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' m2 D = − g2 2π2 � ∞ 0 dp p2 dfiso(p) dp ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (17) In the limit µ/T ≪ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' we have m2 D = 4παsT 2 �Nc 3 + Nf 6 + µ2 T 2 Nf 2π2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (18) and the strong coupling constant [39],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' αs = g2/4π at finite chemical potential is given as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' αs = 6π − 18π(153−19Nf ) ln � 2 ln ��� T ΛT �2+ µ2 π2T 2 �� (33−2Nf )2 ln ��� T ΛT �2+ µ2 π2T 2 � (33 − 2Nf) ln ��� T ΛT �2 + µ2 π2T 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (19) 5 where ΛT ≪ T is the QCD scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Here, Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='155 GeV T=2Tc T=4Tc 0 20 40 60 80 100 0 10 20 30 40 μ (GeV) mD[μ,T] (GeV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Variation of screening mass with respect to chemical potential at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (17) is fully solved numerically and the variation of screening mass with respect to the chemical poten- tial at different temperatures is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' It has been found that mD increases with the chemical poten- tial whereas the effect of different values of temperature is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Next, we shall discuss the derivation of the gluon propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Gluon Propagator To obtain the gluon propagator, we initiate with Maxwell’s equation in the Fourier space, −ikνF νµ(K) = Jµ ind(K) + Jµ ext(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (20) Here, Jµ ext(K) is the external current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The induced cur- rent, Jµ ind(K) can be described in terms of self-energy, Πµν(K) as follows, Jµ ind(K) = Πµν(K)Aν(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (21) The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ( 20) can also be noted as, [K2gµν − kµkν + Πµν(K)]Aν(K) = −Jµ ext(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (22) The quantity in the square bracket is the needed gluon propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Now assuming the temporal gauge, the above equation can be written as, [∆−1(K)]ijEj = [(k2 − ω2)δij − kikj + Πij(K)]Ej = iωJi ext(k), (23) where Ej is the physical electric field and [∆−1(K)]ij = (k2 − ω2)δij − kikj + Πij(K), (24) is the inverse of the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The dispersion equations for collective modes can be obtained from the poles of the propagator, [∆(K)]ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Since, it is a tensorial equation and hence, can not be simply integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' To solve it we first need to construct an analytical form of Πij using the available symmetric tensors such as, Πij = αP ij T + βP ij L + γP ij n + δP ij kn, (25) where, P ij T = δij − kikj/k2, P ij L = kikj/k2, P ij n = ˜ni˜nj/˜n2 and P ij kn = ki˜nj + kj˜ni [31, 33, 40], where, ˜ni = (δij − kikj k2 )ˆnj is a vector orthogonal to, ki ,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', ˜n · k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The structure functions α, β, γ and δ, can be determined by taking the appropriate projections of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (25) as follows, α = (P ij T − P ij n )Πij, β = P ij L Πij, γ = (2P ij n − P ij T )Πij, δ = 1 2k2˜n2 P ij knΠij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (26) The structure functions mainly depend on k, ω, ξ, k · ˆn = cos θn, and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We confined our analysis in the small anisotropy limit, ξ < 1, where all the structure functions can be calculated analytically up to linear or- der in ξ, given as α = (P ij T − P ij n )Πij, β = P ij L Πij, γ = (2P ij n − P ij T )Πij, δ = 1 2k2˜n2 P ij knΠij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (27) The structure functions mainly depend on k, ω, ξ ν and k · ˆn = cos θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In the small anisotropy limit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ξ < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' all the structure functions can be calculated analytically up to linear order in ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' given as α (K) = m2 D 48k � 24kz2 − 2kξ � 9z4 − 13z2 + 4 � + 2iνz � ξ � 9z2 − 7 � − 12 � − 2ξ cos 2θn � k � 15z4 − 19z2 + 4 � +iνz � 13 − 15z2�� + � z2 − 1 � � 3ξ � kz � 5z2 − 3 � + iν � 1 − 5z2�� cos 2θn + kz � −7ξ + 9ξz2 − 12 � +iν � ξ − 9ξz2 + 12 � � ln z + 1 z − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (28) 6 β (K) = −m2 D k 2(kz − iν)2 ν ln z+1 z−1 + 2ik � 1 − 1 2z ln z + 1 z − 1 + 1 12ξ (1 + 3 cos 2θn) � 2 − 6z2 + � 3z2 − 2 � z ln z + 1 z − 1 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (29) γ (K) = −m2 D 12k ξ � k � z2 − 1 � − iνz � � 4 − 6z2 + 3 � z2 − 1 � z ln z + 1 z − 1 � sin2 θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (30) δ (K) = m2 D 24k2 ξ (kz − iν) 2k − iν ln z+1 z−1 � k � 88z − 96z3� + 8iν � 6z2 − 1 � + ln z + 1 z − 1 × � 12k � 4z4 − 5z2 + 1 � − 10iνz − 3 iν � 4z4 − 5z2 + 1 � ln z + 1 z − 1 � � cos θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (31) where z = ω+iν k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' and ln z + 1 z − 1 = ln |z + 1| |z − 1| + i � arg �z + 1 z − 1 � + 2πN � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (32) where, N- corresponds to the number of Riemannian sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Now, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (24) as, [∆−1(K)]ij = (k2 − ω2 + α)P ij T + (−ω2 + β)P ij L +γP ij n + δP ij kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (33) Next, we know that both a tensor and its inverse lie in a space spanned by the same basis vectors or projection operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Hence, [∆(K)]ij can also be expanded as its inverse, as, [∆(K)]ij = aP ij L + bP ij T + cP ij n + dP ij kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (34) Now, using the relation [∆−1(K)]ij[∆(K)]jl = δil, we obtained the following expressions for the coefficients a, b, c and d, a = ∆G(k2 − ω2 + α + γ), b = ∆A c = ∆G(β − ω2) − ∆A, d = −∆Gδ (35) where, ∆A = (k2 − ω2 + α)−1 ∆G = [(k2 − ω2 + α + γ)(β − ω2) − k2˜n2δ2]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (36) Considering the linear ξ, approximation we ignore δ2, as it will be of order ξ2, we have ∆G = [(k2 − ω2 + α + γ)(β − ω2)]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (37) Rewriting, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (34) as, [∆(K)]ij = ∆A(P ij T − P ij n ) + ∆G � (k2 − ω2 + α + γ)P ij L +(β − ω2)P ij n − δP ij kn � , (38) This is a simplified structure of the gluon propagator that could be easily used to obtain the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Modes dispersion relation As noted earlier, the dispersion relation of the collec- tive modes can be acquired from the poles of the gluon propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The poles of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (38) are given as, ∆−1 A (K) = k2 − ω2 + α = 0, (39) ∆−1 G (K) = (k2 − ω2 + α + γ)(β − ω2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (40) ∆−1 G (K), can further splits as, ∆−1 G (K) = ∆−1 G1(k) ∆−1 G2(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (41) Thus, we have two more dispersion equations, ∆−1 G1(K) = k2 − ω2 + α + γ = 0, (42) ∆−1 G2(K) = β − ω2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (43) Note that we have got three dispersion equations (39), (42), and (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We call these A-, G1- and G2-mode dispersion equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Now, we have three dispersion relations for the collective modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Based on their solutions, ωk we can identify if the modes are real or imaginary, stable or unstable which we shall discuss in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS We shall examine the results by solving the disper- sion equations (39), (42) and (43) numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We normalize the frequency ω and wave vector k by the plasma frequency in the absence of chemical potential (ωp = mD(µ = 0)/ √ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' To avoid the bulk of plots that are already available in the literature [19, 20, 27, 28, 31], we shall primarily focus on the effects of finite chemical potential and to have a comparative study we shall also highlight the anisotropy and medium particles collisions effects in the context of collective modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We have fixed the temperature as T = 2Tc, where Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='155 GeV the 7 μ = 0 μ = 5 GeV μ = 10GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0 2 4 6 8 10 12 ω/ωp ℛ[α]/ωp k = ωp, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 8 6 4 2 0 ω/ωp ℑ[α]/ωp k = ωp, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0 5 10 ω/ωp ℛ[β]/ωp k = ωp, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 14 12 10 8 6 4 2 0 ω/ωp ℑ[β]/ωp k = ωp, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 ω/ωp ℛ[γ]/ωp k = ωp, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='20 ω/ωp ℑ[γ]/ωp k = ωp, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 ω/ωp ℛ[δ]/ωp k = ωp, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 ω/ωp ℑ[δ]/ωp k = ωp, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Variation of real and imaginary parts of structure functions at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π/6 and T = 2Tc where Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='155GeV at different µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' direction, θn = π/6, the strength of anisotropy, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, and the finite collision frequency, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 8 μ = 0 μ = 5 GeV μ = 10GeV 0 2 4 6 8 10 0 2 4 6 8 10 k/ωp ℛ[ωA]/ωp ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10 GeV 0 2 4 6 8 10 0 2 4 6 8 10 k/ωp ℛ[ωG1]/ωp ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10 GeV 0 1 2 3 4 5 0 1 2 3 4 5 k/ωp ℛ[ωG2]/ωp ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Variation of real stable modes at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π/6 and T = 2Tc where Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='155GeV at different µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We shall start with the discussion of structure func- tions given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (28) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (31) as they are the important parts of the gluon selfenergy, gluon propaga- tor and hence, important for collective modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We have plotted all the structure functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', α, β, γ and δ with respect to ω normalized by ωp at different values of chemical potential but at a fixed collisions frequency and momentum anisotropy as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' It can be no- ticed that with the increase in the chemical potential, the magnitudes of both the real and imaginary parts of the structure functions grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The behavior of all the struc- ture functions are changing at ω ∼ ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' At very high ω, only the real parts of structure functions survive whereas, their imaginary parts vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The stable modes are long-range modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 3, the real stable modes are found to increase with the k at mentioned parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The stable G2- mode is highly, whereas the G1- mode is marginally suppressed as compared to stable A- mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The magnitude of these modes rise with the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The imaginary parts are emerging only because of the collisional effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Here, the finite ν generates the imag- inary modes with Im(ω(k)) < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', the modes are damped and their amplitudes exponentially decay with time as e−Im(ω(k))t whereas it does not further satisfy the over-damped condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' If we consider the collision- less plasma, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', ν = 0 these modes disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' This fact has already been shown in the studies by different groups [19, 27, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' These results for the collisionless plasma remain untouched in our case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 4 the imaginary stable modes are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The magnitude of imaginary A- and G1- modes are growing with the chemical potential and their magnitudes are not very dis- tinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' An explanation for that is the difference between the A- and G1- modes are only the structure functions, γ whose magnitude as compared to α is very small and hence does not carry a high impact that could make it distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The imaginary G2- mode on the other hand shows the opposite behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' This is because G2- mode depends on β whose magnitude at µ = 10 GeV is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' A crucial aspect in the current study is the unsta- ble modes that are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 5 for weakly squeezed plasma,ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3 and non-zero collisions frequency, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp at a fixed angle, θn = π/6 for different values of chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We have encountered only two un- stable modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', unstable A- and G1- modes whereas, the unstable G2- mode is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' This is because the unstable modes are the imaginary solutions of ω, in the dispersion equations (39), (42) and (43) and to obtain their solutions we consider ω, to be purely imaginary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', ω = iΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In this substitution, we marked that β > 0 and consequently Γ2 + β = 0 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (43) will never be sat- isfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' A similar case was discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' [19, 27, 28, 31] in the absence of medium particle collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' These modes are short-lived though the presence of chemical potential enhances their magnitude which may further support the fast thermalization of the created hot medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' To comprehend the dependence of instability on anisotropy and chemical potential and medium particle collisions, we obtained kmax and νmax at which the un- stable modes were completely suppressed at fixed values of the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The values of kmax, at which the unstable A- and G1- modes are completely suppressed are obtained by substituting ω = 0, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (39) and (42), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 6 for A- mode (left panel) and G1- mode (right panel), it is shown that kmax grows with the chemical po- tential and the existence of anisotropy further enhances it at fixed θn and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The corresponding values of G1- mode are suppressed as compared to A- mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' This sup- ports our prior discussion that the unstable modes exist up to higher values of k at a large chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Following the similar procedure discussed above, we obtained νmax, at the point where the unstable A- and G1- modes are fully suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', the behavior of νmax/ωp, for A- mode (left panel) and G1- mode (right panel) with respect to µ are shown for different ξ (ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6) at fixed θn and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Again it is found that with the increase in chemical potential, the maximum values of collision frequency increase, and the results are further enhanced with the momentum anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 9 μ = 0 μ = 5 GeV μ = 10GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='00 k/ωp ℑ[ωA]/ωp ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='00 k/ωp ℑ[ωG1]/ωp ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 k/ωp ℑ[ωG2]/ωp ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Variation of imaginary stable modes at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π/6 and T = 2Tc where Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='155GeV at different µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' μ = 0 μ = 5 GeV μ = 10GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='04 k/ωp ΓA/ωp ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 μ = 0 μ = 5 GeV μ = 10 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='04 k/ωp ΓG1/ωp ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Variation of unstable modes at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3, θn = π/6 and T = 2Tc where Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='155GeV at different µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 μ (GeV) kmax/ωp ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, θn = π 6 ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 μ (GeV) kmax/ωp ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, θn = π 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Variation of the maximum values of propagation vector for unstable A- mode (left) and G1- mode (right) at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp, θn = π/6 and T = 2Tc where Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='155GeV at different ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Moreover, it can be seen that in both the cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', kmax and νmax, respectively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 7, the results for G1- mode are suppressed as compared to A- mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' It is mainly because of the behavior of the struc- ture function, γ that tries to suppress the G1- mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' This indicates that if we have a high baryon density, the collision frequency among the medium particles also in- creases, and hence, one can infer that at a large baryon 10 ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 μ (GeV) νmax/ωp k = ωp, θn = π 6 ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='4 μ (GeV) νmax/ωp k = ωp, θn = π 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Variation of the maximum values of collision frequency for unstable A- mode (left) and G1- mode (right) at k = ωp, θn = π/6 and T = 2Tc where Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='155GeV at different ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' density, the medium will thermalize faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS In this manuscript, we attempted to assimilate the fi- nite chemical potential in the study of collective excita- tion present in the hot QGP medium and examine its effect in the presence of medium particle collision and small but finite momentum anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We begin with the derivation of gluon self-energy in terms of structure functions where we included the finite chemical potential at the particle distribution level along with the momen- tum anisotropy straightforwardly either by stretching or squeezing the distribution function in one of the direc- tions (denoted as ˆn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We solved the Boltzman equation after incorporating the medium particle collision via the BGK collisional kernel to obtain the change in the distri- bution function of each species viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', quarks, anti-quarks and gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Next, using the Yang-Mills equation or clas- sical Maxwell’s equation, we formulated the gluon propa- gator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' From the poles of this propagator, we fetched the dispersion equation for the collective modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' As men- tioned in the introduction section, we classified the modes as real or imaginary and stable or unstable based on the solution of the dispersion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In the present formalism, one can analyze the modes at any angular direction by changing θn, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=',) the an- gle between k and ˆn at different values of anisotropic strength in the given range, −1 < ξ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Just to avoid the bulk of plots that are available in the prior literature, we did not display the variation of θn and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Entertain- ing the anisotropy in the analysis is important otherwise there will be no unstable modes appearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' [28] that the maximum possible value of collision frequency could be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='62 mD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='36 ωp, we fixed the value of collision frequency, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content='3ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We first investigate the structure functions of the self- energy and found that they intensely depend on the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We noticed that at θn = 0, δ dis- appears whereas at θn = π/2, γ vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' From the poles of the propagator, we got three dispersion equations for collective modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Based on their solutions, we found three real and imaginary stable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Whereas, only two unstable modes were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The imaginary stable modes disappear in the absence of medium particle col- lision whereas the unstable modes vanish in the absence of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Also, the two unstable modes overlap at θn = π/2 as the structure function γ goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In the collisionless isotropic medium, we obtain only three real stable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' It has been found that all kinds of modes rely strongly on the chemical potential and mainly en- hance their magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' We also investigate the suppression of unstable modes through kmax and νmax via plotting them against µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' It has been observed that with µ the maximum values of the wave vector and collision frequency increase which further enhance with anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' In the near future, the current project can be extended through the inclusion of a non-local BGK kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' More- over, an essential aspect that is missing in the study of modes is the study of group velocity as these modes are nothing but plasma waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' The non-abelian term of the field strength tensor could also be included to make the analysis more realistic and closer to the experimental ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' ACKNOWLEDGEMENTS I would like to that SERB for providing NPDF (project ID 2022/F/SKD/014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' I would also like to acknowledge the people of INDIA for their generous support for the research in fundamental sciences in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' 11 [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfXvcJ/content/2301.00187v1.pdf'} +page_content=' (STAR Collaboration), Nucl.' metadata={'source': 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b/hdFAT4oBgHgl3EQf9R5b/content/tmp_files/2301.08755v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5b8c9d5005b119a419ab3d87ac928f90e32033c --- /dev/null +++ b/hdFAT4oBgHgl3EQf9R5b/content/tmp_files/2301.08755v1.pdf.txt @@ -0,0 +1,1129 @@ +Draft version January 24, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Tidal Distortions as a Bottleneck on Constraining Exoplanet Compositions +David Berardo∗ and Julien de Wit† +ABSTRACT +Improvements in the number of confirmed planets and the precision of observations implies a need +to better understand subtle effects which may bias interpretations of exoplanet observations. One +such effect is the distortion of a short period planet by its host star, affecting its derived density. We +extend the work of Burton et al. (2014); Correia (2014) and others, using a gravitational potential +formulation to a sample of nearly 200 planets with periods less than three days. We find five planets +exhibiting density variations of > 10%, and as many as twenty planets with deviations > 5%. We +derive an analytic approximation for this deviation as a function of the orbital period, transit depth, +and mass ratio between the planet and host star, allowing for rapid determination of such tidal effects. +We find that current density error-bars are typically larger than tidal deviations, but that reducing the +uncertainty on transit depth and RV amplitude by a factor of three causes tidal effects to dominate +density errors (> 50%) in >40% of planets in our sample, implying that in the near future upgraded +observational precision will cause shape deviations to become a bottleneck with regards to analysis of +exoplanet compositions. These two parameters are found to dominate uncertainties compared to errors +on stellar mass and radius. We identify a group of eight planets (including WASP-19 b, HAT-P-7 b, +and WASP-12 b) for which current density uncertainties are as much as four times smaller than the +potential shift due to tides, implying a possible 4σ bias on their density estimates. +1. INTRODUCTION +As the list of confirmed exoplanets grows we continu- +ously expand the sampled space of known planetary pa- +rameters. Categories of planets such as those with ultra- +short orbital periods have gone from containing a hand- +ful of planets to hundreds of planets thanks to missions +such as Kepler (Borucki et al. 2010) and TESS (Ricker +et al. 2014). In addition to this increase in population, +the precision of instruments has continued to reach new +heights, reducing the uncertainty in quantities such as +transit depth or planetary mass. This trend will acceler- +ate further with the next generation of observatories and +instruments such as JWST and PLATO (Heras et al. +2020), as well as high precision RV instruments such +as CARMENES (Reiners et al. 2018) and ESPRESSO +(Schmidt et al. 2021). This increase in both the size and +Corresponding author: David Berardo +berardo@mit.edu +∗ Department of Physics and Kavli Institute for Astrophysics +and Space Research, Massachusetts Institute of Technology, +Cambridge, MA 02139, USA +FRQNT Doctoral Research Scholarship +† Department of Earth, Atmospheric and Planetary Sciences, +Massachusetts Institute of Technology, Cambridge, MA 02139, +USA +quality of our sample implies that subtle effects which +in the past where either too small to be detectable or +which affected a single digit number of planets may no +longer be disregarded. An example of this behaviour is +the ‘Transit Light Source’ effect (Rackham et al. 2018), +in which variability of the stellar surface causes biases +in atmospheric characterisation by mimicking or muting +effects which produce similar results, acting a bottleneck +towards properly understanding a planets atmosphere. +The focus of this work is on effects which alter the +shape of an exoplanet, which is often considered to be +a perfect sphere such as in the commonly used models +of Mandel & Agol (2002), implemented in the widely +used batman package (Kreidberg 2015). For short pe- +riod planets close to their host star, one such effect are +tidal distortions which can cause a planet to bulge out +towards its host star (Leconte et al. 2011). This effect +in particular has the potential to introduce a significant +bias on the density of a planet since its sky projection +remains close to a perfect circle. When considering for +example a planet which is deformed due to rotation caus- +ing ts equator to bulge, its projection becomes elliptical +(Seager & Hui 2002; Barnes & Fortney 2003). In this +case, subtle difference in the shape of ingress / egress +of the transit lightcurve may be used to break the de- +generacy between a spherical and oblate planet (Carter +arXiv:2301.08755v1 [astro-ph.EP] 20 Jan 2023 + +2 +Berardo & de Wit 2022 +Figure 1: An illustration of the process by which the surface of the sphere is constructed. Starting from an icosahedron +on the left, triangular faces are continually subdivided. Finally, the points are normalized to generate a uniformly +sampled sphere. +& Winn 2010; Berardo & de Wit 2022). For tidally de- +formed planets, phase curve observations which observe +the planet from different directions could in principle +determine these so called ‘ellipsoidal variations’ through +lightcurve deviations (Correia 2014; Kreidberg et al. +2018), however full phase curve observations require a +significant amount of observing time to obtain, and at +high precision there is likely to be a significant amount +of degeneracy between the orbit, shape, and brightness +distribution of a planet (de Wit et al. 2012). +Tidal distortions imply an underestimate of the vol- +ume of a planet, which in turn implies an overestimate +of its bulk density. Theoretical considerations of the ef- +fect of this have previously been studied in Leconte et al. +(2011) This effect has already been considered, primar- +ily in the work of Burton et al. (2014), which calculated +the magnitude of the distortion and the degree to which +it altered the density measurement for a sample of just +over 30 planets. Additionally, Correia (2014) expanded +on this work using a more detailed model to derive an +analytic expression for the change in density as a func- +tion of distance to the host star. +In this work we aim to expand on these efforts in sev- +eral ways. Our primary effort is to increase the sample of +planets analysed using a gravitational potential model, +which has been found to provide similar results to more +complicated structural models. In the time since these +previous studies were published, roughly 6x as many +planets have now been found to be in the space of pa- +rameters which are susceptible to tidal distortion effects +(i.e. planets with orbital periods below three days on +circular orbits). +In section 2 we briefly outline the theory of tidal de- +formation and describe our method for calculating the +effects of tidal interactions, and thus altered planetary +densities. In section 3 we first highlight our sample of +planets to be analysed, followed by the results of our +analyses. We highlight trends as a function of various +system parameters and derive an approximation which +accurately describes the changes in density without the +need for a full simulation. In section 4 we first highlight +the biases that may be introduced when attempting to +retrieve the interior composition of a planet using mass- +radius relations under the assumption of being perfectly +spherical. We then compare the changes in density to +current density uncertainties, and we also analyse the +relative contributions to these uncertainties from five +parameters underlying parameters. This allows us to de- +termine how upcoming improvements in quantities such +as planet mass and stellar parameters will affect the abil- +ity to ignore such effects, for example through extreme +precision radial velocity efforts (Crass et al. 2021). +2. CALCULATING THE DENSITY OF A TIDALLY +DEFORMED PLANET +2.1. Physical description of scenario +To model the shape of the planet, we follow a similar +methodology as that of Burton et al. (2014), where the +surface of the planet is assumed to be on a gravitational +equipotential. The value of the gravitational potential +generated by a rotating planet and its host star is cal- +culated using the Roche approximation (Chandrasekhar +1987): +Φ1 = − +GM1 +((x + a)2 + y2 + x2)1/2 +(1) +Φ2 = − +GM2 +(x2 + y2 + x2)1/2 +(2) +Φ3 = −1 +2Ω2 � +(x + µ1a)2 + y2� +(3) +where G is the gravitational constant, M1 is the mass + +o Subdivisions +2 Subdivisions +3 Subdivisions + NormalizingDensity Variations +3 +Figure 2: This figure shows two views of the surface of WASP-19 b. Points in black show the spherical planet which +matches the observed transit depth, while points in red show the surface generated by fitting for an equipotential while +also matching the observed transit depth. On the left we see a top down view of the orbital plane. On the right we +see the view along the line of sight between the centers of mass of the planet and star. +of the host star, M2 is the mass of the planet, a is the +separation between the host star and planet (i.e. the +semi-major axis of a circular orbit), µ1 = M1/(M1+M2) +and Ω = 2π/P where P is the orbital period of the +planet. The coordinate system is such that the origin +is placed at the center of the planet. The x coordinate +points along the line connecting the center of masses of +the two bodies, the z axis points along the orbital plane +in the direction of motion of the planet, and the y axis +points normal to the orbital plane. +In order to use such an approximation to model the +distortion of a planets surface, we assume the planet is +both tidally locked as well on a non-eccentric orbit. As +we shall see in later sections, the effect of the distortion +is strongest for low period planets (p < 3 days) which +are most likely to be tidally locked and be on circular +orbits(Barnes 2017). +2.2. Calculating the volume of a deformed planet +We first calculate the surface of a deformed planet +and then ‘measure’ its volume in order to determine +the amount by which its density is altered. +In order +to generate the surface of our planet, we first construct +a geodesic icosahedron as an approximation of a sphere. +This is an object commonly used in computer graph- +ics and 3D rendering software which has the benefit of +having its points uniformly spread out across its sur- +face. We begin with the vertices of an icosahedron and +then iteratively subdivide each of its faces into smaller +triangles (as shown in figure 1). After the last round +of subdivisions we normalize the length of each vertex +from the origin to generate a tiled sphere. +This process leaves us with a collection of triangular +faces which allows us to calculate two necessary quan- +tities, the total projected surface area visible to an ob- +server as well as the enclosed volume of each tetrahedron +generated by the origin and any given triangular face. +An additional benefit of this method is that we can ad- +just the number of iterations in order to achieve any level +of precision we desire. We find that after 5 subdivisions +the calculated volume of our icosphere differs from that +of a perfect sphere by only 0.05%, while the calculated +projected area varies by only 0.03%. We use this as a +benchmark for the accuracy of our method and fix all +further calculations to 5 subdivisions, which gives us a +surface of 10242 triangular tiles. +We next scale each vertex radially until all points have +the same gravitational potential, which requires us to +pick a value of the equipotential Φ. We choose Φ such +that the projected surface area matches the observed +transit depth, similar to what is done in Burton et al. +(2014). We first evaluate the equipotential function for +a range of radii centered on the spherical planet radius. +For each value of Φ generated this way, we then calculate + +Top Down View +Observer View +Spherical +Spherical +0.20 +Towards Star +0.20 +.Distorted +Distorted +0.15 +0.15 +0.10 +0.10 +0.05 +0.05 +z(Ro) +z(Ro) +0.00 +0.00 +-0.05 +-0.05 +-0.10 +-0.10 +0.15 +-0.15 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +x(Ro) +y(Ro)4 +Berardo & de Wit 2022 +the radius of each vertex using a least squares regression +in order to find the surface of constant potential. For +this surface, we then calculate the projected planet area. +This gives us a mapping between gravitational potential +and transit depth, which we use to select the value of Φ +which corresponds to any depth value of our choosing. +The result of this process is shown in figure 2, where +we have calculated the deformation of WASP-19 b (Hebb +et al. 2009) using the described process. This example +highlights the potential for tidal deformation to alter a +planets measured density. +In the left panel we see a +significant deviation from a pure sphere, as the planet is +pulled towards its host star. However in the right panel +we see that the observer-projected shape of the planet +remains nearly perfectly circular. +3. DENSITY VARIATIONS OF CONFIRMED +PLANETS +3.1. Planet Sample +We begin with the full list of confirmed planets found +in the exoplanet archive (NASA Exoplanet Archive +2019) which currently contains just over 5000 exoplan- +ets. As mentioned in the previous section, as well as +motivated by the results of Burton et al. (2014), we fo- +cus our efforts on short period planets, specifically plan- +ets with orbital period of less than 3 days. We do also +analyze planets with periods in the range of 3-5 days, +but those were found to have negligible tidal distortion +effects, consistent with expectations. +We additionally focus only on planets which have re- +ported mass values. In principle, relative variations in +density can be measured based on just changes in planet +volume which is the focus of this work. However we also +consider the magnitude of such a difference relative to +the uncertainty in the measured density, for which a +mass value (along with an error-bar) is required. We +also cut for planets with eccentricity values below e = +0.05. This leaves us with a final sample of 196 planets, +just over 6 times larger than the sample of planets used +in Burton et al. (2014). +3.2. Density Variation Results +We apply the process described in section 2.2 to each +of the planets in our sample. For each planet, we calcu- +late its volume under tidal deformation that produces a +depth value which matches the median reported value to +within 0.1% in order to minimize differences caused by +truncation or any other numerical effects. All analyses +in this section use these values in order to compare the +spherical and tidal planet densities. +3.2.1. Absolute changes in density & trends +Figure 3: Relative change in the density of planets with +orbital periods less than three days. The curves show the +functional dependence of equation 8 for representative +values of planet mass and radius. +We first look at the percent difference in the density +of each planet under the assumptions of being perfectly +spherical or tidally deformed +∆ρ +ρsph += ρsph − ρtide +ρsph +(4) +where we calculate the value of ρsph ourselves using +the reported values of mass, depth, and stellar radius. +This is done to ensure a fair comparison in order to +accurately represent the amount by which density can +shift due to changes in volume. As we will see in the +next section, this quantity is often comparable to the +density uncertainty, which is set by the underlying un- +certainties from transit depth and RV semi-amplitude +measurements used to calculate density. Using the re- +ported value of planet density thus suffers the risk of +including measurement uncertainty (depending on how +density is reported which varies between analyses) when +at this stage we only wish to determine intrinsic dif- +ferences. Thus we ensure that both our measurements +correspond to identical values of depth and planet mass. +The results of this are shown in figure 3. We show the +variation as a function of orbital period, where we note +that the variation decreases as period increases. This is +not surprising given the factor of 1/p2 which appears in +the potential equation, and acts as an additional con- +firmation that our code is accurately calculating planet +deformations (a similar trend with fewer planets was also +seen in Burton et al. (2014)). +We truncate the plot at an upper limit of p = 3 days, +but note that we calculated the deformation out to a + +0.16 +Mp=1Mj,Rp=1 R +0.14 +Mp = 0.5Mj, Rp = 2 Rj +Mp = 10Mj, Rp = 0.5 R +0.12 +0.10 +△p/p +0.08 +0.06 +0.04 +0.02 +0.00 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Period (days)Density Variations +5 +period of 5 days and found that the trend continued, +in particular the upper envelope which flattens out at a +maximal deviation of ∼ 2%. We find that for planets +with orbital periods below 1.5 days, the tidal density +may deviate by as much as 15% compared to the density +which comes from assuming a perfectly spherical planet. +3.2.2. Functional Approximation of Density Variations +The scatter in figure 3 implies that orbital period is +not the sole factor in determining density variations, +which is also apparent from the additional terms in equa- +tion 1. We attempt to derive a functional form of the +variation in density by comparing the full tidal poten- +tial to that of an isolated spherically symmetric body, +given by Φsph = −GMp/r. We first assume that points +along the surface of a tidally distorted planet are at a +similar distance to the center of the planet as for a non- +distorted planet, i.e. no part of the planet is distorted by +a factor of say two or more. Thus in the Roche approx- +imation we may replace quantities such as x2 + y2 + z2 +with r2 +p or equivalently +√ +δRs where δ is the observed +transit depth and Rs is the stellar radius. We also as- +sume that a >> Rp (in our sample we always have at +least a/Rp > 10), and also that that solar mass is much +larger than the planetary mass (for our sample we al- +ways have Ms/Mp > 102). +Under these assumptions +the three terms from equation 1 become: +Φ1 ∼ −GMs +a +, Φ2 ∼ −GMp +rp +, Φ3 ∼ −1 +2Ω2a2 +(5) +which we then combine and scale by Φsph to get +Φsph − Φtide +Φsph += 3 +2 +Mp +Ms +rp +a +(6) +where we’ve used Kepler’s 3rd law to combine the or- +bital period and semi-major axis terms. +We now have an equation for the change in gravita- +tional potential, which must be converted to a change in +density. Given that the potential is treated as a radial +1D function, a reasonable assumption might be that the +scaling term (rp/a) needs to be cubed in order to obtain +a relationship for density. To confirm this, we parame- +terize the change in density as +∆ρ +ρsph += α +�Mp +Ms +�β �rp +a +�γ +(7) +and fit for α, β, γ against the calculated values for +∆ρ/ρ. We do indeed find that γ ∼ 3, as well as α ∼ 2 +and β ∼ 1. We present the final effect on the change in +density (having re-converted to orbital period) as +∆ρ +ρsph += 0.01428 +� P +day +�−2 �Rp +RJ +�3 �Mp +MJ +�−1 +(8) +We plot this function for representative values of +planet radius and planet mass in our sample in figure +3, where we find good agreement particularly in the up- +per envelope of the data points which closely follows an +inverse square dependence on the period. +Figure 4: We show here agreement between the func- +tional form the of the density perturbation we derive +in section 3.2.2 (x-axis) against the values calculated in +section 3.2.1 (y-axis). The black line a linear relation- +ship with a slope of 2 passing through the origin. The +coloring represents the mass of each planet on a log scale. +We additionally compare this analytic description of +the change in density directly to the values calculated +in section 3.2.1 and show the results in figure 4. +We +find that the bulk of the data points follow a linear rela- +tionship with a slope of ∼2, although we do still note a +certain amount of scatter above the line. Planets which +deviate significantly from the trend tend to have smaller +masses (closer to being earth sized). This implies that +one or more of the assumption we have made in deriving +this relationship breaks down for sufficiently low mass +planets. At the scales involved, our approximation de- +viates by at most a factor of ten from the true relative +density change. This implies an underestimate of the +true volume change by at most 10%, which in turn cor- +responds to an error on the linear scale of the planet +by +1.11/3 ∼ 3%. The true deviation is almost always +larger than our functional approximation. Thus equa- +tion 8 represents a fairly robust metric to determine if a +planet may be susceptible to tidal deformations, without +needing to run a full gravitational potential calculation. +A similar metric was derived in Correia (2014) (eq. +27), using a different approach considering the Love +number and fluid displacement of an exoplanet (Love + +0.14 +3.5 +0.12 +3.0 +Mass (Me) +0.10 +2.5 +△p/psph +0.08 +2.0 +Planet +0.06 +1.5 +Log10 +1.0 +0.04 +0.5 +0.02 +0.0 +0.00 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +(rp/a)3Mp/Ms6 +Berardo & de Wit 2022 +Figure 5: Mass-Radius relationships along with data points (and error-bars) for the sample of planets considered +in this work. The left and right panels refer to low and high mass planets respectively. Black curves in the left +figure are taken from Zeng et al. (2019). Colored bands represent variations of these curves by up to 4.7% in radius, +corresponding to density variations of up to 15%. The right hand panel shows a similar phenomenon for high mass +planets, where we show constant-density relations corresponding to solar system gas giants giant densities, as well as +a planet with half the density of Saturn and a planet with a density of 0.2g/cm3 representative of ‘super puffs’. The +planets highlighted in red are those mentioned in Table 2 whose density uncertainty is less than the deviations caused +by tides. +1911). +The result they obtain is similar in that it is +proportional to the ratio of planet to stellar mass, as +well to the third power of planet size to orbital semi- +major axis. While we find a constant scaling factor of +two, they obtain a scaling factor of 7hf/4, where hf is +the fluid second Love number. Estimating hf using the +Darwin-Radau relation (Bourda & Capitaine 2004) and +a value of ∼ 0.27 for the moment of inertia of Jupiter (Ni +2018) gives a prefactor of 2.5. This difference of 25% in +estimated tidal density is well below the measurement +uncertainty on planet density, and using either equation +would indicate weather or not the planet of density may +be significantly different from that of a spherical planet. +An additional consideration of Correia (2014) is the +effect of inclination on the derived density, which intro- +duces a correction term in their equation 27 proportional +to cos2i. We find that for the planets in our sample, the +effect of this correction term is at most 2% for a handful +of planets, and more typically well below a 1% correc- +tion. Thus in our analysis we have chosen to neglect +the effects of inclination, in order to provide a simple +framework which still captures the bulk of the devia- +tions. Even when considering the maximum inclination +that would allow for a transit to occur, the correction +term is at most 2% for the shortest period planets in our +sample, and for most planets is much less than 1%. +4. DISCUSSION +In the previous section we considered the absolute +changes in density a planet may experience under the +effects of tidal forces. We now focus on contextualiz- +ing these results with regards to measurement accuracy, +and biases that may be introduced in measuring planet +compositions to high precision. +4.1. Uncertainty in Mass-Radius Relations +During transit a tidally distorted planet will still have +a nearly circular projection while having a larger than +expected volume as shown in figure 2. The implication +of this is that a spherical transiting planet could have +the same density as a deformed planet with a smaller +projected area, due to the ‘hidden’ extra volume. Thus +when considering mass-radius composition curves, there +is in-fact a degeneracy wherein a single curve could ac- +tually correspond to a range of projected radii, which + +1.8 +Density (g/cm3) +2.0 +1.7 +1.33 (Jupiter) +0.69 (Saturn) +WASP-12 p +1.68 (Neptune) +1.6 +(RE) +(Rj) +1.8 +0.2 +Spherical Radius ( +K2-141 b +1.5 +Spherical Radius ( +Kepler-10b +1.6 +WASP-103 b +1.4 +HAT-P-7 b +1.3 +1.4 +WASP-19 +NASP-4 b +Composition +1.2 +75%fe +50%fe +1.2 +1.1 +20%fe +rocky +25%h20 +1.0 +1.0 +1 +2 +3 +4 +5 +6 +7 +0.5 +1.0 +1.5 +Planet Mass (Me) +Planet Mass (M)Density Variations +7 +Figure 6: This figure illustrates the relative contribution of five underlying factors to the derived measurement error +of a planets density. The x-axis represents a minimum amount that a given parameter contributes to the overall +uncertainty on density. +Solid colored lines represent directly observable quantities (period, transit depth and RV +amplitude), while dashed colored lines refer to model-dependant quantities (stellar mass and radius). The black line +(‘shape’) shows the ratio of tidally-induced variation to measurement error (∆ρ/σρ). The grey line shows a similar +value, after having artificially reduced the overall uncertainty on density by a factor of three to highlight the effect of +future measurement improvements. +we show in figure 5. The implication of this is that even +if a planet had no error whatsoever on its transit depth, +there would still remain uncertainty on its composition +due to a lack of knowledge of its shape, becoming a +bottleneck when attempting to measure planetary com- +positions to high precision. +We separate our sample of planets into low mass +(Earth-sized) and high mass (Jupiter-size) planets, and +for each we show a selection of various composition +curves. For the low mass planets we show curves taken +from Zeng et al. (2019), for a range of iron fractions as +well as an Earth-like composition and a planet with a +25% water composition. For gas giant planets, we show +a range of densities corresponding to the solar system +gas giants, as well as a lower density of 0.2g/cm3 as a +representative value of large planets with low densities, +so called ‘super-puffs’(Masuda 2014; Lopez & Fortney +2014). For each curve, we a plot a range of values (the +colored regions) corresponding to a radius difference of +∼ 5%, which corresponds to a maximal density variation +of ∼ 15%. This represents the range of projected radii +which could all correspond to the same density. +The effect of these considerations is that, for example, +a composition of 20% iron and one of pure-rock become a +near continuous region of parameter space, and a planet +such as Kepler-10 b, which we note in the next section +has a relatively low measurement error, could now be +equally described by either model. We additionally see +planets which fall between models of 25% water and +one of pure rock. While their own uncertainties make +the distinction clear, with one model being two or three +standard deviations away, it becomes much less obvious +which model is correct once the additional uncertainty +from shape variations (colored bands) is considered. +4.2. Uncertainty of Density Measurements +In the previous section we considered the limiting case +of perfect transit depth and mass radius knowledge and +their effect on compositional analysis. +We now focus +on current measurement errors, how they compare to +changes induced by tidal variations, and how upcoming +improvement in precision of the quantities used to cal- +culate density, namely transit depth, stellar radius, and +planet mass (which itself depends on the stellar mass, +RV semi-major amplitude, and orbital period) will in +turn affect the uncertainty on density. + +100 +Period +StellarRadius +Transit Depth +Shape (Ap/op) +RV Amplitude +Shape (△p/(op/3)) +Stellar Mass +80 +Planets in Sample (%) +60 +40 +20 +0 +0 +20 +40 +60 +80 +100 +Minimum Error contribution (%)8 +Berardo & de Wit 2022 +For a function f which depends on independent vari- +ables xi, we can write the uncertainty of f (denoted σf) +as: +σ2 +f = +� +i +�∂f +∂xσxi +�2 +(9) +which for a density calculated using planet mass (Mp), +transit depth (δ) and stellar radius (Rs) becomes +σρ = +�� ρ +Mp +σMp +�2 ++ +�3 +2 +ρ +δ σδ +�2 ++ +� +3 ρ +Rs +σRs +�2 +(10) +We note that most of the planets in our sample have +reported values for their density along with an error-bar +in their entries in the exoplanet archive. We additionally +calculate the uncertainty by ourselves using equation 10 +and the reported uncertainties for the involved quanti- +ties, and find a good agreement between the two values. +An additional consideration is that the planet mass it- +self is dependant on the radial velocity semi-major am- +plitude (K), stellar mass (Ms), and orbital period (p), +which allows us to write the uncertainty on the planet +mass as: +σMp = +��Mp +K σK +�2 ++ +�3 +2 +Mp +Ms +σMs +�2 ++ +�1 +3 +Mp +P σP +�2 +(11) +The benefit of calculating the uncertainty directly in +this way is that we are then able to compare the relative +contribution of each term to the overall uncertainty. We +quantify the relative contribution of a variable xi as: +� ∂ρ +∂xi +σxi +�2 +/σ2 +ρ +(12) +such that the sum of the contributions of each variable +is 100%. The results of this breakdown are shown in fig- +ure 6, where we illustrate how often a given parameter +contributes a minimum amount to the uncertainty. We +find for example that in our sample of planets the orbital +period never contributes more than 0.0001% relative to +the other parameters, which is unsurprising given that +the orbital period of a transiting planet is typically mea- +sured to extremely high precision. +For the remaining four parameters we can categorise +them as being either measurement dependant (transit +depth and RV amplitude) or model dependant (stellar +mass and stellar radius). We find that it is the +measurement parameters which more often contribute +the largest amount of uncertainty, with the RV +amplitude alone contributing at least 60% of the +Error Contributor +Min % +Max % +Median % +1 +31 +100 +59 +2 +0 +45 +25 +3 +0 +28 +9 +4 +0 +18 +3 +Table 1: Summary of the ranked contributions to den- +sity error across all planets where 1 = largest contribut- +ing factor and 4 = smallest. We find that the largest +source of error (regardless of which underlying parame- +ter it comes from) comprises anywhere from 31%-100% +of the uncertainty on a planets density, with a median +value of 59%. +relative uncertainty for ∼20% of the planets in our +sample, and in some case it even contributes almost +the entirety of the uncertainty. Transit depth similarly +can contribute as much as 90% in some cases, whereas +the model dependant parameters never contribute +more than 80% of the relative uncertainty. In table 1 +we show the ranked breakdown of error contributions, +in order of largest to smallest contributor (regardless of +which parameter it comes from). We find for example +that for a given planet the largest source of uncertainty +always contributes at least 31% of the error and +potentially the entire uncertainty, with a median +contribution of 59%. This indicates that for most +planets there is a single parameter which contributes +more than half of the density uncertainty. +When we consider the overall uncertainty on the +spherical density, we find that for most planets the +calculated difference between the spherical and tidal +density (∆ρ) is smaller than the uncertainty (σρ), +illustrated by the black line in figure 6 which shows the +ratio between the two. This implies that with current +data precision assuming a planet to be spherical in +most cases does not introduce a significant statistical +bias, but may be causing density error uncertainties to +be underestimated. +Given this result, we can then ask by how much the +error on planet density needs to be reduced before we +have σρ = ρsphere − ρtidal, which we show in figure 7. +We see that the peak lies around an improvement of +roughly 3-10x, although for many planets the required +improvement is much smaller. For planets where the +radial velocity amplitude or transit depth are the +largest contributing factor, this implies that at a +reduction in their uncertainties by 3-10x, tidal effects +on density will begin to become relevant and the planet +can no longer safely assume to be spherical. This is +shown by the grey curve in figure 6, where we reduce +the density error by a factor of three and show the + +Density Variations +9 +Planet Name +Period (days) +∆ρ/σρ +Reference +WASP-19 b +0.8 +3.8 +Hebb et al. (2009) +HAT-P-7 b +2.2 +3.6 +P´al et al. (2008) +WASP-12 b +1.1 +3.0 +Hebb et al. (2009) +WASP-121 b +1.3 +1.9 +Bourrier et al. (2020) +WASP-4 b +1.3 +1.4 +Bouma et al. (2019) +WASP-103 b +0.9 +1.4 +Gillon et al. (2014) +Kepler-10 b +0.8 +1.4 +Esteves et al. (2015) +K2-141 b +0.3 +1.1 +Malavolta et al. (2018) +Table 2: List of planets with density uncertainties less +than the potential deviation due to tidal effects. +relative value of tidal effects, which is comparable to +∼ 50% of the measurement error on density for > 35% +of planets. +We note that there is a small sample of planets for +which current measurement errors are in fact less than +the calculated deviation on their densities due to tidal +effects (the highlighted orange part of Figure 7). We +report these planets in Table 2, sorted by the +multiplicative factor by which tidal deviations +outweigh measurement uncertainties. In the worst +case, we find that for WASP-19 b this is almost a +factor of four, implying that the reported precision on +its density is significantly underestimated. Again we +note that grey curve of figure 6 shows that after +reducing the total error on density by a factor of three, +we find ∼ 20% of our sample or almost 40 planets for +which tidal variations on density would become larger +than measurement errors. +5. CONCLUSIONS +Using a gravitational potential framework to determine +the shape of a planet under the influence of tidal +distortions, we have expanded on the work of Burton +et al. (2014) and calculated the amount by which such +effects may bias the estimated density of an exoplanet +with orbital periods of less than three days. In +comparison to an assumption of being perfectly +spherical, tidal effects serve to increase the perceived +volume of an exoplanet (and thus decrease its density) +by an amount of up to 15% for the shortest period +planets, which agrees with the values reported in +Burton et al. (2014) and Leconte et al. (2011), which +reported their results as a change in effective planet +radius.Similarly, Akinsanmi et al. (2019) considered a +framework of a planet with ellipsoidal variations how +that would affect their lightcurves and identified many +of the same planets as we do in table 2 as being those +which would exhibit the strongest signal of shape +deformation. +Figure 7: The factor by which the uncertainty on a +planets density needs to be improved such that it is equal +to the change in density due to tidal deformations. The +black dotted line highlights a decrease by a factor of 3 in +the uncertainty on spherical planet density. The orange +region highlights planets whose density uncertainty is +currently less than the difference between the spherical +and tidal density values. +We quantify this change more precisely in terms of the +semi-major axis, planet to star mass radio, and planet +radius for which we are able to derive a robust +relationship (eq. 8). This allows for a rapid estimate of +the magnitude of such variations, and whether or not +an analysis of a planets density (and thus its internal +composition) will be significantly biased by assuming +the planet to be perfectly spherical. In Correia (2014) +a similar expression was derived through an alternate +analytic consideration, including the effect of +inclination as well as the fluid Love number of the +planet. We find the inclination effect to alter the +density perturbation by at most 2% for the planets in +our sample, although in most cases the effect is much +less than 1%. The additional consideration of the fluid +response of the planet implies potential variations of +∼ 20% between our results (i.e. a 15% density +perturbation could change by a factor of 0.8-1.2), +however this is strongly affected by uncertainty in the +Love number. A more detailed analysis of the fluid +response of planets in Wahl et al. (2021) identified +WASP-12 b, WASP-103 b, and WASP-121b as those +with the potential for the greatest variation in tidal +response, which we also found to be among planets +with the highest deviation in derived densities. + +20 +18 +3 += +16 +14 +Number of Planets +12 +10 +8 +6 +4 +2 +0 +1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +logioDensityUncertaintyImprovement10 +Berardo & de Wit 2022 +For planets with orbital periods beyond 2.5 days we +measured variations of no more than 2%, well below +current measurement errors (σρ/ρ > 2.7% for p > 2.5 +days). We find also that for most planets, even those +with shorter orbital periods, measured uncertainties +are currently too large to be affected by such +deviations, however we identify a sample of planets +whose uncertainties may be as much as four times +smaller than the potential change caused by shape +distortions. One such planet, WASP-103 b, was +recently found to show tentative tidal deformations +using multiple transit observations (Barros et al. 2022), +where it is reported that the volumetric radius of a fit +derived using an ellipsoidal planet model is 5-6% larger +than the radius derived from a spherical planet model. +This further strengthens the notion that for such +planets susceptible to tidal deformation, any attempts +to characterise their interior composition based on +their density derived using spherical planet models are +likely to be under-estimating their errors, and that +there is a wall of accuracy which is limited by a lack of +knowledge of their true shape. For other short period +planets however we calculate that an overall +improvement by a factor of three in density error +would cause ∼ 25 planets to have density errors +comparable to tidal distortions, and for ∼ 50 planets +tidal distortions would compare to at least 50% of the +measured density uncertainty. +With this in mind, we find that radius values in a +range of up to a 5% deviation could in fact correspond +to planets with the same density. This implies that +composition curves are not just one-to-one functions +but rather correspond to a family of mass-radius +relationships, where there is a degeneracy induced by a +lack of knowledge of the shape of a planet. This +highlights a fundamental limit in the precision of +characterising the composition of an exoplanet when +disregarding tidal variations, which will become more +severe as measurement errors decrease. +Finally, we break down the uncertainty on a planets +density further as a contribution of five underlying +factors, three of which are directly observable and two +which are model-derived. This breakdown highlights +the fact that it is the directly observable quantities +(specifically RV amplitude and transit depth) which +are in most cases responsible for the bulk of the error +in a planets density (in some cases contributing almost +the entirety of the error budget). We also find that the +median contribution of the largest piece of the +uncertainty budget is 59%, implying that for most +planets there is a single key parameter contributing the +bulk of the uncertainty. Thus upcoming extreme +precision RV measurements as well as high SNR transit +observations such as those from JWST and PLATO +imply that biases due to tidally-induced shape +deformations will become a significant and unavoidable +bottleneck when attempting to measure the density of +planet to a high level of accuracy as the error in these +key contributing factors is reduced. +6. ACKNOWLEDGEMENTS +DB acknowledges support from an FRQNT Doctoral +Research Scholarship. +REFERENCES +Akinsanmi, B., Barros, S. C. C., Santos, N. C., et al. 2019, +A&A, 621, A117, doi: 10.1051/0004-6361/201834215 +Barnes, J. W., & Fortney, J. J. 2003, ApJ, 588, 545, +doi: 10.1086/373893 +Barnes, R. 2017, Celestial Mechanics and Dynamical +Astronomy, 129, 509, doi: 10.1007/s10569-017-9783-7 +Barros, S. C. C., Akinsanmi, B., Bou´e, G., et al. 2022, +A&A, 657, A52, doi: 10.1051/0004-6361/202142196 +Berardo, D., & de Wit, J. 2022, Arxiv e-prints. +https://arxiv.org/abs/2207.07670 +Borucki, W. 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D., et al. 2019, +Proceedings of the National Academy of Science, 116, +9723, doi: 10.1073/pnas.1812905116 + diff --git a/hdFAT4oBgHgl3EQf9R5b/content/tmp_files/load_file.txt b/hdFAT4oBgHgl3EQf9R5b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c97d5bebfb745caa29209458ccf8fe19293e2494 --- /dev/null +++ b/hdFAT4oBgHgl3EQf9R5b/content/tmp_files/load_file.txt @@ -0,0 +1,620 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf,len=619 +page_content='Draft version January 24, 2023 Typeset using LATEX twocolumn style in AASTeX631 Tidal Distortions as a Bottleneck on Constraining Exoplanet Compositions David Berardo∗ and Julien de Wit† ABSTRACT Improvements in the number of confirmed planets and the precision of observations implies a need to better understand subtle effects which may bias interpretations of exoplanet observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' One such effect is the distortion of a short period planet by its host star, affecting its derived density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We extend the work of Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Correia (2014) and others, using a gravitational potential formulation to a sample of nearly 200 planets with periods less than three days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find five planets exhibiting density variations of > 10%, and as many as twenty planets with deviations > 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We derive an analytic approximation for this deviation as a function of the orbital period, transit depth, and mass ratio between the planet and host star, allowing for rapid determination of such tidal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find that current density error-bars are typically larger than tidal deviations, but that reducing the uncertainty on transit depth and RV amplitude by a factor of three causes tidal effects to dominate density errors (> 50%) in >40% of planets in our sample, implying that in the near future upgraded observational precision will cause shape deviations to become a bottleneck with regards to analysis of exoplanet compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' These two parameters are found to dominate uncertainties compared to errors on stellar mass and radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We identify a group of eight planets (including WASP-19 b, HAT-P-7 b, and WASP-12 b) for which current density uncertainties are as much as four times smaller than the potential shift due to tides, implying a possible 4σ bias on their density estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' INTRODUCTION As the list of confirmed exoplanets grows we continu- ously expand the sampled space of known planetary pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Categories of planets such as those with ultra- short orbital periods have gone from containing a hand- ful of planets to hundreds of planets thanks to missions such as Kepler (Borucki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2010) and TESS (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In addition to this increase in population, the precision of instruments has continued to reach new heights, reducing the uncertainty in quantities such as transit depth or planetary mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This trend will acceler- ate further with the next generation of observatories and instruments such as JWST and PLATO (Heras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2020), as well as high precision RV instruments such as CARMENES (Reiners et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2018) and ESPRESSO (Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This increase in both the size and Corresponding author: David Berardo berardo@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='edu ∗ Department of Physics and Kavli Institute for Astrophysics and Space Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' USA FRQNT Doctoral Research Scholarship † Department of Earth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Atmospheric and Planetary Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' USA quality of our sample implies that subtle effects which in the past where either too small to be detectable or which affected a single digit number of planets may no longer be disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' An example of this behaviour is the ‘Transit Light Source’ effect (Rackham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2018), in which variability of the stellar surface causes biases in atmospheric characterisation by mimicking or muting effects which produce similar results, acting a bottleneck towards properly understanding a planets atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The focus of this work is on effects which alter the shape of an exoplanet, which is often considered to be a perfect sphere such as in the commonly used models of Mandel & Agol (2002), implemented in the widely used batman package (Kreidberg 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For short pe- riod planets close to their host star, one such effect are tidal distortions which can cause a planet to bulge out towards its host star (Leconte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This effect in particular has the potential to introduce a significant bias on the density of a planet since its sky projection remains close to a perfect circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' When considering for example a planet which is deformed due to rotation caus- ing ts equator to bulge, its projection becomes elliptical (Seager & Hui 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Barnes & Fortney 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In this case, subtle difference in the shape of ingress / egress of the transit lightcurve may be used to break the de- generacy between a spherical and oblate planet (Carter arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='08755v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='EP] 20 Jan 2023 2 Berardo & de Wit 2022 Figure 1: An illustration of the process by which the surface of the sphere is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Starting from an icosahedron on the left, triangular faces are continually subdivided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Finally, the points are normalized to generate a uniformly sampled sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' & Winn 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Berardo & de Wit 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For tidally de- formed planets, phase curve observations which observe the planet from different directions could in principle determine these so called ‘ellipsoidal variations’ through lightcurve deviations (Correia 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2018), however full phase curve observations require a significant amount of observing time to obtain, and at high precision there is likely to be a significant amount of degeneracy between the orbit, shape, and brightness distribution of a planet (de Wit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Tidal distortions imply an underestimate of the vol- ume of a planet, which in turn implies an overestimate of its bulk density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Theoretical considerations of the ef- fect of this have previously been studied in Leconte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2011) This effect has already been considered, primar- ily in the work of Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014), which calculated the magnitude of the distortion and the degree to which it altered the density measurement for a sample of just over 30 planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Additionally, Correia (2014) expanded on this work using a more detailed model to derive an analytic expression for the change in density as a func- tion of distance to the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In this work we aim to expand on these efforts in sev- eral ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Our primary effort is to increase the sample of planets analysed using a gravitational potential model, which has been found to provide similar results to more complicated structural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In the time since these previous studies were published, roughly 6x as many planets have now been found to be in the space of pa- rameters which are susceptible to tidal distortion effects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' planets with orbital periods below three days on circular orbits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In section 2 we briefly outline the theory of tidal de- formation and describe our method for calculating the effects of tidal interactions, and thus altered planetary densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In section 3 we first highlight our sample of planets to be analysed, followed by the results of our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We highlight trends as a function of various system parameters and derive an approximation which accurately describes the changes in density without the need for a full simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In section 4 we first highlight the biases that may be introduced when attempting to retrieve the interior composition of a planet using mass- radius relations under the assumption of being perfectly spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We then compare the changes in density to current density uncertainties, and we also analyse the relative contributions to these uncertainties from five parameters underlying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This allows us to de- termine how upcoming improvements in quantities such as planet mass and stellar parameters will affect the abil- ity to ignore such effects, for example through extreme precision radial velocity efforts (Crass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' CALCULATING THE DENSITY OF A TIDALLY DEFORMED PLANET 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Physical description of scenario To model the shape of the planet, we follow a similar methodology as that of Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014), where the surface of the planet is assumed to be on a gravitational equipotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The value of the gravitational potential generated by a rotating planet and its host star is cal- culated using the Roche approximation (Chandrasekhar 1987): Φ1 = − GM1 ((x + a)2 + y2 + x2)1/2 (1) Φ2 = − GM2 (x2 + y2 + x2)1/2 (2) Φ3 = −1 2Ω2 � (x + µ1a)2 + y2� (3) where G is the gravitational constant, M1 is the mass o Subdivisions 2 Subdivisions 3 Subdivisions + NormalizingDensity Variations 3 Figure 2: This figure shows two views of the surface of WASP-19 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Points in black show the spherical planet which matches the observed transit depth, while points in red show the surface generated by fitting for an equipotential while also matching the observed transit depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' On the left we see a top down view of the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' On the right we see the view along the line of sight between the centers of mass of the planet and star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' of the host star, M2 is the mass of the planet, a is the separation between the host star and planet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' the semi-major axis of a circular orbit), µ1 = M1/(M1+M2) and Ω = 2π/P where P is the orbital period of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The coordinate system is such that the origin is placed at the center of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The x coordinate points along the line connecting the center of masses of the two bodies, the z axis points along the orbital plane in the direction of motion of the planet, and the y axis points normal to the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In order to use such an approximation to model the distortion of a planets surface, we assume the planet is both tidally locked as well on a non-eccentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' As we shall see in later sections, the effect of the distortion is strongest for low period planets (p < 3 days) which are most likely to be tidally locked and be on circular orbits(Barnes 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Calculating the volume of a deformed planet We first calculate the surface of a deformed planet and then ‘measure’ its volume in order to determine the amount by which its density is altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In order to generate the surface of our planet, we first construct a geodesic icosahedron as an approximation of a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This is an object commonly used in computer graph- ics and 3D rendering software which has the benefit of having its points uniformly spread out across its sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We begin with the vertices of an icosahedron and then iteratively subdivide each of its faces into smaller triangles (as shown in figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' After the last round of subdivisions we normalize the length of each vertex from the origin to generate a tiled sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This process leaves us with a collection of triangular faces which allows us to calculate two necessary quan- tities, the total projected surface area visible to an ob- server as well as the enclosed volume of each tetrahedron generated by the origin and any given triangular face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' An additional benefit of this method is that we can ad- just the number of iterations in order to achieve any level of precision we desire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find that after 5 subdivisions the calculated volume of our icosphere differs from that of a perfect sphere by only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05%, while the calculated projected area varies by only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='03%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We use this as a benchmark for the accuracy of our method and fix all further calculations to 5 subdivisions, which gives us a surface of 10242 triangular tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We next scale each vertex radially until all points have the same gravitational potential, which requires us to pick a value of the equipotential Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We choose Φ such that the projected surface area matches the observed transit depth, similar to what is done in Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We first evaluate the equipotential function for a range of radii centered on the spherical planet radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For each value of Φ generated this way, we then calculate Top Down View Observer View Spherical Spherical 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='20 Towards Star 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='Distorted Distorted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05 z(Ro) z(Ro) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='15 x(Ro) y(Ro)4 Berardo & de Wit 2022 the radius of each vertex using a least squares regression in order to find the surface of constant potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For this surface, we then calculate the projected planet area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This gives us a mapping between gravitational potential and transit depth, which we use to select the value of Φ which corresponds to any depth value of our choosing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The result of this process is shown in figure 2, where we have calculated the deformation of WASP-19 b (Hebb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2009) using the described process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This example highlights the potential for tidal deformation to alter a planets measured density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In the left panel we see a significant deviation from a pure sphere, as the planet is pulled towards its host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' However in the right panel we see that the observer-projected shape of the planet remains nearly perfectly circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' DENSITY VARIATIONS OF CONFIRMED PLANETS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Planet Sample We begin with the full list of confirmed planets found in the exoplanet archive (NASA Exoplanet Archive 2019) which currently contains just over 5000 exoplan- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' As mentioned in the previous section, as well as motivated by the results of Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014), we fo- cus our efforts on short period planets, specifically plan- ets with orbital period of less than 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We do also analyze planets with periods in the range of 3-5 days, but those were found to have negligible tidal distortion effects, consistent with expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We additionally focus only on planets which have re- ported mass values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In principle, relative variations in density can be measured based on just changes in planet volume which is the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' However we also consider the magnitude of such a difference relative to the uncertainty in the measured density, for which a mass value (along with an error-bar) is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We also cut for planets with eccentricity values below e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This leaves us with a final sample of 196 planets, just over 6 times larger than the sample of planets used in Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Density Variation Results We apply the process described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2 to each of the planets in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For each planet, we calcu- late its volume under tidal deformation that produces a depth value which matches the median reported value to within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1% in order to minimize differences caused by truncation or any other numerical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' All analyses in this section use these values in order to compare the spherical and tidal planet densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Absolute changes in density & trends Figure 3: Relative change in the density of planets with orbital periods less than three days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The curves show the functional dependence of equation 8 for representative values of planet mass and radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We first look at the percent difference in the density of each planet under the assumptions of being perfectly spherical or tidally deformed ∆ρ ρsph = ρsph − ρtide ρsph (4) where we calculate the value of ρsph ourselves using the reported values of mass, depth, and stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This is done to ensure a fair comparison in order to accurately represent the amount by which density can shift due to changes in volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' As we will see in the next section, this quantity is often comparable to the density uncertainty, which is set by the underlying un- certainties from transit depth and RV semi-amplitude measurements used to calculate density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Using the re- ported value of planet density thus suffers the risk of including measurement uncertainty (depending on how density is reported which varies between analyses) when at this stage we only wish to determine intrinsic dif- ferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Thus we ensure that both our measurements correspond to identical values of depth and planet mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The results of this are shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We show the variation as a function of orbital period, where we note that the variation decreases as period increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This is not surprising given the factor of 1/p2 which appears in the potential equation, and acts as an additional con- firmation that our code is accurately calculating planet deformations (a similar trend with fewer planets was also seen in Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We truncate the plot at an upper limit of p = 3 days, but note that we calculated the deformation out to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='16 Mp=1Mj,Rp=1 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='14 Mp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5Mj, Rp = 2 Rj Mp = 10Mj, Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 △p/p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 Period (days)Density Variations 5 period of 5 days and found that the trend continued, in particular the upper envelope which flattens out at a maximal deviation of ∼ 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find that for planets with orbital periods below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 days, the tidal density may deviate by as much as 15% compared to the density which comes from assuming a perfectly spherical planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Functional Approximation of Density Variations The scatter in figure 3 implies that orbital period is not the sole factor in determining density variations, which is also apparent from the additional terms in equa- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We attempt to derive a functional form of the variation in density by comparing the full tidal poten- tial to that of an isolated spherically symmetric body, given by Φsph = −GMp/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We first assume that points along the surface of a tidally distorted planet are at a similar distance to the center of the planet as for a non- distorted planet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' no part of the planet is distorted by a factor of say two or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Thus in the Roche approx- imation we may replace quantities such as x2 + y2 + z2 with r2 p or equivalently √ δRs where δ is the observed transit depth and Rs is the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We also as- sume that a >> Rp (in our sample we always have at least a/Rp > 10), and also that that solar mass is much larger than the planetary mass (for our sample we al- ways have Ms/Mp > 102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Under these assumptions the three terms from equation 1 become: Φ1 ∼ −GMs a , Φ2 ∼ −GMp rp , Φ3 ∼ −1 2Ω2a2 (5) which we then combine and scale by Φsph to get Φsph − Φtide Φsph = 3 2 Mp Ms rp a (6) where we’ve used Kepler’s 3rd law to combine the or- bital period and semi-major axis terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We now have an equation for the change in gravita- tional potential, which must be converted to a change in density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Given that the potential is treated as a radial 1D function, a reasonable assumption might be that the scaling term (rp/a) needs to be cubed in order to obtain a relationship for density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' To confirm this, we parame- terize the change in density as ∆ρ ρsph = α �Mp Ms �β �rp a �γ (7) and fit for α, β, γ against the calculated values for ∆ρ/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We do indeed find that γ ∼ 3, as well as α ∼ 2 and β ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We present the final effect on the change in density (having re-converted to orbital period) as ∆ρ ρsph = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='01428 � P day �−2 �Rp RJ �3 �Mp MJ �−1 (8) We plot this function for representative values of planet radius and planet mass in our sample in figure 3, where we find good agreement particularly in the up- per envelope of the data points which closely follows an inverse square dependence on the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Figure 4: We show here agreement between the func- tional form the of the density perturbation we derive in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2 (x-axis) against the values calculated in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1 (y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The black line a linear relation- ship with a slope of 2 passing through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The coloring represents the mass of each planet on a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We additionally compare this analytic description of the change in density directly to the values calculated in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1 and show the results in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find that the bulk of the data points follow a linear rela- tionship with a slope of ∼2, although we do still note a certain amount of scatter above the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Planets which deviate significantly from the trend tend to have smaller masses (closer to being earth sized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This implies that one or more of the assumption we have made in deriving this relationship breaks down for sufficiently low mass planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' At the scales involved, our approximation de- viates by at most a factor of ten from the true relative density change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This implies an underestimate of the true volume change by at most 10%, which in turn cor- responds to an error on the linear scale of the planet by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='11/3 ∼ 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The true deviation is almost always larger than our functional approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Thus equa- tion 8 represents a fairly robust metric to determine if a planet may be susceptible to tidal deformations, without needing to run a full gravitational potential calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' A similar metric was derived in Correia (2014) (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 27), using a different approach considering the Love number and fluid displacement of an exoplanet (Love 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 Mass (Me) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 △p/psph 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 Planet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 Log10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='05 (rp/a)3Mp/Ms6 Berardo & de Wit 2022 Figure 5: Mass-Radius relationships along with data points (and error-bars) for the sample of planets considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The left and right panels refer to low and high mass planets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Black curves in the left figure are taken from Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Colored bands represent variations of these curves by up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='7% in radius, corresponding to density variations of up to 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The right hand panel shows a similar phenomenon for high mass planets, where we show constant-density relations corresponding to solar system gas giants giant densities, as well as a planet with half the density of Saturn and a planet with a density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2g/cm3 representative of ‘super puffs’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The planets highlighted in red are those mentioned in Table 2 whose density uncertainty is less than the deviations caused by tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 1911).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The result they obtain is similar in that it is proportional to the ratio of planet to stellar mass, as well to the third power of planet size to orbital semi- major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' While we find a constant scaling factor of two, they obtain a scaling factor of 7hf/4, where hf is the fluid second Love number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Estimating hf using the Darwin-Radau relation (Bourda & Capitaine 2004) and a value of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='27 for the moment of inertia of Jupiter (Ni 2018) gives a prefactor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This difference of 25% in estimated tidal density is well below the measurement uncertainty on planet density, and using either equation would indicate weather or not the planet of density may be significantly different from that of a spherical planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' An additional consideration of Correia (2014) is the effect of inclination on the derived density, which intro- duces a correction term in their equation 27 proportional to cos2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find that for the planets in our sample, the effect of this correction term is at most 2% for a handful of planets, and more typically well below a 1% correc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Thus in our analysis we have chosen to neglect the effects of inclination, in order to provide a simple framework which still captures the bulk of the devia- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Even when considering the maximum inclination that would allow for a transit to occur, the correction term is at most 2% for the shortest period planets in our sample, and for most planets is much less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' DISCUSSION In the previous section we considered the absolute changes in density a planet may experience under the effects of tidal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We now focus on contextualiz- ing these results with regards to measurement accuracy, and biases that may be introduced in measuring planet compositions to high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Uncertainty in Mass-Radius Relations During transit a tidally distorted planet will still have a nearly circular projection while having a larger than expected volume as shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The implication of this is that a spherical transiting planet could have the same density as a deformed planet with a smaller projected area, due to the ‘hidden’ extra volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Thus when considering mass-radius composition curves, there is in-fact a degeneracy wherein a single curve could ac- tually correspond to a range of projected radii, which 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='8 Density (g/cm3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='33 (Jupiter) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='69 (Saturn) WASP-12 p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='68 (Neptune) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='6 (RE) (Rj) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2 Spherical Radius ( K2-141 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 Spherical Radius ( Kepler-10b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='6 WASP-103 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='4 HAT-P-7 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='4 WASP-19 NASP-4 b Composition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2 75%fe 50%fe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1 20%fe rocky 25%h20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 1 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 Planet Mass (Me) Planet Mass (M)Density Variations 7 Figure 6: This figure illustrates the relative contribution of five underlying factors to the derived measurement error of a planets density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The x-axis represents a minimum amount that a given parameter contributes to the overall uncertainty on density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Solid colored lines represent directly observable quantities (period, transit depth and RV amplitude), while dashed colored lines refer to model-dependant quantities (stellar mass and radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The black line (‘shape’) shows the ratio of tidally-induced variation to measurement error (∆ρ/σρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The grey line shows a similar value, after having artificially reduced the overall uncertainty on density by a factor of three to highlight the effect of future measurement improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' we show in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The implication of this is that even if a planet had no error whatsoever on its transit depth, there would still remain uncertainty on its composition due to a lack of knowledge of its shape, becoming a bottleneck when attempting to measure planetary com- positions to high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We separate our sample of planets into low mass (Earth-sized) and high mass (Jupiter-size) planets, and for each we show a selection of various composition curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For the low mass planets we show curves taken from Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2019), for a range of iron fractions as well as an Earth-like composition and a planet with a 25% water composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For gas giant planets, we show a range of densities corresponding to the solar system gas giants, as well as a lower density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2g/cm3 as a representative value of large planets with low densities, so called ‘super-puffs’(Masuda 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Lopez & Fortney 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For each curve, we a plot a range of values (the colored regions) corresponding to a radius difference of ∼ 5%, which corresponds to a maximal density variation of ∼ 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This represents the range of projected radii which could all correspond to the same density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The effect of these considerations is that, for example, a composition of 20% iron and one of pure-rock become a near continuous region of parameter space, and a planet such as Kepler-10 b, which we note in the next section has a relatively low measurement error, could now be equally described by either model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We additionally see planets which fall between models of 25% water and one of pure rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' While their own uncertainties make the distinction clear, with one model being two or three standard deviations away, it becomes much less obvious which model is correct once the additional uncertainty from shape variations (colored bands) is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Uncertainty of Density Measurements In the previous section we considered the limiting case of perfect transit depth and mass radius knowledge and their effect on compositional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We now focus on current measurement errors, how they compare to changes induced by tidal variations, and how upcoming improvement in precision of the quantities used to cal- culate density, namely transit depth, stellar radius, and planet mass (which itself depends on the stellar mass, RV semi-major amplitude, and orbital period) will in turn affect the uncertainty on density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 100 Period StellarRadius Transit Depth Shape (Ap/op) RV Amplitude Shape (△p/(op/3)) Stellar Mass 80 Planets in Sample (%) 60 40 20 0 0 20 40 60 80 100 Minimum Error contribution (%)8 Berardo & de Wit 2022 For a function f which depends on independent vari- ables xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' we can write the uncertainty of f (denoted σf) as: σ2 f = � i �∂f ∂xσxi �2 (9) which for a density calculated using planet mass (Mp),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' transit depth (δ) and stellar radius (Rs) becomes σρ = �� ρ Mp σMp �2 + �3 2 ρ δ σδ �2 + � 3 ρ Rs σRs �2 (10) We note that most of the planets in our sample have reported values for their density along with an error-bar in their entries in the exoplanet archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We additionally calculate the uncertainty by ourselves using equation 10 and the reported uncertainties for the involved quanti- ties, and find a good agreement between the two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' An additional consideration is that the planet mass it- self is dependant on the radial velocity semi-major am- plitude (K), stellar mass (Ms), and orbital period (p), which allows us to write the uncertainty on the planet mass as: σMp = ��Mp K σK �2 + �3 2 Mp Ms σMs �2 + �1 3 Mp P σP �2 (11) The benefit of calculating the uncertainty directly in this way is that we are then able to compare the relative contribution of each term to the overall uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We quantify the relative contribution of a variable xi as: � ∂ρ ∂xi σxi �2 /σ2 ρ (12) such that the sum of the contributions of each variable is 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The results of this breakdown are shown in fig- ure 6, where we illustrate how often a given parameter contributes a minimum amount to the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find for example that in our sample of planets the orbital period never contributes more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0001% relative to the other parameters, which is unsurprising given that the orbital period of a transiting planet is typically mea- sured to extremely high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For the remaining four parameters we can categorise them as being either measurement dependant (transit depth and RV amplitude) or model dependant (stellar mass and stellar radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find that it is the measurement parameters which more often contribute the largest amount of uncertainty, with the RV amplitude alone contributing at least 60% of the Error Contributor Min % Max % Median % 1 31 100 59 2 0 45 25 3 0 28 9 4 0 18 3 Table 1: Summary of the ranked contributions to den- sity error across all planets where 1 = largest contribut- ing factor and 4 = smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find that the largest source of error (regardless of which underlying parame- ter it comes from) comprises anywhere from 31%-100% of the uncertainty on a planets density, with a median value of 59%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' relative uncertainty for ∼20% of the planets in our sample, and in some case it even contributes almost the entirety of the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Transit depth similarly can contribute as much as 90% in some cases, whereas the model dependant parameters never contribute more than 80% of the relative uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In table 1 we show the ranked breakdown of error contributions, in order of largest to smallest contributor (regardless of which parameter it comes from).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find for example that for a given planet the largest source of uncertainty always contributes at least 31% of the error and potentially the entire uncertainty, with a median contribution of 59%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This indicates that for most planets there is a single parameter which contributes more than half of the density uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' When we consider the overall uncertainty on the spherical density, we find that for most planets the calculated difference between the spherical and tidal density (∆ρ) is smaller than the uncertainty (σρ), illustrated by the black line in figure 6 which shows the ratio between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This implies that with current data precision assuming a planet to be spherical in most cases does not introduce a significant statistical bias, but may be causing density error uncertainties to be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Given this result, we can then ask by how much the error on planet density needs to be reduced before we have σρ = ρsphere − ρtidal, which we show in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We see that the peak lies around an improvement of roughly 3-10x, although for many planets the required improvement is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For planets where the radial velocity amplitude or transit depth are the largest contributing factor, this implies that at a reduction in their uncertainties by 3-10x, tidal effects on density will begin to become relevant and the planet can no longer safely assume to be spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This is shown by the grey curve in figure 6, where we reduce the density error by a factor of three and show the Density Variations 9 Planet Name Period (days) ∆ρ/σρ Reference WASP-19 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='8 Hebb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2009) HAT-P-7 b 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='6 P´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2008) WASP-12 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 Hebb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2009) WASP-121 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='9 Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2020) WASP-4 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='4 Bouma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2019) WASP-103 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='4 Gillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014) Kepler-10 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='4 Esteves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2015) K2-141 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1 Malavolta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2018) Table 2: List of planets with density uncertainties less than the potential deviation due to tidal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' relative value of tidal effects, which is comparable to ∼ 50% of the measurement error on density for > 35% of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We note that there is a small sample of planets for which current measurement errors are in fact less than the calculated deviation on their densities due to tidal effects (the highlighted orange part of Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We report these planets in Table 2, sorted by the multiplicative factor by which tidal deviations outweigh measurement uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In the worst case, we find that for WASP-19 b this is almost a factor of four, implying that the reported precision on its density is significantly underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Again we note that grey curve of figure 6 shows that after reducing the total error on density by a factor of three, we find ∼ 20% of our sample or almost 40 planets for which tidal variations on density would become larger than measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' CONCLUSIONS Using a gravitational potential framework to determine the shape of a planet under the influence of tidal distortions, we have expanded on the work of Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014) and calculated the amount by which such effects may bias the estimated density of an exoplanet with orbital periods of less than three days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In comparison to an assumption of being perfectly spherical, tidal effects serve to increase the perceived volume of an exoplanet (and thus decrease its density) by an amount of up to 15% for the shortest period planets, which agrees with the values reported in Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2014) and Leconte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2011), which reported their results as a change in effective planet radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='Similarly, Akinsanmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2019) considered a framework of a planet with ellipsoidal variations how that would affect their lightcurves and identified many of the same planets as we do in table 2 as being those which would exhibit the strongest signal of shape deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Figure 7: The factor by which the uncertainty on a planets density needs to be improved such that it is equal to the change in density due to tidal deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The black dotted line highlights a decrease by a factor of 3 in the uncertainty on spherical planet density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The orange region highlights planets whose density uncertainty is currently less than the difference between the spherical and tidal density values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We quantify this change more precisely in terms of the semi-major axis, planet to star mass radio, and planet radius for which we are able to derive a robust relationship (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This allows for a rapid estimate of the magnitude of such variations, and whether or not an analysis of a planets density (and thus its internal composition) will be significantly biased by assuming the planet to be perfectly spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' In Correia (2014) a similar expression was derived through an alternate analytic consideration, including the effect of inclination as well as the fluid Love number of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find the inclination effect to alter the density perturbation by at most 2% for the planets in our sample, although in most cases the effect is much less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' The additional consideration of the fluid response of the planet implies potential variations of ∼ 20% between our results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' a 15% density perturbation could change by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='8-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='2), however this is strongly affected by uncertainty in the Love number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' A more detailed analysis of the fluid response of planets in Wahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' (2021) identified WASP-12 b, WASP-103 b, and WASP-121b as those with the potential for the greatest variation in tidal response, which we also found to be among planets with the highest deviation in derived densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 20 18 3 = 16 14 Number of Planets 12 10 8 6 4 2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 logioDensityUncertaintyImprovement10 Berardo & de Wit 2022 For planets with orbital periods beyond 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 days we measured variations of no more than 2%, well below current measurement errors (σρ/ρ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='7% for p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='5 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We find also that for most planets, even those with shorter orbital periods, measured uncertainties are currently too large to be affected by such deviations, however we identify a sample of planets whose uncertainties may be as much as four times smaller than the potential change caused by shape distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' One such planet, WASP-103 b, was recently found to show tentative tidal deformations using multiple transit observations (Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' 2022), where it is reported that the volumetric radius of a fit derived using an ellipsoidal planet model is 5-6% larger than the radius derived from a spherical planet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This further strengthens the notion that for such planets susceptible to tidal deformation, any attempts to characterise their interior composition based on their density derived using spherical planet models are likely to be under-estimating their errors, and that there is a wall of accuracy which is limited by a lack of knowledge of their true shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' For other short period planets however we calculate that an overall improvement by a factor of three in density error would cause ∼ 25 planets to have density errors comparable to tidal distortions, and for ∼ 50 planets tidal distortions would compare to at least 50% of the measured density uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' With this in mind, we find that radius values in a range of up to a 5% deviation could in fact correspond to planets with the same density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This implies that composition curves are not just one-to-one functions but rather correspond to a family of mass-radius relationships, where there is a degeneracy induced by a lack of knowledge of the shape of a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This highlights a fundamental limit in the precision of characterising the composition of an exoplanet when disregarding tidal variations, which will become more severe as measurement errors decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Finally, we break down the uncertainty on a planets density further as a contribution of five underlying factors, three of which are directly observable and two which are model-derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' This breakdown highlights the fact that it is the directly observable quantities (specifically RV amplitude and transit depth) which are in most cases responsible for the bulk of the error in a planets density (in some cases contributing almost the entirety of the error budget).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' We also find that the median contribution of the largest piece of the uncertainty budget is 59%, implying that for most planets there is a single key parameter contributing the bulk of the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content=' Thus upcoming extreme precision RV measurements as well as high SNR transit observations such as those from JWST and PLATO imply that biases due to tidally-induced shape deformations will become a significant and unavoidable bottleneck when attempting to measure the density of planet to a high level of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} +page_content='1812905116' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFAT4oBgHgl3EQf9R5b/content/2301.08755v1.pdf'} diff --git a/j9AyT4oBgHgl3EQfkvhK/content/tmp_files/2301.00438v1.pdf.txt b/j9AyT4oBgHgl3EQfkvhK/content/tmp_files/2301.00438v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c86d3314e6e0d9baba7eef29a185921c1e11e509 --- /dev/null +++ b/j9AyT4oBgHgl3EQfkvhK/content/tmp_files/2301.00438v1.pdf.txt @@ -0,0 +1,534 @@ +arXiv:2301.00438v1 [math.CA] 1 Jan 2023 +ON THE HARMONIC CONTINUATION OF THE RIEMANN XI +FUNCTION +ALEXANDER E. PATKOWSKI +Abstract. We generalize the harmonic continuation of the Riemann xi-function +to the n-dimension case, to obtain the solution to the Dirichlet problem on +Rn. We also provide a new expansion for the harmonic continuation of the +Riemann xi-function using an expansion given by R.J. Duffin. +Keywords: Fourier Integrals; Riemann xi function +2010 Mathematics Subject Classification 11M06, 33C05. +1. Introduction and main result +Recall the Riemann xi-function is given as ξ(s) = 1 +2s(s − 1)π− s +2 Γ( s +2)ζ(s), where +ζ(s) is the Riemann zeta function [11], and Γ(s) is the gamma function. In a recent +paper [9], we presented an application of classical integrals involving the Riemann +xi-function to the Dirichlet problem in the half plane. Let L(s) denote the self-dual +principal automorphic L-functions and ¯γ(s) its associated gamma factor, then our +main theorem presented therein is given in the following. +Theorem 1.1. ([9, Theorem 3.1]) The solution of Dirichlet’s problem in the half +plane, +∂2w +∂y2 + ∂2w +∂x2 = 0, +where y ∈ R, x ≥ 0, initial condition w(y, 0) = h(y) := ¯γ( 1 +2 + iy)L( 1 +2 + iy), +w(y, x) → 0, as |y| → ∞, is given by the Poisson integral. Furthermore, the solution +also satisfies the condition w(0, s− 1 +2) = h(−i(s− 1 +2))−r(s), for an analytic function +r(s) that satisfies h(−i(s − 1/2)) = C/(s(s − 1)) + r(s) + r(1 − s). +The proof relied on extending a classical integral found in Titchmarsh [11, eq.(2.16.1)] +(1.1) +� ∞ +0 +Ξ(t) +t2 + 1 +4 +cos(xt)dt = π +2 +� +ex/2 − 2e−x/2ψ(e−2x) +� +, +1 + +2 +ALEXANDER E. PATKOWSKI +to self-dual principal automorphic L-functions, and then taking the Laplace trans- +form. Here, as usual, Ξ(x) = ξ( 1 +2 + ix) and ψ(x) = � +n=1 e−πn2x. In the special +case of the Riemann xi-function, this was offered in [9] as the following corollary. +Let Γ(s, x) denote the incomplete gamma function over the interval (x, ∞). +Corollary 1.1.1. ([9, Corollary 1.2]) When ℜ(s) > 1, the Riemann ξ-function +satisfies the integral equation, +(1.2) +Υ(s) := (s − 1 +2) +� ∞ +0 +Ξ(t) +(t2 + 1 +4)(t2 + (s − 1 +2)2)dt += π +2 + + +1 +s − 1 − +ξ(s) +s(s − 1) + π−s/2 � +n≥1 +n−sΓ +�s +2, πn2� + + . +Many papers have utilized Fourier integrals of this type to obtain interesting iden- +tities and applications [1, 2, 3, 6, 7, 8, 10]. +The purpose of this paper is to generalize [9, Corollary 1.2] to obtain the harmonic +continuation that solves the n-dimensional Dirichlet problem. In this way we offer +a general result which isn’t captured in [9, Theorem 3.1]. +We also offer a new +expansion of the harmonic continuation of the Riemann xi-function by appealing +to the work of R.J. Duffin [4], which will be presented in the last section. To state +our main result we will need to define a kernel. From [5, pg.93, eq.(2.1.13)], +K(x) = +Γ( n+1 +2 ) +π +n+1 +2 (1 + |x|2) +n+1 +2 . +Theorem 1.2. Let g(x) = �n +l=1 +Ξ(xl) +x2 +l + 1 +4 and set Ky(x) = y−nK(y−1x). Then func- +tion (convolution) u(x, y) = (g ∗ Ky)(x), solves the Dirichlet problem +∂2 +yu + +n +� +l=1 +∂2 +l u = 0, +u(x, 0) = g(x). +Furthermore, we have +u(0, y) = (g ∗ Ky)(0) = y1−n +� +Rn e−2π|yz| +n +� +l=1 +� +ezl/2 − 2e−zl/2ψ(e−2zl) +� +dz. +We note that the case n = 1 reduces to a special case of Theorem 1.1 and contains +[9, Corollary 1.2], being that u(0, y) = (g ∗ Ky)(0) essentially becomes the Laplace +transform of (1.1). + +ON THE HARMONIC CONTINUATION OF THE RIEMANN XI FUNCTION +3 +2. Proof of the n-dimensional case and related comments +Let R+ denote the positive reals. From [5, pg.92], we see that u(x, y) = (g ∗ Ky)(x) +solves the Dirichlet problem +∂2 +yu + +n +� +l=1 +∂2 +l u = 0, +u(x, 0) = g(x). +Then we can say that u(x, y) is harmonic on Rn+1 ++ +. Recall the n-dimensional Fourier +transform [5, pg.108] is given by +ˆf(y) = +� +Rn f(x)e−2πix·ydx, +where x·y = �n +l=1 xlyl. Consider g(x) = �n +l=1 +Ξ(xl) +x2 +l + 1 +4 , and note [5, pg.112, Theorem +2.2.14] +(2.1) +� +Rn f(x)ˆg(x)dx = +� +Rn +ˆf(x)g(x)dx. +Invoking [5, pg.118, Exercise 2.2.11] +y1−n +� +Rn e−2π|yz|e−2πiz·xdz = y−nK(y−1x) = Ky(x), +with our chosen g(x) in (2.1) gives u(0, y), +y1−n +� +Rn e−2π|yz| +n +� +l=1 +� +ezl/2 − 2e−zl/2ψ(e−2zl) +� +dz = +� +Rn g(z)Ky(z)dz, +since +ˆg(y) = +� +Rn g(x)e−2πix·ydx += +n +� +l=1 +� +eyl/2 − 2e−yl/2ψ(e−2yl) +� +. +The right hand side is now precisely u(0, y) = (g ∗ Ky)(0). +It is possible to take another direction to obtain a formula which contains [9, Corol- +lary 1.2] as a special case. If we set yl = yj = y for l ̸= j, and l, j ≤ n, then we can +write +ˆg(y) = +� +ey/2 − 2e−y/2ψ(e−2y) +�n +(2.2) +n +� +k=0 +�n +k +� +ey(n−k)/2 � +−2e−y/2ψ(e−2y) +�k +. +(2.3) + +4 +ALEXANDER E. PATKOWSKI +We write the sum of squares function to be generated by +∞ +� +n=0 +rk(n)qn = +� +∞ +� +n=−∞ +qn2 +�k += +k +� +l=0 +�k +l +� � +2 +∞ +� +n=1 +qn2 +�l += +k +� +l=0 +�k +l +� +2k +∞ +� +m=1 +r′ +l(m)qm, +for |q| < 1 say, and note that r0(0) = 1, and r0(n) = 0 if n > 1. Equating coefficients +of qn we find that for n > 1, +rk(n) = +k +� +l=0 +�k +l +� +2kr′ +l(n), +which gives us 1 +2r1(n) = r′ +1(n). +Note that taking the Laplace transform of (2.2)–(2.3) similar to the method we +employed in [9] gives a different kernel. Namely, +� ∞ +0 +ˆg(y)e−sydy +� +Rn +sg(x) +s2 + (�n +l=1 xl) +2 dx += +1 +s − 1 +2 ++ +n +� +k=1 +�n +k +� +(−2)k +∞ +� +m=1 +r′ +k(m) +� 1 +0 +ts−1+(n−k)/2+k/2e−mt2dt += +1 +s − 1 +2 ++ 1 +2 +n +� +k=1 +�n +k +� +(−2)k +∞ +� +m=1 +r′ +k(m)Γ +�s + (2k − n)/2 +2 +� +1 +ms/2+(2k−n)/4 +− 1 +2 +n +� +k=1 +�n +k +� +(−2)k +∞ +� +m=1 +r′ +k(m)Γ +�s + (2k − n)/2 +2 +, πm +� +1 +ms/2+(2k−n)/4 , +which again gives [9, Corollary 1.2] when n = 1. +3. The Duffin expansion of the harmonic continuation of the +Riemann xi function +In Duffin’s paper [4, pg.275], we find an interesting theorem for the harmonic con- +tinuation of a function. As usual, let µ(n) denote the M¨obius function [11]. For a +definition of P summable see [4, pg.273]. + +ON THE HARMONIC CONTINUATION OF THE RIEMANN XI FUNCTION +5 +Theorem 3.1. ([4, Theorem 2]) Let f(x) ∈ L1(0, ∞) and let C(x) be its transform +relative to the kernel cos(2πx). Then the harmonic continuation is given by +f(x, y) := 1 +π +� +R +y +y2 + (x − t)2 f(t)dt = +∞ +� +m=1 +µ(m) +m +∞ +� +n=1 +e−2nπy/mxC +� n +mx +� +where x, y > 0, and the sum over m is evaluated by P summability. In particular, +f(x, y) → f(x) a.e. when y → 0. +By the integral (1.1) it readily follows that we have the following result. +Theorem 3.2. The harmonic continuation of +f(x) = +Ξ(x) +x2 + 1 +4 +, +is given by +f(x, y) = π +2 +∞ +� +m=1 +µ(m) +m +∞ +� +n=1 +e−2nπy/mx � +e +n +mx2 − 2e− +n +m2x ψ(e−2 +n +mx ) +� +, +where x, y > 0. Furthermore, +Ξ(x) +x2 + 1 +4 += π +2 lim +y→0 +∞ +� +m=1 +µ(m) +m +∞ +� +n=1 +e−2nπy/mx � +e +n +mx2 − 2e− +n +m2x ψ(e−2 +n +mx ) +� +, +and +π +2 + + +1 +y − 1 +2 +− ξ(y + 1 +2)(y − 1 +2) +y + 1 +2 ++ π−(y− 1 +2 )/2 � +n≥1 +n−y+ 1 +2 Γ +�y + 1 +2 +2 +, πn2 +� + += π +2 lim +x→0 +∞ +� +m=1 +µ(m) +m +∞ +� +n=1 +e−2nπy/mx � +e +n +mx2 − 2e− +n +m2x ψ(e−2 +n +mx ) +� +. +Proof. Clearly our choice of f(x) is L1(0, ∞) since Ξ(x) = O(xNe−πx/4) by [11, +pg.257]. The second limit case in the theorem, which is all that remains to prove, +follows directly from [9, Corollary 1.2]. +□ +A criteria for the Riemann Hypothesis may be formulated from this theorem by +putting x = γ where ℑ(ρ) =: γ with ρ := 1 +2 +iγ, the non-trivial zeros of ζ(s), in the +first limit case. Now note that x > 0, and non-trivial zeros come in pairs (ρ, 1 − ρ). +It follows that if for every γ > 0, +(3.1) +lim +y→0 +∞ +� +m=1 +µ(m) +m +∞ +� +n=1 +e−2nπy/mγ � +e +n +m2γ − 2e− +n +m2γ ψ(e−2 +n +mγ ) +� += 0, + +6 +ALEXANDER E. PATKOWSKI +then the Riemann hypothesis would be true since we are assuming all the non- +trivial zeros have ℜ(s) = 1 +2. That is, if (3.1) is true for every γ > 0, then it’s also +true that Ξ(−γ) = 0 and the Riemann Hypothesis follows. The second limit case +doesn’t appear to offer a simple criteria due to the condition that y > 0. +References +[1] G. Csordas, Fourier Transforms of Positive Definite Kernels and the Riemann ξ-function, +Computational Methods and Function Theory, Volume 15, Issue 3, pp 373–391 (2015). +[2] A. Dixit, Series transformations and integrals involving the Riemann Ξ-function, J. Math. +Anal. Appl., 368, (2010), 358–373. +[3] A. Dixit, Character analogues of Ramanujan type integrals involving the Riemann Ξ-function, +Pacific J. Math., 255, No. 2 (2012), 317–348 +[4] R. J. Duffin, Representation of Fourier integrals as sums, III. Proc. Amer. Math. Soc. 8 (1957), +272–277. +[5] L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics. New York: Springer, +Third Edition 2008. +[6] N. S. Koshlyakov, Investigation of some questions of the analytic theory of a rational and +quadratic field. I Izv. Akad. Nauk SSSR Ser. Mat., 18:2 (1954), 113–144. +[7] N. S. Koshlyakov, Investigation of some questions of the analytic theory of a rational and +quadratic field. II Izv. Akad. Nauk SSSR Ser. Mat. 18 No. 3, 213–260 (1954). +[8] A. Kuznetsov, Expansion of the Riemann Ξ function in Meixner-Pollaczek polynomials, Cana- +dian Math. Bulletin 51 (2008), No. 4, 561–569. +[9] A. E. Patkowski, A new integral equation and integrals associated with number theory, Analysis +Mathematica 47, 881–892 (2021). +[10] S. Ramanujan, New expressions for Riemann’s functions ξ(s) and Ξ(t), Quart. J. Math., 46: +253–260, 1915. +[11] E. C. Titchmarsh, The theory of the Riemann zeta function, Oxford University Press, 2nd +edition, 1986. +1390 Bumps River Rd. +Centerville, MA 02632 +USA +E-mail: alexpatk@hotmail.com, alexepatkowski@gmail.com + diff --git a/j9AyT4oBgHgl3EQfkvhK/content/tmp_files/load_file.txt b/j9AyT4oBgHgl3EQfkvhK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..122d6c526729460af8d691cce9b2ea9c5b2f4be0 --- /dev/null +++ b/j9AyT4oBgHgl3EQfkvhK/content/tmp_files/load_file.txt @@ -0,0 +1,180 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf,len=179 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='00438v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='CA] 1 Jan 2023 ON THE HARMONIC CONTINUATION OF THE RIEMANN XI FUNCTION ALEXANDER E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' PATKOWSKI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' We generalize the harmonic continuation of the Riemann xi-function to the n-dimension case, to obtain the solution to the Dirichlet problem on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' We also provide a new expansion for the harmonic continuation of the Riemann xi-function using an expansion given by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Duffin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Keywords: Fourier Integrals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Riemann xi function 2010 Mathematics Subject Classification 11M06, 33C05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Introduction and main result Recall the Riemann xi-function is given as ξ(s) = 1 2s(s − 1)π− s 2 Γ( s 2)ζ(s), where ζ(s) is the Riemann zeta function [11], and Γ(s) is the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' In a recent paper [9], we presented an application of classical integrals involving the Riemann xi-function to the Dirichlet problem in the half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Let L(s) denote the self-dual principal automorphic L-functions and ¯γ(s) its associated gamma factor, then our main theorem presented therein is given in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' ([9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1]) The solution of Dirichlet’s problem in the half plane, ∂2w ∂y2 + ∂2w ∂x2 = 0, where y ∈ R, x ≥ 0, initial condition w(y, 0) = h(y) := ¯γ( 1 2 + iy)L( 1 2 + iy), w(y, x) → 0, as |y| → ∞, is given by the Poisson integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Furthermore, the solution also satisfies the condition w(0, s− 1 2) = h(−i(s− 1 2))−r(s), for an analytic function r(s) that satisfies h(−i(s − 1/2)) = C/(s(s − 1)) + r(s) + r(1 − s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' The proof relied on extending a classical integral found in Titchmarsh [11, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1)] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1) � ∞ 0 Ξ(t) t2 + 1 4 cos(xt)dt = π 2 � ex/2 − 2e−x/2ψ(e−2x) � , 1 2 ALEXANDER E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' PATKOWSKI to self-dual principal automorphic L-functions, and then taking the Laplace trans- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Here, as usual, Ξ(x) = ξ( 1 2 + ix) and ψ(x) = � n=1 e−πn2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' In the special case of the Riemann xi-function, this was offered in [9] as the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Let Γ(s, x) denote the incomplete gamma function over the interval (x, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' ([9, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2]) When ℜ(s) > 1, the Riemann ξ-function satisfies the integral equation, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2) Υ(s) := (s − 1 2) � ∞ 0 Ξ(t) (t2 + 1 4)(t2 + (s − 1 2)2)dt = π 2 \uf8eb \uf8ed 1 s − 1 − ξ(s) s(s − 1) + π−s/2 � n≥1 n−sΓ �s 2, πn2� \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Many papers have utilized Fourier integrals of this type to obtain interesting iden- tities and applications [1, 2, 3, 6, 7, 8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' The purpose of this paper is to generalize [9, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2] to obtain the harmonic continuation that solves the n-dimensional Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' In this way we offer a general result which isn’t captured in [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' We also offer a new expansion of the harmonic continuation of the Riemann xi-function by appealing to the work of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Duffin [4], which will be presented in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' To state our main result we will need to define a kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' From [5, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='93, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='13)], K(x) = Γ( n+1 2 ) π n+1 2 (1 + |x|2) n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Let g(x) = �n l=1 Ξ(xl) x2 l + 1 4 and set Ky(x) = y−nK(y−1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Then func- tion (convolution) u(x, y) = (g ∗ Ky)(x), solves the Dirichlet problem ∂2 yu + n � l=1 ∂2 l u = 0, u(x, 0) = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Furthermore, we have u(0, y) = (g ∗ Ky)(0) = y1−n � Rn e−2π|yz| n � l=1 � ezl/2 − 2e−zl/2ψ(e−2zl) � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' We note that the case n = 1 reduces to a special case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1 and contains [9, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2], being that u(0, y) = (g ∗ Ky)(0) essentially becomes the Laplace transform of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' ON THE HARMONIC CONTINUATION OF THE RIEMANN XI FUNCTION 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Proof of the n-dimensional case and related comments Let R+ denote the positive reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' From [5, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='92], we see that u(x, y) = (g ∗ Ky)(x) solves the Dirichlet problem ∂2 yu + n � l=1 ∂2 l u = 0, u(x, 0) = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Then we can say that u(x, y) is harmonic on Rn+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Recall the n-dimensional Fourier transform [5, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='108] is given by ˆf(y) = � Rn f(x)e−2πix·ydx, where x·y = �n l=1 xlyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Consider g(x) = �n l=1 Ξ(xl) x2 l + 1 4 , and note [5, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='112, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='14] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1) � Rn f(x)ˆg(x)dx = � Rn ˆf(x)g(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Invoking [5, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='118, Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='11] y1−n � Rn e−2π|yz|e−2πiz·xdz = y−nK(y−1x) = Ky(x), with our chosen g(x) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1) gives u(0, y), y1−n � Rn e−2π|yz| n � l=1 � ezl/2 − 2e−zl/2ψ(e−2zl) � dz = � Rn g(z)Ky(z)dz, since ˆg(y) = � Rn g(x)e−2πix·ydx = n � l=1 � eyl/2 − 2e−yl/2ψ(e−2yl) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' The right hand side is now precisely u(0, y) = (g ∗ Ky)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' It is possible to take another direction to obtain a formula which contains [9, Corol- lary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2] as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' If we set yl = yj = y for l ̸= j, and l, j ≤ n, then we can write ˆg(y) = � ey/2 − 2e−y/2ψ(e−2y) �n (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2) n � k=0 �n k � ey(n−k)/2 � −2e−y/2ψ(e−2y) �k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='3) 4 ALEXANDER E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' PATKOWSKI We write the sum of squares function to be generated by ∞ � n=0 rk(n)qn = � ∞ � n=−∞ qn2 �k = k � l=0 �k l � � 2 ∞ � n=1 qn2 �l = k � l=0 �k l � 2k ∞ � m=1 r′ l(m)qm, for |q| < 1 say, and note that r0(0) = 1, and r0(n) = 0 if n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Equating coefficients of qn we find that for n > 1, rk(n) = k � l=0 �k l � 2kr′ l(n), which gives us 1 2r1(n) = r′ 1(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Note that taking the Laplace transform of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='3) similar to the method we employed in [9] gives a different kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Namely, � ∞ 0 ˆg(y)e−sydy � Rn sg(x) s2 + (�n l=1 xl) 2 dx = 1 s − 1 2 + n � k=1 �n k � (−2)k ∞ � m=1 r′ k(m) � 1 0 ts−1+(n−k)/2+k/2e−mt2dt = 1 s − 1 2 + 1 2 n � k=1 �n k � (−2)k ∞ � m=1 r′ k(m)Γ �s + (2k − n)/2 2 � 1 ms/2+(2k−n)/4 − 1 2 n � k=1 �n k � (−2)k ∞ � m=1 r′ k(m)Γ �s + (2k − n)/2 2 , πm � 1 ms/2+(2k−n)/4 , which again gives [9, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2] when n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' The Duffin expansion of the harmonic continuation of the Riemann xi function In Duffin’s paper [4, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='275], we find an interesting theorem for the harmonic con- tinuation of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' As usual, let µ(n) denote the M¨obius function [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' For a definition of P summable see [4, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='273].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' ON THE HARMONIC CONTINUATION OF THE RIEMANN XI FUNCTION 5 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' ([4, Theorem 2]) Let f(x) ∈ L1(0, ∞) and let C(x) be its transform relative to the kernel cos(2πx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Then the harmonic continuation is given by f(x, y) := 1 π � R y y2 + (x − t)2 f(t)dt = ∞ � m=1 µ(m) m ∞ � n=1 e−2nπy/mxC � n mx � where x, y > 0, and the sum over m is evaluated by P summability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' In particular, f(x, y) → f(x) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' when y → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' By the integral (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1) it readily follows that we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' The harmonic continuation of f(x) = Ξ(x) x2 + 1 4 , is given by f(x, y) = π 2 ∞ � m=1 µ(m) m ∞ � n=1 e−2nπy/mx � e n mx2 − 2e− n m2x ψ(e−2 n mx ) � , where x, y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Furthermore, Ξ(x) x2 + 1 4 = π 2 lim y→0 ∞ � m=1 µ(m) m ∞ � n=1 e−2nπy/mx � e n mx2 − 2e− n m2x ψ(e−2 n mx ) � , and π 2 \uf8eb \uf8ed 1 y − 1 2 − ξ(y + 1 2)(y − 1 2) y + 1 2 + π−(y− 1 2 )/2 � n≥1 n−y+ 1 2 Γ �y + 1 2 2 , πn2 �\uf8f6 \uf8f8 = π 2 lim x→0 ∞ � m=1 µ(m) m ∞ � n=1 e−2nπy/mx � e n mx2 − 2e− n m2x ψ(e−2 n mx ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Clearly our choice of f(x) is L1(0, ∞) since Ξ(x) = O(xNe−πx/4) by [11, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='257].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' The second limit case in the theorem, which is all that remains to prove, follows directly from [9, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' □ A criteria for the Riemann Hypothesis may be formulated from this theorem by putting x = γ where ℑ(ρ) =: γ with ρ := 1 2 +iγ, the non-trivial zeros of ζ(s), in the first limit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' Now note that x > 0, and non-trivial zeros come in pairs (ρ, 1 − ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' It follows that if for every γ > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1) lim y→0 ∞ � m=1 µ(m) m ∞ � n=1 e−2nπy/mγ � e n m2γ − 2e− n m2γ ψ(e−2 n mγ ) � = 0, 6 ALEXANDER E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' PATKOWSKI then the Riemann hypothesis would be true since we are assuming all the non- trivial zeros have ℜ(s) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' That is, if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content='1) is true for every γ > 0, then it’s also true that Ξ(−γ) = 0 and the Riemann Hypothesis follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' The second limit case doesn’t appear to offer a simple criteria due to the condition that y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfkvhK/content/2301.00438v1.pdf'} +page_content=' References [1] G.' 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a/jNFQT4oBgHgl3EQflTY-/content/tmp_files/2301.13361v1.pdf.txt b/jNFQT4oBgHgl3EQflTY-/content/tmp_files/2301.13361v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0467a8f551bcd06675063576de5c5333e36a5ac --- /dev/null +++ b/jNFQT4oBgHgl3EQflTY-/content/tmp_files/2301.13361v1.pdf.txt @@ -0,0 +1,1854 @@ +Iterative Loop Learning Combining Self-Training and Active Learning for +Domain Adaptive Semantic Segmentation +Licong Guan1 +Xue Yuan1 � +1School of Electronic and Information Engineering, Beijing Jiaotong University +{lcguan941,xyuan}@bjtu.edu.cn +Abstract +Recently, self-training and active learning have been +proposed to alleviate this problem. Self-training can im- +prove model accuracy with massive unlabeled data, but +some pseudo labels containing noise would be generated +with limited or imbalanced training data. And there will +be suboptimal models if human guidance is absent. Active +learning can select more effective data to intervene, while +the model accuracy can not be improved because the mas- +sive unlabeled data are not used. And the probability of +querying sub-optimal samples will increase when the do- +main difference is too large, increasing annotation cost. +This paper proposes an iterative loop learning method com- +bining Self-Training and Active Learning (STAL) for do- +main adaptive semantic segmentation. The method first uses +self-training to learn massive unlabeled data to improve +model accuracy and provide more accurate selection mod- +els for active learning. Secondly, combined with the sample +selection strategy of active learning, manual intervention is +used to correct the self-training learning. Iterative loop to +achieve the best performance with minimal label cost. Ex- +tensive experiments show that our method establishes state- +of-the-art performance on tasks of GTAV → Cityscapes, +SYNTHIA → Cityscapes, improving by 4.9% mIoU and +5.2% mIoU, compared to the previous best method, respec- +tively. Code will be available. +1. Introduction +Semantic segmentation can understand image scenes at +the pixel level and is crucial for various real-world appli- +cations. Thanks to the rapid development of deep learn- +ing, many advanced segmentation methods have been pro- +posed and achieved great breakthroughs in various tasks +such as autonomous driving [24], scene parsing [13, 50], +medical analysis [2], and human-computer interaction [47]. +�Corresponding author +75 +0 1.0 +2.2 +3.4 +5.0 +Percentage of labeled images (%) +100% +35 +40 +45 +50 +55 +60 +70 +65 +80 +mIOU (%) +GTAV→Cityscapes +Source Only +Ours +RIPU [CVPR’22] +MADA [ICCV’21] +AADA [WACV’20] +ALDA (V3+) +DAP [CVPR’22] +UDA (V2) +SAC [CVPR’21] +DLDM [CVPR’21] +SSDA (V2) +ASS [CVPR’20] +MME [ICCV’19] +Better +Figure 1. Performance comparison between our method and other +methods on GTAV → Cityscapes. +V2 and V3+ are based on +DeepLab-v2 and DeepLab-V3+, respectively. +However, the traditional fully-supervised scenes [43,76,78] +are very eager for images carefully annotated by human +annotators, especially in aforementioned areas, a large +number of annotated images are very expensive, time- +consuming [13, 43], or even infeasible, which greatly hin- +ders their widespread application. Therefore, it remains a +major challenge to guarantee good generalization for differ- +ent domain scenarios with minimal annotation cost. +Nowadays, many research works have proved that do- +main adaptation is one of the powerful means to address +the above issues [23,37,44,60,63]. Among them, unsuper- +vised domain adaptation (UDA) [26,28,33,41,69,77] aims +to solve this problem by leveraging the knowledge of label- +rich data (source data) and transferring it to unlabeled data +(target data) [52]. While it can avoid the intensive workload +of manual annotation, the performance still lags far behind +fully supervised models [56]. Furthermore, active learn- +ing can significantly improve performance on both classi- +1 +arXiv:2301.13361v1 [cs.CV] 31 Jan 2023 + +fication and detection tasks by introducing a few additional +manual annotations to a few selected samples from the tar- +get domain [57]. Nevertheless, active learning does not uti- +lize the massive unlabeled data in the target domain, and +only improves the accuracy through manual intervention, +which makes the labeling cost difficult to control. And the +probability of querying sub-optimal samples will increase +when the domain difference is too large, increasing annota- +tion cost. Moreover, in the process of semantic segmenta- +tion data labeling, the outline labeling of objects is usually +completed by clicking with a mouse. Some previous studies +on active learning domain adaptation are based on pixel se- +lectors [56], which are difficult to generalize to real-world +applications using a single image as the minimum selection +unit and labeling based on object contours. Therefore, how +to provide an accurate selection model for active learning +and design a practical selection strategy is one the urgent +problems to be solved. +Recently, self-training has greatly facilitated domain +adaptation, as it can retrain the network with pseudo-labels +generated from massive unlabeled data [11, 45, 65, 77, 79– +81]. +However, due to the limited and imbalanced train- +ing data, the pseudo-labels generated by self-training usu- +ally contain noise. +This disadvantageous experience not +only can not improve the accuracy of pseudo-labels but also +further affect the performance of machine learning models +without timely manual intervention and guidance. There- +fore, how discovering and correcting false pseudo-labels in +the self-training learning process is one of the problems to +be solved urgently. +To solve the above problems, this paper proposes an it- +erative loop learning method combining self-training and +active learning for domain-adaptive semantic segmentation. +It learns massive unlabeled data through self-training to im- +prove the accuracy in the target domain and provides an ac- +curate selection model for active learning. Then, the active +learning sample selection strategy is used to further correct +the false pseudo labels in the self-training process through +manual intervention. Self-training and active learning com- +plement each other and help the model achieve the best per- +formance with minimal annotation cost. (Fig. 1). +In a nutshell, our contributions can be summarized as: +• We propose a method combining self-training with ac- +tive learning for domain-adaptive semantic segmenta- +tion, termed STAL. By complementing the advantages +of self-training and active learning, we achieve the best +performance for domain-adaptive semantic segmenta- +tion with minimal label cost. +• We propose an iterative loop learning strategy to opti- +mize the performance of semantic segmentation mod- +els through three stages: warm-up learning, active se- +lection, and incremental learning. +2. Related Work +Domain adaptation (DA) can transfer knowledge from a +label-rich source domain to a label-scarce target domain, +and recent work has achieved great success on a range of +tasks. Such as classification [38,40,72], detection [10,62], +and segmentation [41,42]. Most previous works have used +adversarial learning [30, 39, 63, 68] in an attempt to re- +duce the domain gap between source and target features +from the image level or feature level [34]. Recent work on +domain-adaptive semantic segmentation can be mainly di- +vided into two categories: adversarial training-based meth- +ods [63,64] and self-training-based methods [31,77,79,81]. +For the first branch, most works tend to learn domain- +invariant representations based on min-max adversarial op- +timization games by tricking the domain discriminator to +obtain aligned feature distributions [63, 64]. The second +branch focuses on how to generate high-quality pseudo- +labels for target domain data for further model optimiza- +tion [77,79], which drives the development of self-training +techniques. +Semi-supervised learning (SSL) involves two typical +paradigms: consistency regularization [3, 48, 73] and en- +tropy minimization [5, 6, 27, 49]. Consistency regulariza- +tion forces the model to produce stable and consistent pre- +dictions on the same unlabeled data under various perturba- +tions [71]. On the other hand, entropy minimization, gen- +eralized by self-training pipelines [1,12,15], exploits unla- +beled target domain data in a way that uses pseudo-labels +for training. For example, Wang et al. [67] propose the +Semantic-Level Shift (ASS) framework, which introduces +an additional semantic-level adaptation module by adver- +sarial training on the corresponding outputs of the source +and target labeled inputs. However, adversarial loss makes +training unstable due to weak supervision. Zou et al. [80] +proposed an iterative learning strategy with class-balanced +and spatial priors for target instances. Tranheden et al. [59] +proposed a domain-mixed self-training pipeline to improve +training stability. Wang et al. [66] exploit unreliable pixels +by adding a contrastive learning loss on top of self-training. +The above studies can improve the model accuracy by using +massive unlabeled data, while the noise problem of pseudo- +labels can not be effectively solved. Therefore, we incorpo- +rate a sample selection strategy of active learning and em- +ploy human intervention to correct self-training learning in +this paper. +Active learning (AL) aims to maximize the model perfor- +mance with the least labeling cost of the dataset. Query +rules are the core content of active learning, and commonly +used query strategies are divided into uncertainty-based +methods [4, 16] and diversity-based methods [22]. In the +field of active learning domain adaptation research, previ- +ous work has mainly focused on classification tasks [21,51]. +2 + +Ning et al. [46] and Shin et al. [56] were the first to adopt +active learning domains for semantic segmentation tasks. +Among them, [46] proposed a multi-anchor strategy to ac- +tively select image subsets, which can be inefficient. [56] +proposed a more efficient point-based annotation. However, +the selected points ignore the pixel spatial continuity of the +image. Recently, Xie et al. [70] greatly improved the seg- +mentation performance in the target domain by exploring +the consistency of the image space and selecting the most +diverse and uncertain image regions. However, most ac- +tive learning research works ignore the utilization of mas- +sive unlabeled data in the target domain, resulting in high +labeling costs. This paper uses self-training to learn mas- +sive unlabeled data to improve the accuracy of the model, +provide an accurate selection model for active learning, and +reduce the cost of labeling. +3. Approach +In this section, we establish our problem mathematically +and first outline our proposed method in § 3.1. Our strate- +gies for self-training and active learning are presented in +§ 3.2 and § 3.3, respectively. +3.1. Overview +In domain-adapted semantic segmentation, we have a set +of labeled source domain data Ds = {(xs, ys)} and incom- +pletely labeled target data Dt = {(xt, yt)}, where ys is the +pixel label belonging to one of the C known classes in the +label space Y. Our goal is to learn a function f ◦ h : x → y +(a semantic segmentation network parameterized by θ) that +achieves good segmentation performance on the target do- +main with a small amount of labeled target data and a large +amount of labeled source data and unlabeled target data. +In general, the success of CNN-based methods bene- +fits from a large amount of manually labeled data and the +assumption of independent and identical data distributions +between training and testing samples. However, when a +model trained on the training set (source domain) is di- +rectly applied to an unseen test scene (target domain), the +performance drops significantly. To transfer knowledge ef- +ficiently, recent advances employ self-training techniques +and optimize the cross-entropy loss using target pseudo- +labels �yt. Due to limited training data and class imbalance, +the pseudo-labels generated by self-training often contain +noise, and when lacking human intervention, inferior ex- +perience during training can lead to sub-optimal models. +Further, active learning can pick out more effective data to +intervene. However, active learning does not use massive +amounts of unlabeled data, and it relies too much on picking +models. And the probability of querying sub-optimal sam- +ples will increase when the domain difference is too large, +resulting in an increase in annotation cost. +To solve the above problems, we propose an iterative +loop learning method that combines self-training and ac- +tive learning. The proposed framework consists of three +stages: the first stage is self-training learning (Fig. 2a). Use +a very small amount of labeled data and a large amount +of unlabeled data to perform self-training learning to ob- +tain a warm-up model. The second stage is active selection +(Fig. 2b). Use the current model to predict the unlabeled tar- +get domain data and send it to the acquisition function. The +third stage is image labeling (Fig. 2c). The selected samples +are manually labeled and added to the labeled dataset of the +target domain, and the first stage of self-training is repeated +to obtain the final model. +3.2. Self-Training Learning +STAL follows a typical self-training framework, which +consists of a student model and a teacher model. +The +teacher model and the student model have the same schema. +The two models differ only in updating their weights, the +student’s weight θs is updated by convention, while the +teacher model’s weight θt is updated by the exponential +moving average (EMA) of the student model. For labeled +images of source and target domains, we use standard cross- +entropy loss on them. And for each unlabeled target domain +image, we bring it into the teacher model for prediction and +obtain the pseudo-label according to the pixel prediction en- +tropy. Subsequently, the student model is trained on unla- +beled target domain data and corresponding pseudo-labels. +Our optimization objective is to minimize the overall loss, +which can be expressed as: +L = Ls + λuLu + λcLc , +(1) +where Ls (Eq. 2) and Lu (Eq. 3) represent the supervised +and unsupervised loss applied to labeled and unlabeled im- +ages, respectively. Lc (Eq. 4) is the contrastive learning +loss [66]. λu is the weight of the unsupervised loss, λc is +the weight of the contrastive loss, and Ls and Lu are both +cross-entropy (CE) loss. +Ls = 1 +Nl +Nl +� +i=1 +1 +WH +W H +� +j=1 +ℓce(f ◦ h(xl +i,j; θ), yl +i,j) , +(2) +Lu = +1 +Nu +Nu +� +i=1 +1 +WH +W H +� +j=1 +ℓce(f ◦ h(xu +i,j; θ), ˆyu +i,j) , +(3) +Lc = − +1 +C × M +C−1 +� +c=0 +M +� +i=1 +log + + +e⟨aci,a+ +ci⟩/ω +e⟨aci,a+ +ci⟩/ω + �N +j=1 e⟨aci,a− +cij⟩/ω + + , +(4) +Among them, f ◦ h is the composition function of h and f, +which means that the image xi,j is first sent to h to extract +features, and then sent to f to obtain segmentation results. +yl +i,j is the manually annotated mask label for the j-th pixel +3 + +Labeled target image +Student +Teacher +EMA Update +S +L +Labeled source image +Copy +Prob map +Unlabeled target image +Segmentation network +Prob map +Prob map +U +C +L +L + +Pseudo Label ˆ +tY +Unlabeled target image +Acquisition +function +Human participants +(a) Self-Training +(b) Active Selection +(c) Image Labeling +For re-training +Prob map +Corresponding original +image +Select +Figure 2. The overview of the proposed STAL. The proposed framework consists of three stages: the first stage (a) is self-training +learning. Use a very small amount of labeled data and a large amount of unlabeled data to perform self-training learning to obtain a +warm-up model. The second stage (b) is active selection. Use the current model to predict the unlabeled target domain data and send it to +the acquisition function. The third stage (c) is image labeling. The selected samples are manually labeled and added to the labeled dataset +of the target domain, and the first stage of self-training is repeated to obtain the final model. +in the i-th labeled image, ˆyu +i,j is the pseudo-label for the j- +th pixel in the i-th unlabeled image, Nl and Nu represents +the number of labeled and unlabeled images in the train- +ing batch. W and H are the width and height of the input +image. ℓce is the standard cross-entropy loss. In the con- +trast loss Eq. 4, M is the total number of anchor pixels, and +aci represents the representation of the i-th anchor of class +c. The positive and negative samples corresponding to each +anchor pixel are denoted as a+ +ci and a− +ci, ⟨·, ·⟩ is the cosine +similarity between features from two different pixels, whose +range is limited between -1 to 1 according to ω. Following +[66], we set M = 50, N = 256, and ω = 0.5. +During training, some tail categories will introduce more +pseudo-label noise due to insufficient training, which in- +directly leads to the degradation of underperforming cate- +gories. We dynamically record the performance of each cat- +egory during training by maintaining a confidence library, +and for the underperforming categories, the confidence met- +ric is shown in Eq. 5. +Confc = 1 +Nl +Nl +� +i=1 +1 +N c +i +Nc +i +� +j=1 +(f ◦ h(xc +i,j; θ)), c ∈ {1, . . . , C} , (5) +Among them, C is the category number, N c +i indicates the +number of pixels belonging to category c according to the +ground truth label yi,j, and f ◦ h(xc +i,j; θ) indicates the c-th +channel prediction result of the j-th pixel in the i-th image. +We use EMA to update the confidence by class at each +training step, and the update criterion is shown in Eq. 6: +Confc +n ← αConfc +n−1 + (1 − α)Confc +n, c ∈ {1, . . . , C} , +(6) +where n represents the n-th iteration and α ∈ [0, 1) is the +momentum coefficient, which we set to 0.999 in our experi- +ments. For underperforming categories, we improve by two +data augmentation techniques, Copy Paste [25] and Cut- +mix [75]. We calculate the sampling probability according +4 + +to Eq. 7, which converts the class confidence in the confi- +dence base into the normalized sampling probability s. +s = Softmax(1 − Conf). +(7) +3.3. Active Learning +Inferior experience during self-training can have an im- +pact on the performance of machine learning models in the +absence of human intervention. Therefore, we utilize active +learning to correct self-training erroneous pseudo-labels. +Our sample acquisition strategy is as follows: Given an +unlabeled target image xt and a warm-up model θw, the +acquisition function A is a function that the active learning +system uses to query. First, the softmax output Pt of the un- +labeled image xt is obtained by warm-up model θw. Since +the prediction Pt carries semantic relation information, we +adopt the prediction entropy H of each pixel to measure the +uncertainty. For a single image with a C classification, we +evaluate the uncertainty �xu by averaging all entropies of all +pixels in the image. Calculated as follows: +�xu = − +1 +WH +W H +� +j=1 +C +� +c=1 +Pi,j,c +t +log Pi,j,c +t +(8) +Among them, W and H are the width and height of the fea- +ture map, respectively, and Pi,j,c +t +represents the c-th channel +prediction result of the j-th pixel in the i-th image. After +obtaining uncertainty results for all unlabeled images, we +preferentially select the most uncertain images S for anno- +tation according to Eq. 9. +S = argmax A(�xu) +(9) +4. Experiments +Dataset. +To verify the effectiveness of the proposed +method, we evaluate our method on two popular scenarios, +transferring information from synthetic images GTAV [53] +and SYNTHIA [54] to the real domain, the Cityscapes [13] +dataset. +GTAV is a synthetic image dataset contain- +ing 24,966 1914×1052 images, sharing 19 classes as +Cityscapes. SYNTHIA is a synthetic urban scene dataset +containing 9,400 1280×760 images, sharing 16 classes as +Cityscapes. Cityscapes is an autonomous driving dataset of +real urban scenes, containing 2,975 training images and 500 +validation images, each with a resolution of 2048×1024. +Implementation details. All experiments are performed +on NVIDIA A100 GPU with Pytorch. +We adopt +DeepLabv2 [7] and DeepLab-v3+ [8] architectures with +ResNet-101 [29] pre-trained on ImageNet [14] as backbone. +Regarding the training, we use the SGD optimizer with an +initial learning rate of 0.0025, weight decay of 0.0001, and +momentum of 0.9. For all experiments, we train about 200K +iterations with batch size of 12, and data are resized into +769×769. +Evaluation metric. As a common practice [45,46,56,70], +we report the mean Intersection-over-Union (mIoU) [17] on +the Cityscapes validation set. Specifically, we report the +mIoU on the shared 19 classes for GTAV → Cityscapes and +report the results on 13 (mIoU*) and 16 (mIoU) common +classes for SYNTHIA → Cityscapes. We also add the 19- +class evaluation for the SYNTHIA → Cityscapes task in +§ 4.4, and we believe that the extremely small amount of +target domain data is sufficient to optimize the three classes +missing from SYNTHIA. +Annotation budget. Previous active learning-based selec- +tion strategies use pixels or regions as selection units, and +they select a fixed proportion (2.2% or 5%) of each sample +in the dataset. While our sample selection strategy takes a +single image as the smallest unit, for a fair comparison, we +only select the corresponding percentage of images. The +selection process is divided into two rounds, the first round +we randomly select 1% (30 images) from the target domain +dataset for self-training learning. In the second round, we +selected 1.2% (35 images) or 4% (120 images). Therefore, +we label 2.2% or 5% of the target domain data in total for +self-training learning to obtain the final evaluation model. +4.1. Comparisons with the state-of-the-arts +Table 1 and Table 2 are the domain adaptation results +of GTAV → Cityscapes and SYNTHIA → Cityscapes, re- +spectively, and it can be seen that our method greatly out- +performs the previous leading unsupervised domain adap- +tation and active learning domain adaptation methods. At +the limit, our results using only 1% of the data are also sub- +stantial improvements over previous state-of-the-art unsu- +pervised methods (DAP+ProDA [33]). +For the GTAV → Cityscapes task, based on using +the same backbone (DeepLab-v3+), we can easily beat +AADA [57] and MADA [46] with an annotation budget of +1%. Compared to the state-of-the-art model, using the same +annotation budget (5%), our method achieves 4.9% mIoU +improvement over RIPU [70]. Notably, our method signifi- +cantly outperforms the contrastive methods in some specific +categories, namely the tail category of Cityscapes (such as +“traffic light”, “traffic sign”, “rider”, “bus”, “train”, “mo- +torcycle”, and “bicycle”), which indicates that the proposed +method can effectively alleviate the long-tailed distribution +problem to outperform the adversary. +Our STAL is still competitive for the SYNTHIA → +Cityscapes task. +On the basis of using the same back- +bone (DeepLab-v3+), our method can beat all methods us- +ing 1% of the target data. Compared to the state-of-the-art +model, our method achieves 5.2% mIoU improvement over +RIPU [70] if the same annotation budget (5%) is used. Like- +wise, our method outperforms RIPU [70] on the tail cate- +gories of Cityscapes (such as “traffic light”, “traffic sign”, +“rider”, “bus”, “motorcycle”, and “bicycle”). +5 + +Table 1. Comparison with previous results on task GTAV → Cityscapes. We report the mIoU and best results are shown in bold. +Method +Net. +road +side. +buil. +wall +fence +pole +light +sign +veg. +terr. +sky +pers. +rider +car +truck +bus +train +motor +bike +mIoU +Source Only +V2 +75.8 +16.8 +77.2 +12.5 +21.0 +25.5 +30.1 +20.1 +81.3 +24.6 +70.3 +53.8 +26.4 +49.9 +17.2 +25.9 +6.5 +25.3 +36.0 +36.6 +CBST [80] +91.8 +53.5 +80.5 +32.7 +21.0 +34.0 +28.9 +20.4 +83.9 +34.2 +80.9 +53.1 +24.0 +82.7 +30.3 +35.9 +16.0 +25.9 +42.8 +45.9 +MRKLD [81] +91.0 +55.4 +80.0 +33.7 +21.4 +37.3 +32.9 +24.5 +85.0 +34.1 +80.8 +57.7 +24.6 +84.1 +27.8 +30.1 +26.9 +26.0 +42.3 +47.1 +SIM [68] +90.6 +44.7 +84.8 +34.3 +28.7 +31.6 +35.0 +37.6 +84.7 +43.3 +85.3 +57.0 +31.5 +83.8 +42.6 +48.5 +1.9 +30.4 +39.0 +49.2 +SAC [1] +90.4 +53.9 +86.6 +42.4 +27.3 +45.1 +48.5 +42.7 +87.4 +40.1 +86.1 +67.5 +29.7 +88.5 +49.1 +54.6 +9.8 +26.6 +45.3 +53.8 +ProDA [77] +87.8 +56.0 +79.7 +46.3 +44.8 +45.6 +53.5 +53.5 +88.6 +45.2 +82.1 +70.7 +39.2 +88.8 +45.5 +59.4 +1.0 +48.9 +56.4 +57.5 +DAP+ProDA [33] +94.5 +63.1 +89.1 +29.8 +47.5 +50.4 +56.7 +58.7 +89.5 +50.2 +87.0 +73.6 +38.6 +91.3 +50.2 +52.9 +0.0 +50.2 +63.5 +59.8 +LabOR (2.2%) [56] +V2 +96.6 +77.0 +89.6 +47.8 +50.7 +48.0 +56.6 +63.5 +89.5 +57.8 +91.6 +72.0 +47.3 +91.7 +62.1 +61.9 +48.9 +47.9 +65.3 +66.6 +RIPU (2.2%) [70] +96.5 +74.1 +89.7 +53.1 +51.0 +43.8 +53.4 +62.2 +90.0 +57.6 +92.6 +73.0 +53.0 +92.8 +73.8 +78.5 +62.0 +55.6 +70.0 +69.6 +Ours (2.2%) +96.4 +74.6 +91.1 +45.9 +52.4 +59.4 +67.9 +68.3 +91.4 +50.0 +92.8 +76.2 +57.2 +93.6 +78.2 +81.3 +69.5 +58.4 +72.1 +72.5 +AADA (5%) [57] +V3+ +92.2 +59.9 +87.3 +36.4 +45.7 +46.1 +50.6 +59.5 +88.3 +44.0 +90.2 +69.7 +38.2 +90.0 +55.3 +45.1 +32.0 +32.6 +62.9 +59.3 +MADA (5%) [46] +95.1 +69.8 +88.5 +43.3 +48.7 +45.7 +53.3 +59.2 +89.1 +46.7 +91.5 +73.9 +50.1 +91.2 +60.6 +56.9 +48.4 +51.6 +68.7 +64.9 +RIPU (5%) [70] +97.0 +77.3 +90.4 +54.6 +53.2 +47.7 +55.9 +64.1 +90.2 +59.2 +93.2 +75.0 +54.8 +92.7 +73.0 +79.7 +68.9 +55.5 +70.3 +71.2 +Ours (1%) +95.2 +67.0 +90.9 +47.4 +49.6 +60.9 +68.2 +67.5 +90.9 +44.6 +91.5 +81.3 +60.5 +93.9 +67.2 +76.6 +47.9 +54.7 +74.8 +70.0 +Ours (2.2%) +96.5 +75.6 +91.2 +46.7 +53.6 +62.1 +70.3 +76.0 +91.4 +52.1 +94.1 +82.0 +60.8 +94.4 +83.1 +86.4 +71.9 +61.2 +75.8 +75.0 +Ours (5%) +96.9 +77.8 +91.6 +46.7 +56.0 +63.2 +70.8 +77.4 +91.9 +54.9 +94.5 +82.3 +61.2 +94.9 +79.3 +88.1 +75.3 +65.8 +77.6 +76.1 +Methods with V2 are based on DeepLab-v2 [7] and methods with V3+ are based on DeepLab-v3+ [8] for a fair comparison. +Table 2. Comparisons with previous results on task SYNTHIA → Cityscapes. We report the mIoUs in terms of 13 classes (excluding +the “wall”, “fence”, and “pole”) and 16 classes. Best results are shown in bold. +Method +Net. +road +side. +buil. +wall* +fence* +pole* +light +sign +veg. +sky +pers. +rider +car +bus +motor +bike +mIoU +mIoU* +Source Only +V2 +55.6 +23.8 +74.6 +9.2 +0.2 +24.4 +6.1 +12.1 +74.8 +79.0 +55.3 +19.1 +39.6 +23.3 +13.7 +25.0 +33.5 +38.6 +CBST [80] +68.0 +29.9 +76.3 +10.8 +1.4 +33.9 +22.8 +29.5 +77.6 +78.3 +60.6 +28.3 +81.6 +23.5 +18.8 +39.8 +42.6 +48.9 +MRKLD [81] +67.7 +32.2 +73.9 +10.7 +1.6 +37.4 +22.2 +31.2 +80.8 +80.5 +60.8 +29.1 +82.8 +25.0 +19.4 +45.3 +43.8 +50.1 +SIM [68] +83.0 +44.0 +80.3 +- +- +- +17.1 +15.8 +80.5 +81.8 +59.9 +33.1 +70.2 +37.3 +28.5 +45.8 +- +52.1 +SAC [1] +89.3 +47.2 +85.5 +26.5 +1.3 +43.0 +45.5 +32.0 +87.1 +89.3 +63.6 +25.4 +86.9 +35.6 +30.4 +53.0 +52.6 +59.3 +ProDA [77] +87.8 +45.7 +84.6 +37.1 +0.6 +44.0 +54.6 +37.0 +88.1 +84.4 +74.2 +24.3 +88.2 +51.1 +40.5 +45.6 +55.5 +62.0 +DAP+ProDA [33] +84.2 +46.5 +82.5 +35.1 +0.2 +46.7 +53.6 +45.7 +89.3 +87.5 +75.7 +34.6 +91.7 +73.5 +49.4 +60.5 +59.8 +64.3 +RIPU (2.2%) [70] +V2 +96.8 +76.6 +89.6 +45.0 +47.7 +45.0 +53.0 +62.5 +90.6 +92.7 +73.0 +52.9 +93.1 +80.5 +52.4 +70.1 +70.1 +75.7 +Ours (2.2%) +96.8 +76.1 +89.7 +47.3 +52.8 +56.3 +62.9 +70.1 +91.1 +93.2 +78.4 +59.7 +93.5 +78.2 +58.2 +74.2 +73.7 +78.6 +AADA (5%) [57] +V3+ +91.3 +57.6 +86.9 +37.6 +48.3 +45.0 +50.4 +58.5 +88.2 +90.3 +69.4 +37.9 +89.9 +44.5 +32.8 +62.5 +61.9 +66.2 +MADA (5%) [46] +96.5 +74.6 +88.8 +45.9 +43.8 +46.7 +52.4 +60.5 +89.7 +92.2 +74.1 +51.2 +90.9 +60.3 +52.4 +69.4 +68.1 +73.3 +RIPU (5%) [70] +97.0 +78.9 +89.9 +47.2 +50.7 +48.5 +55.2 +63.9 +91.1 +93.0 +74.4 +54.1 +92.9 +79.9 +55.3 +71.0 +71.4 +76.7 +Ours (1%) +96.8 +74.8 +90.0 +34.0 +46.3 +60.9 +68.0 +74.8 +90.2 +92.5 +81.1 +58.2 +93.0 +72.3 +63.4 +75.6 +73.2 +79.3 +Ours (2.2%) +96.8 +76.3 +90.9 +48.1 +54.2 +62.4 +69.0 +77.3 +91.0 +93.7 +82.2 +60.3 +94.2 +80.0 +63.8 +76.0 +76.0 +80.9 +Ours (5%) +97.4 +80.1 +91.8 +38.6 +55.2 +64.1 +70.9 +78.7 +91.6 +94.5 +82.7 +60.1 +94.4 +81.7 +66.8 +77.2 +76.6 +82.1 +Methods with V2 are based on DeepLab-v2 [7] and methods with V3+ are based on DeepLab-v3+ [8] for a fair comparison. +(a) Target Image +(b) Ground Truth +(c) RIPU (5%) +(d) Ours (5%) +Figure 3. Visualization of segmentation results for the task GTAV → Cityscapes. From left to right: original target image, ground-truth +label, result predicted by RIPU [70], and result predicted by Ours are shown one by one. +4.2. Qualitative results +We visualize the segmentation results predicted by STAL +in GTAV → Cityscapes and compare them with the state-of- +the-art RIPU [70] model prediction results. As can be seen +from Fig. 3, our method has smoother prediction results for +the head category (such as “pole”), and for Cityscapes tail +categories (such as “bus”, “rider”, and “traffic sign”) have a +6 + +7(a) Target Image +(b) Ground Truth +(c) RIPU (5%) +(d) Ours (5%) +Figure 4. Visualization of segmentation results for the task SYNTHIA → Cityscapes. From left to right: original target image, +ground-truth label, result predicted by RIPU [70], and result predicted by Ours are shown one by one. +great improvement. The visualization of the segmentation +results predicted by SYNTHIA → Cityscapes is shown in +Fig. 4. Our method achieves accurate predictions for the +missing class “truck” with only a small amount of target +data. For the head class “person”, STAL can achieve more +detailed contour prediction than adversaries. For the tail +categories of Cityscapes (such as “traffic sign” and “traffic +light”), our predictions are greatly improved. Readers are +referred to Appendix D for more details. +4.3. Ablation Study +To further investigate the efficacy of each component +of our STAL, we perform ablation studies on GTAV → +Cityscapes. We randomly select 5% of the data in the target +domain to train in DeepLab-v3+ as the baseline. As shown +in Table 3, satisfactory and consistent gains from the base- +line to our full method demonstrate the effectiveness of each +component. Compared to the baseline, Lu is able to lever- +age the unlabeled data of the target domain to improve per- +formance by 10.38%. The performance is further improved +by 0.9% and 1.63% after adding Lc and Data aug in self- +training, respectively. We applied both Lc and Data aug +strategies simultaneously and achieved a 4.32% improve- +ment in model performance. Finally, by replacing the sam- +ples in the baseline with the samples picked by active learn- +ing, our performance reaches 76.11%, proving the effective- +ness of the sample selection strategy based on prediction +uncertainty. +4.4. Further Analysis +Extension to open set domain adaptive. +SYNTHIA +shares only 16 classes with Cityscapes, so previous meth- +ods only evaluate mIoU for 16 classes and 13 classes on +this task. In active learning domain adaptation, we will re- +port mIoU for 19 classes on the SYNTHIA → Cityscapes +task due to the addition of data from the target domain. The +evaluation results are shown in Table 4. Judging from the +evaluation results of “terrain”, “truck”, and “train” missing +Table 3. Ablation study on the effectiveness of various compo- +nents in our STAL, including contrastive loss Lu, Lc, Data aug, +and Uncertainty. +Self-Training +Active Learning +GTAV +Lu +Lc +Data aug +Uncertainty +mIoU +59.33 +✓ +69.71 +✓ +✓ +70.61 +✓ +✓ +71.34 +✓ +✓ +✓ +74.03 +✓ +✓ +✓ +✓ +76.11 +Table 4. Experiments with Open Set Domain Adaptation on task +SYNTHIA → Cityscapes. +Method +Net. +terrain +truck +train +mIoU +Ours (1%) +V3+ +41.5 +51.6 +48.6 +69.1 +Ours (2.2%) +53.5 +74.7 +59.9 +73.9 +three categories, we can still produce effects comparable to +a large amount of labeled data under the premise of using a +very small amount of target data. +Comparison with SSDA and SSL methods. +To bet- +ter understand the performance improvement of strategies +combining self-training with active learning, we compare +our method with semi-supervised domain adaptation and +semi-supervised learning methods. +Among them, semi- +supervised learning only uses the data of single-domain +Cityscapes. The results are shown in Table 5. +Extension to source-free scenario. Due to data privacy +and constraints on computing resources, domain adapta- +tion sometimes fails to obtain source domain datasets. We +further evaluate the generalization of STAL by extend- +ing to source-free domain adaptation (SFDA). Comparing +methods include URMA [19], LD [74], SFDA [36], and +7 + +RIPU +Ours +Ground Truth and Image +road +side. +buil. +wall +fence +pole +light +sign +veg. +terr. +sky +person +rider +car +truck +bus +train +motor +bike +ignored +Figure 5. t-SNE analysis [61] of active learning method RIPU [70] and our method. The visualization of embedded features further +demonstrates that our method can exhibit the clearest clustering. +Table 5. Comparisons with the SSDA and SSL methods on task +GTAV → Cityscapes, SYNTHIA → Cityscapes. We report the +mIoUs in terms of 19 classes and 13 classes. +Type +Method +Net. +GTAV +SYNTHIA +mIoU +mIoU* +SSDA +MME (3.4%) [55] +V2 +52.6 +59.6 +ASS (3.4%) [67] +54.2 +62.1 +DlDM (3.4%) [9] +61.2 +68.4 +SSL +Cutmix (3.4%) [20] +V2 +50.8 +61.3 +DST-CBC (3.4%) [18] +48.7 +59.7 +ALDA +Ours (2.2%) +V2 +72.5 +78.6 +SSL +GCT (3.1%) [35] +V3+ +63.2 +- +MT (3.1%) [58] +64.1 +- +CCT (3.1%) [48] +66.4 +- +Cutmix (3.1%) [20] +69.1 +- +AEL (3.1%) [32] +74.3 +- +U2PL+AEL (6.3%) [9] +74.9 +- +ALDA +Ours (2.2%) +V3+ +75.0 +80.9 +Ours (5.0%) +76.1 +82.1 +RIPU [70]. Results in Table 6 validate the effectiveness of +STAL for this challenging DA task. More details about how +STAL works well are provided in Appendix B. +t-SNE Visualization. To better develop intuition, we draw +t-SNE visualization [61] of the learned feature representa- +tions for contrast methods (RIPU [70]) and ours STAL in +Fig. 5. We randomly select an image from the target domain +and then map its high-dimensional latent feature representa- +tion into 2D space. In Fig. 5, our method is able to separate +Table 6. Experiments on source-free domain adaptation sce- +nario (SFDA). +Method +Net. +Budget +GTAV +SYNTHIA +mIoU +mIoU +mIoU* +URMA [19] +V2 +- +45.1 +39.6 +45.0 +LD [74] +- +45.5 +42.6 +50.1 +SFDA [36] +- +53.4 +52.0 +60.1 +RIPU [70] +2.2% +67.1 +68.7 +74.1 +Ours +V2 +2.2% +70.4 +72.0 +78.1 +the features between different categories better, and the de- +cision boundary is clearer than other methods. This shows +the discriminative power of STAL contrasting adaptations. +5. Conclusion +We propose an iterative loop learning algorithm that +combines self-training and active learning. +It success- +fully enhances the potential of combining the self-training +paradigm with active learning to achieve the best perfor- +mance for domain-adaptive semantic segmentation with +minimal label cost. +STAL leverages the ability of self- +training to learn from massive unlabeled data to improve +accuracy in the target domain and provide an accurate se- +lection model for active learning. Then, the disadvantaged +experience in the self-training is further corrected through +active learning. The effectiveness of the proposed method is +validated through extensive experiments and ablation stud- +ies, and our STAL achieves new state-of-the-art results. +8 + +References +[1] Nikita Araslanov and Stefan Roth. Self-supervised augmen- +tation consistency for adapting semantic segmentation. 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In ICCV, +pages 5982–5991, 2019. 2, 6 +11 + diff --git a/jNFQT4oBgHgl3EQflTY-/content/tmp_files/load_file.txt b/jNFQT4oBgHgl3EQflTY-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d1e8043f5c36b9d7924c2b2dce95278148c20f10 --- /dev/null +++ b/jNFQT4oBgHgl3EQflTY-/content/tmp_files/load_file.txt @@ -0,0 +1,1245 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf,len=1244 +page_content='Iterative Loop Learning Combining Self-Training and Active Learning for Domain Adaptive Semantic Segmentation Licong Guan1 Xue Yuan1 � 1School of Electronic and Information Engineering, Beijing Jiaotong University {lcguan941,xyuan}@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='cn Abstract Recently, self-training and active learning have been proposed to alleviate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Self-training can im- prove model accuracy with massive unlabeled data, but some pseudo labels containing noise would be generated with limited or imbalanced training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' And there will be suboptimal models if human guidance is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Active learning can select more effective data to intervene, while the model accuracy can not be improved because the mas- sive unlabeled data are not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' And the probability of querying sub-optimal samples will increase when the do- main difference is too large, increasing annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' This paper proposes an iterative loop learning method com- bining Self-Training and Active Learning (STAL) for do- main adaptive semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The method first uses self-training to learn massive unlabeled data to improve model accuracy and provide more accurate selection mod- els for active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Secondly, combined with the sample selection strategy of active learning, manual intervention is used to correct the self-training learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Iterative loop to achieve the best performance with minimal label cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Ex- tensive experiments show that our method establishes state- of-the-art performance on tasks of GTAV → Cityscapes, SYNTHIA → Cityscapes, improving by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9% mIoU and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2% mIoU, compared to the previous best method, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Code will be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Introduction Semantic segmentation can understand image scenes at the pixel level and is crucial for various real-world appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Thanks to the rapid development of deep learn- ing, many advanced segmentation methods have been pro- posed and achieved great breakthroughs in various tasks such as autonomous driving [24], scene parsing [13, 50], medical analysis [2], and human-computer interaction [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' �Corresponding author 75 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='0 Percentage of labeled images (%) 100% 35 40 45 50 55 60 70 65 80 mIOU (%) GTAV→Cityscapes Source Only Ours RIPU [CVPR’22] MADA [ICCV’21] AADA [WACV’20] ALDA (V3+) DAP [CVPR’22] UDA (V2) SAC [CVPR’21] DLDM [CVPR’21] SSDA (V2) ASS [CVPR’20] MME [ICCV’19] Better Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Performance comparison between our method and other methods on GTAV → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' V2 and V3+ are based on DeepLab-v2 and DeepLab-V3+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' However, the traditional fully-supervised scenes [43,76,78] are very eager for images carefully annotated by human annotators, especially in aforementioned areas, a large number of annotated images are very expensive, time- consuming [13, 43], or even infeasible, which greatly hin- ders their widespread application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Therefore, it remains a major challenge to guarantee good generalization for differ- ent domain scenarios with minimal annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Nowadays, many research works have proved that do- main adaptation is one of the powerful means to address the above issues [23,37,44,60,63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Among them, unsuper- vised domain adaptation (UDA) [26,28,33,41,69,77] aims to solve this problem by leveraging the knowledge of label- rich data (source data) and transferring it to unlabeled data (target data) [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' While it can avoid the intensive workload of manual annotation, the performance still lags far behind fully supervised models [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Furthermore, active learn- ing can significantly improve performance on both classi- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='13361v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='CV] 31 Jan 2023 fication and detection tasks by introducing a few additional manual annotations to a few selected samples from the tar- get domain [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Nevertheless, active learning does not uti- lize the massive unlabeled data in the target domain, and only improves the accuracy through manual intervention, which makes the labeling cost difficult to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' And the probability of querying sub-optimal samples will increase when the domain difference is too large, increasing annota- tion cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Moreover, in the process of semantic segmenta- tion data labeling, the outline labeling of objects is usually completed by clicking with a mouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Some previous studies on active learning domain adaptation are based on pixel se- lectors [56], which are difficult to generalize to real-world applications using a single image as the minimum selection unit and labeling based on object contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Therefore, how to provide an accurate selection model for active learning and design a practical selection strategy is one the urgent problems to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Recently, self-training has greatly facilitated domain adaptation, as it can retrain the network with pseudo-labels generated from massive unlabeled data [11, 45, 65, 77, 79– 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' However, due to the limited and imbalanced train- ing data, the pseudo-labels generated by self-training usu- ally contain noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' This disadvantageous experience not only can not improve the accuracy of pseudo-labels but also further affect the performance of machine learning models without timely manual intervention and guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' There- fore, how discovering and correcting false pseudo-labels in the self-training learning process is one of the problems to be solved urgently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' To solve the above problems, this paper proposes an it- erative loop learning method combining self-training and active learning for domain-adaptive semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' It learns massive unlabeled data through self-training to im- prove the accuracy in the target domain and provides an ac- curate selection model for active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Then, the active learning sample selection strategy is used to further correct the false pseudo labels in the self-training process through manual intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Self-training and active learning com- plement each other and help the model achieve the best per- formance with minimal annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In a nutshell, our contributions can be summarized as: We propose a method combining self-training with ac- tive learning for domain-adaptive semantic segmenta- tion, termed STAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' By complementing the advantages of self-training and active learning, we achieve the best performance for domain-adaptive semantic segmenta- tion with minimal label cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We propose an iterative loop learning strategy to opti- mize the performance of semantic segmentation mod- els through three stages: warm-up learning, active se- lection, and incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Related Work Domain adaptation (DA) can transfer knowledge from a label-rich source domain to a label-scarce target domain, and recent work has achieved great success on a range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Such as classification [38,40,72], detection [10,62], and segmentation [41,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Most previous works have used adversarial learning [30, 39, 63, 68] in an attempt to re- duce the domain gap between source and target features from the image level or feature level [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Recent work on domain-adaptive semantic segmentation can be mainly di- vided into two categories: adversarial training-based meth- ods [63,64] and self-training-based methods [31,77,79,81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' For the first branch, most works tend to learn domain- invariant representations based on min-max adversarial op- timization games by tricking the domain discriminator to obtain aligned feature distributions [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The second branch focuses on how to generate high-quality pseudo- labels for target domain data for further model optimiza- tion [77,79], which drives the development of self-training techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Semi-supervised learning (SSL) involves two typical paradigms: consistency regularization [3, 48, 73] and en- tropy minimization [5, 6, 27, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Consistency regulariza- tion forces the model to produce stable and consistent pre- dictions on the same unlabeled data under various perturba- tions [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' On the other hand, entropy minimization, gen- eralized by self-training pipelines [1,12,15], exploits unla- beled target domain data in a way that uses pseudo-labels for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' For example, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' [67] propose the Semantic-Level Shift (ASS) framework, which introduces an additional semantic-level adaptation module by adver- sarial training on the corresponding outputs of the source and target labeled inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' However, adversarial loss makes training unstable due to weak supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' [80] proposed an iterative learning strategy with class-balanced and spatial priors for target instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Tranheden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' [59] proposed a domain-mixed self-training pipeline to improve training stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' [66] exploit unreliable pixels by adding a contrastive learning loss on top of self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The above studies can improve the model accuracy by using massive unlabeled data, while the noise problem of pseudo- labels can not be effectively solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Therefore, we incorpo- rate a sample selection strategy of active learning and em- ploy human intervention to correct self-training learning in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Active learning (AL) aims to maximize the model perfor- mance with the least labeling cost of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Query rules are the core content of active learning, and commonly used query strategies are divided into uncertainty-based methods [4, 16] and diversity-based methods [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In the field of active learning domain adaptation research, previ- ous work has mainly focused on classification tasks [21,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2 Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' [46] and Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' [56] were the first to adopt active learning domains for semantic segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Among them, [46] proposed a multi-anchor strategy to ac- tively select image subsets, which can be inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' [56] proposed a more efficient point-based annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' However, the selected points ignore the pixel spatial continuity of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Recently, Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' [70] greatly improved the seg- mentation performance in the target domain by exploring the consistency of the image space and selecting the most diverse and uncertain image regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' However, most ac- tive learning research works ignore the utilization of mas- sive unlabeled data in the target domain, resulting in high labeling costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' This paper uses self-training to learn mas- sive unlabeled data to improve the accuracy of the model, provide an accurate selection model for active learning, and reduce the cost of labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Approach In this section, we establish our problem mathematically and first outline our proposed method in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Our strate- gies for self-training and active learning are presented in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2 and § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Overview In domain-adapted semantic segmentation, we have a set of labeled source domain data Ds = {(xs, ys)} and incom- pletely labeled target data Dt = {(xt, yt)}, where ys is the pixel label belonging to one of the C known classes in the label space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Our goal is to learn a function f ◦ h : x → y (a semantic segmentation network parameterized by θ) that achieves good segmentation performance on the target do- main with a small amount of labeled target data and a large amount of labeled source data and unlabeled target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In general, the success of CNN-based methods bene- fits from a large amount of manually labeled data and the assumption of independent and identical data distributions between training and testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' However, when a model trained on the training set (source domain) is di- rectly applied to an unseen test scene (target domain), the performance drops significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' To transfer knowledge ef- ficiently, recent advances employ self-training techniques and optimize the cross-entropy loss using target pseudo- labels �yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Due to limited training data and class imbalance, the pseudo-labels generated by self-training often contain noise, and when lacking human intervention, inferior ex- perience during training can lead to sub-optimal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Further, active learning can pick out more effective data to intervene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' However, active learning does not use massive amounts of unlabeled data, and it relies too much on picking models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' And the probability of querying sub-optimal sam- ples will increase when the domain difference is too large, resulting in an increase in annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' To solve the above problems, we propose an iterative loop learning method that combines self-training and ac- tive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The proposed framework consists of three stages: the first stage is self-training learning (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Use a very small amount of labeled data and a large amount of unlabeled data to perform self-training learning to ob- tain a warm-up model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The second stage is active selection (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Use the current model to predict the unlabeled tar- get domain data and send it to the acquisition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The third stage is image labeling (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The selected samples are manually labeled and added to the labeled dataset of the target domain, and the first stage of self-training is repeated to obtain the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Self-Training Learning STAL follows a typical self-training framework, which consists of a student model and a teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The teacher model and the student model have the same schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The two models differ only in updating their weights, the student’s weight θs is updated by convention, while the teacher model’s weight θt is updated by the exponential moving average (EMA) of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' For labeled images of source and target domains, we use standard cross- entropy loss on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' And for each unlabeled target domain image, we bring it into the teacher model for prediction and obtain the pseudo-label according to the pixel prediction en- tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Subsequently, the student model is trained on unla- beled target domain data and corresponding pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Our optimization objective is to minimize the overall loss, which can be expressed as: L = Ls + λuLu + λcLc , (1) where Ls (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2) and Lu (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 3) represent the supervised and unsupervised loss applied to labeled and unlabeled im- ages, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Lc (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 4) is the contrastive learning loss [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' λu is the weight of the unsupervised loss, λc is the weight of the contrastive loss, and Ls and Lu are both cross-entropy (CE) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Ls = 1 Nl Nl � i=1 1 WH W H � j=1 ℓce(f ◦ h(xl i,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' θ), yl i,j) , (2) Lu = 1 Nu Nu � i=1 1 WH W H � j=1 ℓce(f ◦ h(xu i,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' θ), ˆyu i,j) , (3) Lc = − 1 C × M C−1 � c=0 M � i=1 log \uf8ee \uf8f0 e⟨aci,a+ ci⟩/ω e⟨aci,a+ ci⟩/ω + �N j=1 e⟨aci,a− cij⟩/ω \uf8f9 \uf8fb , (4) Among them, f ◦ h is the composition function of h and f, which means that the image xi,j is first sent to h to extract features, and then sent to f to obtain segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' yl i,j is the manually annotated mask label for the j-th pixel 3 Labeled target image Student Teacher EMA Update S L Labeled source image Copy Prob map Unlabeled target image Segmentation network Prob map Prob map U C L L \uf02b Pseudo Label ˆ tY Unlabeled target image Acquisition function Human participants (a) Self-Training (b) Active Selection (c) Image Labeling For re-training Prob map Corresponding original image Select Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The overview of the proposed STAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The proposed framework consists of three stages: the first stage (a) is self-training learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Use a very small amount of labeled data and a large amount of unlabeled data to perform self-training learning to obtain a warm-up model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The second stage (b) is active selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Use the current model to predict the unlabeled target domain data and send it to the acquisition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The third stage (c) is image labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The selected samples are manually labeled and added to the labeled dataset of the target domain, and the first stage of self-training is repeated to obtain the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' in the i-th labeled image, ˆyu i,j is the pseudo-label for the j- th pixel in the i-th unlabeled image, Nl and Nu represents the number of labeled and unlabeled images in the train- ing batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' W and H are the width and height of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' ℓce is the standard cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In the con- trast loss Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 4, M is the total number of anchor pixels, and aci represents the representation of the i-th anchor of class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The positive and negative samples corresponding to each anchor pixel are denoted as a+ ci and a− ci, ⟨·, ·⟩ is the cosine similarity between features from two different pixels, whose range is limited between -1 to 1 according to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Following [66], we set M = 50, N = 256, and ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' During training, some tail categories will introduce more pseudo-label noise due to insufficient training, which in- directly leads to the degradation of underperforming cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We dynamically record the performance of each cat- egory during training by maintaining a confidence library, and for the underperforming categories, the confidence met- ric is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Confc = 1 Nl Nl � i=1 1 N c i Nc i � j=1 (f ◦ h(xc i,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' θ)), c ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' , C} , (5) Among them, C is the category number, N c i indicates the number of pixels belonging to category c according to the ground truth label yi,j, and f ◦ h(xc i,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' θ) indicates the c-th channel prediction result of the j-th pixel in the i-th image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We use EMA to update the confidence by class at each training step, and the update criterion is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 6: Confc n ← αConfc n−1 + (1 − α)Confc n, c ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' , C} , (6) where n represents the n-th iteration and α ∈ [0, 1) is the momentum coefficient, which we set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='999 in our experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' For underperforming categories, we improve by two data augmentation techniques, Copy Paste [25] and Cut- mix [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We calculate the sampling probability according 4 to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 7, which converts the class confidence in the confi- dence base into the normalized sampling probability s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' s = Softmax(1 − Conf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' (7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Active Learning Inferior experience during self-training can have an im- pact on the performance of machine learning models in the absence of human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Therefore, we utilize active learning to correct self-training erroneous pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Our sample acquisition strategy is as follows: Given an unlabeled target image xt and a warm-up model θw, the acquisition function A is a function that the active learning system uses to query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' First, the softmax output Pt of the un- labeled image xt is obtained by warm-up model θw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Since the prediction Pt carries semantic relation information, we adopt the prediction entropy H of each pixel to measure the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' For a single image with a C classification, we evaluate the uncertainty �xu by averaging all entropies of all pixels in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Calculated as follows: �xu = − 1 WH W H � j=1 C � c=1 Pi,j,c t log Pi,j,c t (8) Among them, W and H are the width and height of the fea- ture map, respectively, and Pi,j,c t represents the c-th channel prediction result of the j-th pixel in the i-th image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' After obtaining uncertainty results for all unlabeled images, we preferentially select the most uncertain images S for anno- tation according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' S = argmax A(�xu) (9) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Experiments Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' To verify the effectiveness of the proposed method, we evaluate our method on two popular scenarios, transferring information from synthetic images GTAV [53] and SYNTHIA [54] to the real domain, the Cityscapes [13] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' GTAV is a synthetic image dataset contain- ing 24,966 1914×1052 images, sharing 19 classes as Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' SYNTHIA is a synthetic urban scene dataset containing 9,400 1280×760 images, sharing 16 classes as Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Cityscapes is an autonomous driving dataset of real urban scenes, containing 2,975 training images and 500 validation images, each with a resolution of 2048×1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' All experiments are performed on NVIDIA A100 GPU with Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We adopt DeepLabv2 [7] and DeepLab-v3+ [8] architectures with ResNet-101 [29] pre-trained on ImageNet [14] as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Regarding the training, we use the SGD optimizer with an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='0025, weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='0001, and momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' For all experiments, we train about 200K iterations with batch size of 12, and data are resized into 769×769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' As a common practice [45,46,56,70], we report the mean Intersection-over-Union (mIoU) [17] on the Cityscapes validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Specifically, we report the mIoU on the shared 19 classes for GTAV → Cityscapes and report the results on 13 (mIoU*) and 16 (mIoU) common classes for SYNTHIA → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We also add the 19- class evaluation for the SYNTHIA → Cityscapes task in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4, and we believe that the extremely small amount of target domain data is sufficient to optimize the three classes missing from SYNTHIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Annotation budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Previous active learning-based selec- tion strategies use pixels or regions as selection units, and they select a fixed proportion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2% or 5%) of each sample in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' While our sample selection strategy takes a single image as the smallest unit, for a fair comparison, we only select the corresponding percentage of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The selection process is divided into two rounds, the first round we randomly select 1% (30 images) from the target domain dataset for self-training learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In the second round, we selected 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2% (35 images) or 4% (120 images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Therefore, we label 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2% or 5% of the target domain data in total for self-training learning to obtain the final evaluation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Comparisons with the state-of-the-arts Table 1 and Table 2 are the domain adaptation results of GTAV → Cityscapes and SYNTHIA → Cityscapes, re- spectively, and it can be seen that our method greatly out- performs the previous leading unsupervised domain adap- tation and active learning domain adaptation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' At the limit, our results using only 1% of the data are also sub- stantial improvements over previous state-of-the-art unsu- pervised methods (DAP+ProDA [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' For the GTAV → Cityscapes task, based on using the same backbone (DeepLab-v3+), we can easily beat AADA [57] and MADA [46] with an annotation budget of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Compared to the state-of-the-art model, using the same annotation budget (5%), our method achieves 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9% mIoU improvement over RIPU [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Notably, our method signifi- cantly outperforms the contrastive methods in some specific categories, namely the tail category of Cityscapes (such as “traffic light”, “traffic sign”, “rider”, “bus”, “train”, “mo- torcycle”, and “bicycle”), which indicates that the proposed method can effectively alleviate the long-tailed distribution problem to outperform the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Our STAL is still competitive for the SYNTHIA → Cityscapes task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' On the basis of using the same back- bone (DeepLab-v3+), our method can beat all methods us- ing 1% of the target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Compared to the state-of-the-art model, our method achieves 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2% mIoU improvement over RIPU [70] if the same annotation budget (5%) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Like- wise, our method outperforms RIPU [70] on the tail cate- gories of Cityscapes (such as “traffic light”, “traffic sign”, “rider”, “bus”, “motorcycle”, and “bicycle”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Comparison with previous results on task GTAV → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We report the mIoU and best results are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Method Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' road side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' buil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' wall fence pole light sign veg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' terr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' sky pers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' rider car truck bus train motor bike mIoU Source Only V2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='8 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='8 77.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 Methods with V2 are based on DeepLab-v2 [7] and methods with V3+ are based on DeepLab-v3+ [8] for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Comparisons with previous results on task SYNTHIA → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We report the mIoUs in terms of 13 classes (excluding the “wall”, “fence”, and “pole”) and 16 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Best results are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Method Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' road side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' buil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' wall* fence* pole* light sign veg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' sky pers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' rider car bus motor bike mIoU mIoU* Source Only V2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 23.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='8 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='7 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 Methods with V2 are based on DeepLab-v2 [7] and methods with V3+ are based on DeepLab-v3+ [8] for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' (a) Target Image (b) Ground Truth (c) RIPU (5%) (d) Ours (5%) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Visualization of segmentation results for the task GTAV → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' From left to right: original target image, ground-truth label, result predicted by RIPU [70], and result predicted by Ours are shown one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Qualitative results We visualize the segmentation results predicted by STAL in GTAV → Cityscapes and compare them with the state-of- the-art RIPU [70] model prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 3, our method has smoother prediction results for the head category (such as “pole”), and for Cityscapes tail categories (such as “bus”, “rider”, and “traffic sign”) have a 6 7(a) Target Image (b) Ground Truth (c) RIPU (5%) (d) Ours (5%) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Visualization of segmentation results for the task SYNTHIA → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' From left to right: original target image, ground-truth label, result predicted by RIPU [70], and result predicted by Ours are shown one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' great improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The visualization of the segmentation results predicted by SYNTHIA → Cityscapes is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Our method achieves accurate predictions for the missing class “truck” with only a small amount of target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' For the head class “person”, STAL can achieve more detailed contour prediction than adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' For the tail categories of Cityscapes (such as “traffic sign” and “traffic light”), our predictions are greatly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Readers are referred to Appendix D for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Ablation Study To further investigate the efficacy of each component of our STAL, we perform ablation studies on GTAV → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We randomly select 5% of the data in the target domain to train in DeepLab-v3+ as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' As shown in Table 3, satisfactory and consistent gains from the base- line to our full method demonstrate the effectiveness of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Compared to the baseline, Lu is able to lever- age the unlabeled data of the target domain to improve per- formance by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='38%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The performance is further improved by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='63% after adding Lc and Data aug in self- training, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We applied both Lc and Data aug strategies simultaneously and achieved a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='32% improve- ment in model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Finally, by replacing the sam- ples in the baseline with the samples picked by active learn- ing, our performance reaches 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='11%, proving the effective- ness of the sample selection strategy based on prediction uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Further Analysis Extension to open set domain adaptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' SYNTHIA shares only 16 classes with Cityscapes, so previous meth- ods only evaluate mIoU for 16 classes and 13 classes on this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In active learning domain adaptation, we will re- port mIoU for 19 classes on the SYNTHIA → Cityscapes task due to the addition of data from the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The evaluation results are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Judging from the evaluation results of “terrain”, “truck”, and “train” missing Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Ablation study on the effectiveness of various compo- nents in our STAL, including contrastive loss Lu, Lc, Data aug, and Uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Self-Training Active Learning GTAV Lu Lc Data aug Uncertainty mIoU 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='33 ✓ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='71 ✓ ✓ 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='61 ✓ ✓ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='34 ✓ ✓ ✓ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='03 ✓ ✓ ✓ ✓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='11 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Experiments with Open Set Domain Adaptation on task SYNTHIA → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Method Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' terrain truck train mIoU Ours (1%) V3+ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 Ours (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2%) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9 three categories, we can still produce effects comparable to a large amount of labeled data under the premise of using a very small amount of target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Comparison with SSDA and SSL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' To bet- ter understand the performance improvement of strategies combining self-training with active learning, we compare our method with semi-supervised domain adaptation and semi-supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Among them, semi- supervised learning only uses the data of single-domain Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The results are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Extension to source-free scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Due to data privacy and constraints on computing resources, domain adapta- tion sometimes fails to obtain source domain datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We further evaluate the generalization of STAL by extend- ing to source-free domain adaptation (SFDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Comparing methods include URMA [19], LD [74], SFDA [36], and 7 RIPU Ours Ground Truth and Image road side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' buil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' wall fence pole light sign veg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' terr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' sky person rider car truck bus train motor bike ignored Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' t-SNE analysis [61] of active learning method RIPU [70] and our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The visualization of embedded features further demonstrates that our method can exhibit the clearest clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Comparisons with the SSDA and SSL methods on task GTAV → Cityscapes, SYNTHIA → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We report the mIoUs in terms of 19 classes and 13 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Type Method Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' GTAV SYNTHIA mIoU mIoU* SSDA MME (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4%) [55] V2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 ASS (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4%) [67] 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 DlDM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4%) [9] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4 SSL Cutmix (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4%) [20] V2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='3 DST-CBC (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4%) [18] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='7 ALDA Ours (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2%) V2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 SSL GCT (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1%) [35] V3+ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2 MT (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1%) [58] 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 CCT (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1%) [48] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4 Cutmix (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1%) [20] 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 AEL (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1%) [32] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='3 U2PL+AEL (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='3%) [9] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9 ALDA Ours (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2%) V3+ 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='9 Ours (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='0%) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 RIPU [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Results in Table 6 validate the effectiveness of STAL for this challenging DA task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' More details about how STAL works well are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' t-SNE Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' To better develop intuition, we draw t-SNE visualization [61] of the learned feature representa- tions for contrast methods (RIPU [70]) and ours STAL in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' We randomly select an image from the target domain and then map its high-dimensional latent feature representa- tion into 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 5, our method is able to separate Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Experiments on source-free domain adaptation sce- nario (SFDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Method Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Budget GTAV SYNTHIA mIoU mIoU mIoU* URMA [19] V2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='0 LD [74] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='6 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 SFDA [36] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 RIPU [70] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 Ours V2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='2% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content='1 the features between different categories better, and the de- cision boundary is clearer than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' This shows the discriminative power of STAL contrasting adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Conclusion We propose an iterative loop learning algorithm that combines self-training and active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' It success- fully enhances the potential of combining the self-training paradigm with active learning to achieve the best perfor- mance for domain-adaptive semantic segmentation with minimal label cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' STAL leverages the ability of self- training to learn from massive unlabeled data to improve accuracy in the target domain and provide an accurate se- lection model for active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Then, the disadvantaged experience in the self-training is further corrected through active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The effectiveness of the proposed method is validated through extensive experiments and ablation stud- ies, and our STAL achieves new state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 8 References [1] Nikita Araslanov and Stefan Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Self-supervised augmen- tation consistency for adapting semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In CVPR, pages 15384–15394, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2, 6 [2] Saeid Asgari Taghanaki, Kumar Abhishek, Joseph Paul Co- hen, Julien Cohen-Adad, and Ghassan Hamarneh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Deep se- mantic segmentation of natural and medical images: a re- view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Artif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=', 54(1):137–178, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 1 [3] Philip Bachman, Ouais Alsharif, and Doina Precup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Learn- ing with pseudo-ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' NeurIPS, 27, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2 [4] William H Beluch, Tim Genewein, Andreas N¨urnberger, and Jan M K¨ohler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' The power of ensembles for active learning in image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In CVPR, pages 9368–9377, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2 [5] Paola Cascante-Bonilla, Fuwen Tan, Yanjun Qi, and Vicente Ordonez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Curriculum labeling: Revisiting pseudo-labeling for semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In AAAI, volume 35, pages 6912–6920, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2 [6] Huaian Chen, Yi Jin, Guoqiang Jin, Changan Zhu, and En- hong Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Semisupervised semantic segmentation by im- proving prediction confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Neural Net- works Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 2 [7] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolu- tion, and fully connected crfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' IEEE TPAMI, 40(4):834–848, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 5, 6 [8] Liang-Chieh Chen, Yukun Zhu, George Papandreou, Florian Schroff, and Hartwig Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Encoder-decoder with atrous separable convolution for semantic image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In ECCV, pages 801–818, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' 5, 6 [9] Shuaijun Chen, Xu Jia, Jianzhong He, Yongjie Shi, and Jianzhuang Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' Semi-supervised domain adaptation based on dual-level domain mixing for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} +page_content=' In CVPR, pages 11018–11027, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFQT4oBgHgl3EQflTY-/content/2301.13361v1.pdf'} 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0000000000000000000000000000000000000000..fbca16e9e9a1bfe2cec2e15158abfc9ddf855628 --- /dev/null +++ b/jdAyT4oBgHgl3EQfx_mp/content/tmp_files/2301.00677v1.pdf.txt @@ -0,0 +1,1540 @@ +EARTH AND PLANETARY PHYSICS, VOL. 0, XXXX, DOI:, +The Mars Orbiter Magnetometer of Tianwen-1: In-flight +Performance and First Science Results +Yuming Wang,1,2,∗ Tielong Zhang,2,3 Guoqiang Wang,4 Sudong Xiao,4 Zhuxuan +Zou,1,2 Long Cheng,1,2 Zonghao Pan,1,2 Kai Liu,1,2 Xinjun Hao,1,2 Yiren Li,1,2 +Manming Chen,1,2 Zhoubin Zhang,5 Wei Yan,5 Zhenpeng Su,1,2 Zhiyong Wu,1,2 +Chenglong Shen,1,2 Yutian Chi,6 Mengjiao Xu,6 Jingnan Guo,1,2 and Yang Du7 +1 School of Earth and Space Sciences/Deep Space Exploration Laboratory, University of Science and Technology +of China, Hefei 230026, China +2 CAS Center for Excellence in Comparative Planetology/CAS Key Laboratory of Geospace +Environment/Mengcheng National Geophysical Observatory, University of Science and Technology of China, +Hefei 230026, China +3 Space Research Institute, Austrian Academy of Sciences, Graz, Austria +4 Institute of Space Science and Applied Technology, Harbin Institute of Technology, Shenzhen, China +5 National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China +6 Institute of Deep Space Sciences, Deep Space Exploration Laboratory, Hefei 230026, China +7 Shanghai Institute of Satellite Engineering, Shanghai, China +∗ Corresponding author, Email: ymwang@ustc.edu.cn +Abstract. +Mars Orbiter MAGnetometer (MOMAG) is a scientific instrument onboard +the orbiter of China’s first mission for Mars — Tianwen-1. It started to routinely mea- +sure the magnetic field from the solar wind to magnetic pile-up region surrounding Mars +since November 13, 2021. Here we present its in-flight performance and first science re- +sults based on the first one and a half months’ data. By comparing with the magnetic +field data in the solar wind from the Mars Atmosphere and Volatile EvolutioN (MAVEN), +the magnetic field by MOMAG is at the same level in magnitude, and the same mag- +netic structures with the similar variations in three components could be found in MO- +MAG data. In the first one and a half months, we recognize 158 clear bow shock (BS) +crossings from MOMAG data, whose locations statistically match well with the mod- +eled average BS. We also identify 5 pairs of simultaneous BS crossings of the Tianwen- +1’s orbiter and MAVEN. These BS crossings confirm the global shape of modeled BS +as well as the south-north asymmetry of the Martian BS. Two presented cases in this +paper suggest that the BS is probably more dynamic at flank than near the nose. So +far, MOMAG performs well, and provides accurate magnetic field vectors. MOMAG is +continuously scanning the magnetic field surrounding Mars. These measurements com- +plemented by observations from MAVEN will undoubtedly advance our understanding +of the plasma environment of Mars. +1. Introduction +Tianwen-1 is the first mission of China to explore and +study Mars from its space environment to the surface [Wan +et al., 2020; Zou et al., 2021]. It consists of an orbiter, a +lander and a rover, called Zhurong. Mars Orbiter MAG- +netometer (MOMAG) is one of the scientific instruments +onboard the orbiter[Liu et al., 2020]. +It investigates the +magnetic field environment of Mars by measuring the local +vector magnetic field, and therefore provides some key infor- +mation for the understanding of the history and evolution +of Mars. +The magnetic field surrounding Mars has two sources. +One is the dynamic magnetic field resulted from the cou- +pling between the solar wind and the Martian ionosphere, +and the other is the static crustal magnetic field of Mars +itself. Since Mars has no global intrinsic magnetic field, the +solar wind carrying interplanetary magnetic field directly in- +teracts with Martian ionosphere, and forms the bow shock +(BS) and induced magnetosphere, which consists of mag- +netic pileup region (MPR) and wake region[e.g., Bertucci +et al., 2004; Brain et al., 2006]. Between the BS and MPR, +there is magnetosheath separated from MPR with the mag- +netic pileup boundary (MPB)[e.g., Mazelle et al., 2004]. +The magnetic field in these regions influenced by solar wind +is highly dynamic. +Escape of ions in Martian atmosphere is one of the +core science issues of Tianwen-1, and is closely related to +the magnetic environment. For example, the strong static +crustal magnetic field on the southern hemisphere may reach +upto a high altitude and reconnect with interplanetary mag- +netic field, causing the escape of ions[Brain et al., 2015], just +like the behavior of Venus[Zhang et al., 2012]. Besides, var- +ious waves in the ionosphere may heat particles, causing +ion outflow[Ergun et al., 2006], and when these heated ions +transport outside the MPB, they will interact with magnetic +field carried by solar wind stream to further generate ion +cyclotron wave, boosting the escape of the ions[Russell and +Blanco-Cano, 2007]. The escape rate during storm times +will be one to two orders higher than that during quite +time[Jakosky et al., 2015b]. +1 +arXiv:2301.00677v1 [astro-ph.EP] 2 Jan 2023 + +X - 2 +WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +Tianwen-1 orbiter was running on a highly eccentric or- +bit with the periapsis of about 1.08 Mars radii (rm) and the +apoapsis of about 4.17 rm, and the orbital plane is highly +inclined during November and December in 2021, as shown +in Figure 1. +During that period, the periapsis was right +above the northern pole of Mars, the apoapsis far above +the southern pole in the solar wind, and the orbital period +was about 7.8 hr, with about 50% – 75% of time in so- +lar wind. Thus, MOMAG mainly measured the magnetic +field from solar wind to the MPR on the dawn-dusk side. +Later, the inclination angle of the orbit will decrease to al- +low the orbiter detect the wake region of Mars. These data +will help us understand the structure and evolution of Mar- +tian magnetic field environment and provide clues for ion +escape. Since the Mars Atmosphere and Volatile EvolutioN +(MAVEN, Jakosky et al. 2015a), which also carries a mag- +netometer (MAG, Connerney et al. 2015), is still working, +the successful operation of MOMAG will make us for the +first time to study Martian magnetic field environment from +two points. +In this paper, we show and analyze the data during +November 13 – December 31, 2021. In Section 2, we will +describe the basic information and the current status of MO- +MAG, and present some measured magnetic field. Then we +show the first results of MOMAG about Martian BS with +the comparison with MAVEN/MAG data in Sections 3. In +the last section, we summarize the paper. +2. In-flight Calibration and Performance +MOMAG contains two sensors mounted on a 3.19 m long +boom. The outer sensor accommodates at the top of the +boom and the inner sensor is 0.9 m away (see Liu et al. +2020 for details). Since the orbiter of Tianwen-1 does not +have magnetic cleanliness control, the boom is actually not +long enough to avoid the contaminations of the magnetic +field from the orbiter. Thus, how to remove the magnetic +interference based on the magnetic fields measured by the +two separated sensors become pivotal. +The procedure of the mitigation of the orbiter’s mag- +netic field generally includes two steps, which is similar to +the procedure applied on the magnetometer of Venus Ex- +press[Zhang et al., 2008; Pope et al., 2011]. The first step +is to remove the magnetic interference due to the opera- +tions of instruments. Such interferences behave as jumps +in the magnetic field. +If a real discontinuity in the solar +wind passes the spacecraft, the amplitudes of the jump at +the two sensors should be the same. +However, since the +distances of the two sensors from the instrument are dif- +ferent, an artificial jump will show different amplitudes at +the two sensors, and therefore could be distinguished from +real jumps. For these artificial jumps, we use the method +of Pope et al. [2011] to remove them. The second step is +to remove the static magnetic field of the orbiter and cor- +rect the offset of the fluxgate magnetometer. This step is +mainly based on the property of Alfv´enic waves, of which +the magnetic field almost rotates in a plane without the +change of magnitude[Wang and Pan, 2021]. +We process +the raw data of MOMAG to the level 2 (or level C in +China’s convention) data for scientific use through these +steps. Based on the data of the first several months since +November 13, 2021, when MOMAG started to formally +operate, we iterate the procedure and reach the first ver- +sion of the level 2 data, that have been released at the +Planet Exploration Program Scientific Data Release Sys- +tem (http://202.106.152.98:8081/marsdata/). The data +used in this paper and in our forthcoming papers will also +be put on the official website of the MOMAG team at +University of Science and Technology of China (USTC, +http://space.ustc.edu.cn/dreams/tw1_momag/). A com- +plete description of the in-flight calibration procedure as +well as the demonstration of the reliability of the calibrated +data is given in the separated paper by Zou et al. [2023]. +Figure 2a shows the magnetic field in the Mars-centered +Solar Orbital (MSO) system measured by MOMAG dur- +ing 01:00 – 09:00 UT on 2021 December 30. The orbiter +was running in the magnetosheath before 02:55 UT, and at +around 03:01:40 UT the orbiter crossed the BS where the +amplitude of the magnetic field discontinuity was more than +10 nT (Fig.2h). After about four hours, the orbiter crossed +the BS again where the amplitude of the magnetic field dis- +continuity was about 16 nT (Fig.2i). Around 08:20 UT, the +orbiter even crossed the MPB. The BS during the second +crossing was obviously stronger than that during the first +crossing. The reason is that the BS was compressed during +the second crossing, which can be seen from Figure 2d–g +that the two BS crossings were on the south, the first cross- +ing was outside and further away from the modeled averaged +BS [Edberg et al., 2008] than the second BS crossing. +If we look into the details of the first BS crossing as shown +in Figure 2h, it could be found that the orbiter crossed out +the BS around 02:56:30 UT and crossed in again at 02:57:45 +UT before it finally entered the solar wind. The magnetic +field changes during these preceding crossings suggest that +the BS was slightly stronger than the BS at 03:01:40 UT. +This could also be explained as the compression of the BS. +During these crossings, the orbiter was moving away from +Mars as the indicated by the color-coded orbit in Figure 2d. +The locations of the preceding crossings were closer to Mars +than that of the final crossing at 03:01:40 UT. +The magnetic fields in the solar wind stayed around 9 +– 10 nT, and were much less fluctuated than those in the +magnetosheath. Figure 2b displays the power spectral den- +sity of the magnetic field. It is generated by using a 10-min +window and 1-min running step. It can been seen that the +solar wind was indeed quiet except at very low frequency, +whereas in the magnetosheath the magnetic fluctuation was +enhanced. +Behind the bow shock, weak magnetic waves +right below proton gyro-frequency appeared. In the solar +wind, a small structure can be found between 05:15 and +06:35 UT. Though the total magnetic field only slightly en- +hanced, the most notable change occurred for By, which +decreased from about 4 nT to zero twice. +For comparison, Figure 3 shows the MAVEN measure- +ments of the magnetic field and solar wind during 04:00 +– 07:00 UT on the same day. +MAVEN had a quite dif- +ferent orbit, of which the orbital period was shorter than +4 hours (Fig.3c–g). +Within one hour, it crossed the BS +twice, but the positions of the crossings were both closer +to the shock nose than those of Tianwen-1. Since the two +crossings stay close to the same modeled BS, suggesting the +solar wind conditions during the two crossings are almost +the same, the amplitudes of their magnetic field discontinu- +ities were similar. We also show the detailed BS crossings +in Figure 3h and i. No multiple BS crossings happened at +MAVEN, probably suggesting the different behavior of BS +at different locations. +Though the solar wind region that MAVEN detected was +far away from that by Tianwen-1, a similar structure could +be found between 05:20 and 05:55 UT (Fig.3a), which cor- +responds to the first dip recorded in MOMAG. During that +time, the solar wind speed was about 310 km s−1 and the +number density of protons was about 7 cm−3 (see the blue +and red lines in Fig.3c). Different from MOMAG data, the +magnetic field at MAVEN was highly fluctuated with a no- +table frequency at the proton gyro frequency (Fig.3b). This +might be because MAVEN closer to Mars than Tianwen-1 +when they fly in the solar wind, and the particles escaping +from Martian atmosphere more easily interact with the so- +lar wind and interplanetary magnetic field to generate such +fluctuations at a closer distance. However, according to the +statistical study of MAVEN data [Ruhunusiri et al., 2017], +such a pattern seems not to be evident and needs further +validation. + +WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +X - 3 +Figure 1. The orbits of Tianwen-1’s orbiter (solid lines) and MAVEN (dashed lines) during November +13 – December 31, 2021. The modeled Martian bow shock and MPB [Edberg et al., 2008] are indicated +by thick and thin black lines, respectively. + +a +4- +2021-11-16 TW1 +2021-11-16 MVN +2021-11-26 TW1 +2021-11-26 MVN +2021-12-06 TW1 +3 - +2021-12-06 MVN +2021-12-16 TW1 +2021-12-16 MVN +ryz, MSo (RM) +2021-12-26 TW1 +2021-12-26 MVN +2 +1 +0 +-2 +-1 +0 +1 +2 +3 +4 +-3 +XMSO (RM) +4. +4 +2 +b +d +C +1 +2 +2 +0 +(RM) +0 +1 +0 +ZMSO +2 +-2 +-2 +-3 +-4 +4 +0 +2 +2 +-2 +0 +0 +2 +XMSO (RM) +XMSO (RM) +YMSO (RM)X - 4 +WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +Figure 2. +The magnetic field measured by MOMAG during 01:00 – 09:00 UT on December 30, 2021. Panel +(a) shows the three components of the magnetic field in MSO coordinates with the total magnitude overplotted. +Two purple arrows mark the crossings of the bow shock. +Panel (b) shows the power spectral density of the +magnetic field fluctuations. The green line indicates the proton’s gyro frequency. Panel (c) shows the height of +the Tianwen-1 orbiter, and Panel (d) – (g) display its orbit in the MSO coordinates during the period of interest, +in which the two circled dots mark the positions of the bow shock crossings. Note that Panel (d) is in aberrated +MSO coordinates with the modeled BS and MPB indicated by black lines. Panel (h) and (i) show the two BS +crossings, respectively, in details. + +40 +a +Bx +B朗 +30 +IBI +20 +(lu) +10 +Mag +11 +0 +-10 +-20 +5 +p gyro +4 +10-1 +(ZH) +3 +Freq +2 +10-2 +1 +10-3 +0 +10000 +C +Height (km) +7500 +5000 +2500 +0 +01:00 +02:00 +03:00 +04:00 +05:00 +06:00 +07:00 +08:00 +09:00 +Start time 2021-12-30 01:00:00 UT +(hr) +2 +4 +d +g +e +9 +(Rm) +(RM) +(RM) +0 + elapse +5 +0 +2 +ZMSO +yMSO +ZMSO +-2 +-2 +Time +2 +.4 +0 +.4 +0 +2.5 +2.5 +2.5 +2.5 +2.5 +0.0 +0.0 +0.0 +2.5 +0.0 +2.5 +XMSO' (RM) +XMSO (RM) +XMSO (RM) +yMSo (Rm)30 +h +Bx +By +(lu) +20 +Bz +Mag +[BI +10 +0 +02:59 +03:01 +03:05 +02:55 +02:56 +02:57 +02:58 +03:00 +03:02 +03:03 +03:04 +Start time 2021-12-30 02:55:00 UT30 +！ +Bx +Mag (nT) +20 +Bz +IBI +M +10 +M +0 +07:00 +07:01 +07:02 +07:03 +07:04 +07:05 +07:06 +07:07 +07:08 +07:09 +07:10 +Start time 2021-12-30 07:00:00 UTWANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +X - 5 +Figure 3. The magnetic field and solar wind plasma measured by MAVEN during the same period as +Fig.2. In Panel (c) the solar wind speed and number density of ions are presented with the blue and red +lines, respectively. The arrangements of other panels are the same as those in Fig.2. + +a +Bx +40 +RBI +IBI +30 +20 +(lu) +Mag +10 +0 +-10 +5 +Log PSD (nT2/Hz) +p gryo +4 +10-1 +Freq (Hz) +３ +2 +1 +10-3 +0 +40 +c +4000 +Height +300 +(km) +3 +Ivl +30 +7. +3000 +Number density +cm +s +Height ( +km +2000 +200 +uo!u +> +1000 +10 +100 +0 +30 04:00 +30 04:30 +30 05:30 +30 06:00 +30 06:30 +30 05:00 +30 07:00 +Start time 2021-12-30 04:00:00 UT +elapse (hr) +2 +2 +1 +(Rm) +9 +(RM) +(Rm) +(RM) +2 +0 +ryz, MSo' +2 +ZMSO +yMSO +ZMSO +0 +1 +0 +0 +-1 +0 +2 +0 +0 +0 +1 +XMSO' (RM) +XMSO (RM) +XMSO (RM) +YMSO (RM)30 +h +Bx +By +(lu) +20 +Bz +MM +VM/A +Mag +[BI +10 +LMAA +W +05:00 +05:01 +05:02 +05:03 +05:04 +05:05 +05:06 +05:07 +05:08 +05:09 +05:10 +Start time 2021-12-30 05:00:00 UT30 +Bx +BV +(lu) +20 +Bz +Mag +[BI +10 +W +0 +05:55 +05:56 +05:57 +05:58 +05:59 +06:00 +06:01 +06:02 +06:03 +06:04 +06:05 +Start time 2021-12-30 05:55:00 UTX - 6 +WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +Figure 4. The bow shock (BS) crossings during the period of interest. In Panel (a), the red asterisks +mean the BS crossings of Tianwen-1’s orbiter, and the blue dots the BS crossings of MAVEN. The +modeled average BS [Edberg et al., 2008] is displayed by the black line, and the other dashed lines display +the BS when the uncertainties of the BS model parameters are considered. Panel (b) shows the 5 pairs +of simultaneous (within 2 minutes) BS crossings of the Tianwen-1’s orbiter and MAVEN. Each pair is +indicated with the same colored dots, but the dots of Tianwen-1 are enclosed by black circles. The BS +crossing time of the Tianwen-1’s orbiter is given in the upper-right corner followed by a time interval +with the positive value meaning the later BS crossing of MAVEN. All these data are presented in the +aberrated MSO coordinates. +3. Bow Shock Crossings +Bow shock is one of the notable features in the Mar- +tian space environment. Its shape may reflect the up- +stream solar wind conditions and solar EUV inten- +sity and the interaction processes between solar wind +and Martian atmosphere [Mazelle et al., 2004; Ram- +stad et al., 2017; Hall et al., 2019]. +Thus, studying +Martian BS is our first choice to show the science re- +sults of MOMAG. During 2021 November 13 – Decem- +ber 31, we recognize 158 BS crossings from MOMAG +data by manually checking the magnetic field strength +variation and the fluctuation level which is measured +by the standard deviation of magnetic field within one +minute. In principle, there should be more crossings, +but Tianwen-1 mostly crossed the flank of the BS where +the characteristic of a shock may be too weak to be rec- +ognized. During the same period, we recognize 454 BS +crossings from MAVEN/MAG data. +Figure 4a shows all of the BS crossings in the aber- +rated MSO coordinates (MSO coordinates are rotated +by 4◦ about the z-axis to reduce the effect of the Mars +orbital motion on the solar wind flow direction). Since +the spatial coverage of the crossings is not wide enough, +we do not try to fit these crossings to find the best-fit +BS model, but instead to compare with the previously +established BS model [Edberg et al., 2008]. We can find +in the figure that the crossings statistically match the +model fairly well. +Martian BS position and global shape were derived +from many single crossings. +Now we can check this +based on the joint magnetic field observations from +Tianwen-1/MOMAG and MAVEN/MAG. By assum- +ing that the BS remains unchanged within 2 minutes, +we use two crossings of Tianwen-1 and MAVEN within +2 minutes to exam the BS global shape. +We choose +2 minutes because the upstream solar wind conditions +that determine the BS position and shape, i.e., the fast- +mode Mach number and dynamic pressure, are usually +stable within this time-scale as revealed by the follow- +ing analysis. +Figure 5a shows the characteristic speeds in the solar +wind, which are calculated every minute based on the +MAVEN/SWIA [Halekas et al., 2017] measurements of + +a +TW1 +4 +MVN +3 +2 +-2 +0 +1 +2 +3 +-1 +XMSO' (RM)4.0 +b +2021-11-19 05:05:40,-30s +2021-12-11 13:18:29, 90s +2021-12-12 07:14:39, -90s +3.5 +2021-12-16 19:43:35, -90s +2021-12-17 02:44:42, 8s +3.0 +(RM) +2.5 +ryz, MSo' ( +2.0 +1.5 +1.0 +0.5 +0.0 +-1 +-2 +0 +1 +2 +3 +XMSO° (Rm)WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +X - 7 +the solar wind velocity, ion density and temperature +and MAVEN/MAG measurements of magnetic field +during November 13 – December 31, 2021. The Alfv´en +speed, vA, ranges from almost zero to more than 100 +km s−1 with the peak around 30 km s−1. Since Alfv´en +wave propagates along the magnetic field, if we take +the direction of magnetic field, that mostly concen- +trates around 86◦ with respect to the x-axis in MSO +(as indicated by the black line in Fig.5a), into account, +the Alfv´en speed along the x-axis approaches zero. The +sound speed, vcs, is overall larger than the Alfv´en speed, +and is rarely smaller than 30 km s−1. The fast-mode +magnetoacoustic speed along the x-axis, vf,x, is overall +larger than both the Alfv´en speed and the sound speed +with its peak around 60 km s−1. +Since the solar wind propagates along the x-axis and +vf,x is the fastest among these characteristic speeds, +the fast-mode Mach number in x-axis, Mf,x, is calcu- +lated. The black line in Fig.5b shows the median value +of Mf,x within one minute during the period of interest. +We can read from the line that the dynamic range of +Mf,x is about 7, i.e., ranging from about 2 to 9 with the +peak at about 6.2. We further exam the inhomogeneity +of Mf,x by calculating the difference between the max- +imum and minimum values of Mf,x within a given time +scale varying from one minute to 29 minutes as shown +by the color-coded thin lines in Figure 5b. Each line +presents the distribution of the difference or the range +of the Mf,x in a given time scale. We can see that these +distributions extend toward large values with increas- +ing time scales, suggesting the enhancement of the in- +homogeneity in terms of Mf,x. Then we determine the +middle value of Mf,x for each distribution, at which the +distribution is divided equally, and define the inhomo- +geneity as the ratio of the middle value to the dynamic +range of Mf,x. The dependence of the inhomogeneity +on the time scale is plotted in Figure 5c. If considering +that inhomogeneity of 0.1 is an acceptable level for a +stale solar wind, we may conclude that the time scale of +stable solar wind is about 2 minutes in terms of Mf,x. +The similar analysis is applied on the solar wind dy- +namic pressure, pd, as shown in Figure 5d and e. The +dynamic pressure also shows a single-peak distribution +ranging from about 0.01 nPa to nearly 2.5 nPa with +the peak around 0.3 nPa. +The inhomogeneity of pd +also increases as the time scale increases. By setting +the dynamic range of pd to be 2, we find that the in- +homogeneity is less than 0.1 even at the time scale of +30 minutes. This suggests that Mf,x is much more dy- +namic than pd in the upstream of Martian BS. +Based on the above analysis, we search the BS- +crossing pairs of Tianwen-1 and MAVEN within 2 min- +utes in the period of interest, and plot the results in +Figure 4b. A total of 5 pairs are found. A first impres- +sion is that the global shape of the BS is slightly more +flattened than the model. +But this just reflects the +south-north asymmetry of the Martian BS[e.g., Edberg +et al., 2008; Dubinin et al., 2008], as Tianwen-1 orbiter +crossed the southern flank of the BS, while MAVEN +crossed the BS at low latitude on the northern hemi- +sphere. +Figures 6–10 show the 5 pairs of the BS crossings. +Around 05:05 UT on November 19, both spacecraft +crossed the BS from the solar wind into magnetosheath, +when Tianwen-1 orbiter was far above the southern pole +of Mars and MAVEN was close to the BS nose (see +Fig.6). The magnetic fields in the solar wind measured +before they entered magnetosheath look quite similar. +Between 04:56 and 05:00 UT, we can see the large vari- +ation patterns in the three components of the magnetic +fields without a significant change in the total mag- +nitude, which are probably the features of an Alfv´en +wave. This featured structure arrived at Tianwen-1 or- +biter later than MAVEN by nearly 30 s, which was +roughly the time spent by solar wind travelling from +MAVEN to Tianwen-1. +The magnetic fields measured by the two spacecraft +after they crossed the BS show different patterns. From +the first panel of Figure 6, it seems that the Tianwen-1 +orbiter crossed the BS three times within 7 minutes, +finally returned back to solar wind at 05:11:30 UT, and +at about 05:17:20 UT, the orbiter started to cross the +BS again. Not like Tianwen-1, MAVEN stayed in the +magnetosheath after the crossing except one turning +back at around 05:07:30 UT. It is similar to the case +shown in Figure 2h and 3h, of which the Tianwen-1 +orbiter crossed the BS three times in 7 minutes, but +MAVEN had only one clear crossing. These phenom- +ena suggest that the Martian BS is very dynamic with +the time scale even less than one minute, and the BS +flank is more dynamic than the nose during this time +period. Such multiple-crossings in minutes deserve fur- +ther study, especially for events with Tianwen-1 orbiter +crossing the BS and MAVEN staying in the solar wind +to monitor the upstream condition. +The second pair of the BS crossings is found around +13:19 UT on December 11 as shown in Figure 7. Both +spacecraft were crossing the BS from magnetosheath to +the solar wind. We can see a sharp jump at 13:18:30 UT +in the MOMAG data, and a sharp jump at 13:20:00 UT +in the MAVEN/MAG data. In both data, we also can +find a large dip in the total magnetic field strength. It is +hard to determine if they are correlated. The third pair +was around 07:14 UT on December 12 with one crossing +from the solar wind into magnetosheath and the other +from magnetosheath into the solar wind (Fig.8). The +fourth pair was around 19:43 UT on December 16. Both +spacecraft travelled from the magnetosheath into the +solar wind (Fig.9). The last pair is found around 02:45 +UT on December 17. +Both spacecraft also travelled +from the magnetosheath into the solar wind (Fig.10). +If looking at the total strengths of the magnetic fields +in the magnetosheath for all the BS crossing pairs, we + +X - 8 +WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +Figure 5. +Characteristic speeds and inhomogeneities of solar wind fast Mach number and dynamic pressure +based on MAVEN data. Panel (a) shows the distributions of the minutely-averaged Alfv´en speed (blue), sound +speed (orange) and the fast-mode magnetoacoustic speed along the x direction in MSO coordinates (green) during +November 13 and December 31, 2021. The black line gives the distribution of the absolute value of the angle +between the minutely-averaged magnetic field vector and the x direction. Panel (b) shows the distributions of +the range of the fast-mode Mach number along the x direction, Mf,x, within the various time scales, and the +distribution of the minutely-averaged Mf,x (see the main text for details). Panel (c) gives the inhomogeneity as +a function of time scale. The definition of inhomogeneity here can be found in the main text. Panel (d) and (e) +are for solar dynamic pressure with the same arrangement as Panel (b) and (c). +may find they are more or less similar no matter how +large in distance the two spacecraft are apart, suggest- +ing the consistency of the global magnetic structure +surrounding Mars at the level of large scale. + +Cone angle(°) +0 +30 +60 +90 +120 +150 +180 +a +0.05 +VA +Vcs +Vfast,x +0.04 +N +0.03 +Fraction +0.02 +0.01 +0.00 +50 +100 +150 +200 +250 +300 +0 +Characteristic speed (km/s)1 min +16 min +0.07 +17 min +2 min +3 min +18 min +0.15 +geneity +4 min +19 min +0.06: +5 min +20 min +21 min +6 min +0.10 +9 +22 min +7 min +8 min +23 min +0.05 - +9 min +24 min +105 +25 min +10 min +26 min +0 0.04 - +11 min +12 min +27 min +Fractic +13 min +28 min +5 +10 +15 +20 +25 +14 min +29 min +0 +30 +0.03 +15 min +1-min median +Time scale (min) +0.02 +0.01 +0.00 - +0 +6 +8 +2 +4 +10 +Mf,xd +0.10 +1 min +16 min +e +17 min +2 min +0.14 +3 min +18 min +0.08 +4 min +19 min +5 min +20 min +0.12 +0.06 +21 min +6 min +22 min +7 min +8 min +23 min +0.04 +0.10 +9 min +24 min +10 min +25 min +0.02 +26 min +11 min +0.08 +12 min +27 min +ctic +13 min +0.00 +28 min +Fra +14 min +5 +10 +15 +20 +25 +30 +29 min +0 +0.06 +15 min + 1-min median +Time scale (min) +0.04 +0.02 +0.00 +0.5 +1.5 +0.0 +1.0 +2.0 +2.5 +Pα (nPa)WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +X - 9 +Figure 6. The simultaneous bow shock crossing around 05:05 UT on November 19, 2021. From the top +to bottom, the panels display the magnetic field measured by MOMAG, the magnetic field measured by +MAVEN/MAG, and the solar wind velocity, the number density and temperature of ions measured by +MAVEN/SWIA, and the orbits of Tianwen-1’s orbiter and MAVEN viewed from different angles. The +purple arrows in the first two panels indicate the times of the bow shock crossings, and the markers in +the panels on the bottom indicate the positions of the crossings. + +20 +Bx +(lu) +By +MMi +10 +. Mag +Bz +M +[B| +0 +TW1 +-10 +Bx +MVN Mag (nT) +20 +B +Bz +W +B +0 +-20 +360 +Ivl +400 +270 +MVN v (km s +Ciock angie +Cone angle +180 - +e +Angl +300 +90 +W +0 +3 +15 +Number density +Temperature +7 +10 +IM +5 +04:55 +05:00 +05:05 +05:10 +05:15 +05:20 +05:25 +Start time 2021-11-19 04:55:00 UT +(hr) +4 +(RM) +1 +(RM) +0 +(RM) +(RM) +0 +0.25 +MVN +Iyz, MSo' +0.00 +2 +0 +ZMSO +YMSO +ZMSO +-2 +TW1 +0.25 +1 +0 +0.00 +2.5 +-2.5 +2.5 +2 +-2.5 +2.5 +0.0 +0.0 +0 +0.0 +XMSO' (RM) +XMSO (RM) +XMSO (RM) +yMSO (RM)X - 10 +WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +Figure 7. The simultaneous bow shock crossing around 13:19 UT on December 11, 2021. + +20 +Bx +(lu) +By +10 + Mag ( +Bz +[BI +0 +TW1 +10 +Bx +20 +(nT) +By +MVN Mag ( +10 +Bz +BI +0 +-10 +360 +W +MVN v (km s +350 +270 +Ciock angie +(。) +Cone angle +180 +Angl +300 +90 +250 +0 +15 +2.0 +Number density +(x106 +Temperature +10 +1.5 +Tion +1.0 +MVN +5 +>sS>S +13:05 +13:10 +13:15 +13:20 +13:25 +13:30 +13:35 +Start time 2021-12-11 13:05:00 UT +(hr) +(RM) +(Wy) OsWz +(Rm) +D +(RM) +0.25 +MVN +0 +0 +0 +2 +0.00 +YMSO +ZMSO +TW1 +2 +0.25 +0 +-2 +0.00 +2.5 +-2.5 +2.5 +0.0 +0.0 +0 +1 +-2.5 +0.0 +2.5 +-1 +XMSO' (RM) +XMSO (RM) +XMSO (RM) +YMSO (RM)WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +X - 11 +Figure 8. The simultaneous bow shock crossing around 07:14 UT on December 12, 2021. + +20 +Bx +(lu) +10 +BB +Mag +0 +TW1 +-10 +Bx +20 +(nT) +By +MVN Mag ( +10 +Bz +[BI +0 +w +iwwy +10 +360 +>>> +500 +Iv] +Ciock angie +270 +(。) +450 +Cone angle +180 +Angl +400 +90 +350 +0 +3 +K +Number density +Temperature +20 +2 +10 +07:05 +07:10 +07:20 +07:25 +07:30 +07:15 +07:35 +Start time 2021-12-12 07:05:00 UT +(hr) +2 +(RM) +(Rm) +(RM) +(Rm) +0.25 +MVN +0 +0 +2 +0.00 +ZMSO +ZMSO +W1 +0.25 +-2 +0 +0.00 +2 +0.0 +2.5 +-2 +2.5 +0 +-2.5 +0 +2 +0.0 +XMSO' (RM) +XMSO (RM) +XMSO (RM) +YMSO (RM)X - 12 +WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +Figure 9. The simultaneous bow shock crossing around 19:43 UT on December 16, 2021. + +20 +Bx +TW1 Mag (nT) +By +Bz +10 +[BI +0 +MAA +hM +30 +Bx +By +20 +Bz +[BI +10 +0 +360 +500 +270 +Ciock angie +'s +LNW +。） +(km +Cone angle +Angle +400 +180 +MVN V ( +>>> +90 +300 +W +0 +Number density +Temperature +15 +3 +M +2 +10 +1 +5 +19:30 +19:35 +19:40 +19:45 +19:50 +19:55 +20:00 +Start time 2021-12-16 19:30:00 UT +(hr) +(RM) +(RM) +(RM) +0.25 +MVN +1 +7 +0 +2 +0.00 +ZMSO +0 +YMSO +ZMSO +0 +TW1 +0.25 +-2 +: +0 +0.00 +0 +2 +0 +2 +-2.5 +0.0 +0 +2 +XMSO° (RM) +XMSO (RM) +XMSO (RM) +YMSO (RM)WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +X - 13 +Figure 10. The simultaneous bow shock crossing around 02:44 UT on December 17, 2021. + +Bx +TW1 Mag (nT) +By +10 +Bz +[B +20 +Bx +(lu) +By +10 +Mag +Bz +IB +MVN I +-10 +360 +500 +270 +Ciock angie +(。) +(km +Cone angle +Angle ( +400 +180 +MVN V ( +90 +300 +0 +4 +Number density +Temperature +10 +3 +2 +5 +L +<>?> +02:30 +02:35 +02:40 +02:45 +02:50 +02:55 +03:00 +Start time 2021-12-17 02:30:00 UT +(hr) +(Rm) +(Rm) +(RM) +(RM) +0.25 +MVN +1 +0 +1 +2 +ryz, Mso' +0.00 +ZMSO +yMSO +ZMSO +0 +0 +TW1 +0.25 +1 +2 +1 +0 +0.00 +0 +2 +-2 +0 +2 +-2 +0 +0 +XMSO° (RM) +XMSO (RM) +XMSO (RM) +YMSO (RM)X - 14 +WANG ET AL.: MARS ORBITER MAGNETOMETER OF TIANWEN-1 +4. Summary +we have presented the in-flight performance and first +results of Tianwen-1/MOMAG with the focus on the +most notable structure — Martian BS. Based on the +first one and a half months’ data, we identified 158 clear +BS crossings, whose locations are consistent with the +BS model in statistics. The simultaneous BS crossings +of the Tianwen-1 and MAVEN verified the south-north +asymmetry of the BS, and also showed the similarity +of magnetic field profiles from the two spacecraft. The +first pair of the simultaneous BS crossings along with +the BS crossing case on December 30 suggests that the +BS is probably more dynamic at flank than near the +nose. +By comparing with the MAVEN observations, +we also found similar structures propagating with the +solar wind from MAVEN to the Tianwen-1 orbiter. We +conclude that MOMAG shows an excellent performance +and provides accurate measurements of magnetic field +vectors. Now MOMAG has scanned the magnetic field +in the MPR, magnetosheath and solar wind near the +dawn-dusk side. These measurements along with the +MAVEN data will help us better understand the plasma +environment surrounding Mars. +Acknowledgments. We acknowledge the use of the data +from the MAG and SWIA onboard MAVEN spacecraft, which are +obtained from NASA Planetary Data System (https://pds-ppi. +igpp.ucla.edu/). One may apply for the Tianwen-1/MOMAG +data at CNSA Data Release System (http://202.106.152.98: +8081/marsdata/) or can just download the data used in this +paper from the official website of the MOMAG team (http: +//space.ustc.edu.cn/dreams/tw1_momag/). +The work is sup- +port by the NSFC (Grant Nos 42130204 and 42188101) and the +Strategic Priority Program of the Chinese Academy of Sciences +(Grant No. XDB41000000). Y.W. is particularly grateful to the +support of the Tencent Foundation. +References +Bertucci, C., et al., MGS MAG/ER observations at the mag- +netic pileup boundary of mars: +Draping enhancement and +low frequency waves, Adv. Space Res., 33, 1938–1944, doi: +10.1016/j.asr.2003.04.054, 2004. +Brain, D. A., et al., On the origin of aurorae on Mars, Geophys. +Res. 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Sci., submitted, +2023. + diff --git a/jdAyT4oBgHgl3EQfx_mp/content/tmp_files/load_file.txt b/jdAyT4oBgHgl3EQfx_mp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac8c5dc1c69b4d33eae873a24577b26fe4581ec6 --- /dev/null +++ b/jdAyT4oBgHgl3EQfx_mp/content/tmp_files/load_file.txt @@ -0,0 +1,685 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf,len=684 +page_content='EARTH AND PLANETARY PHYSICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' XXXX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' DOI:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The Mars Orbiter Magnetometer of Tianwen-1: In-flight Performance and First Science Results Yuming Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='∗ Tielong Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='3 Guoqiang Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='4 Sudong Xiao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='4 Zhuxuan Zou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Long Cheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Zonghao Pan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Kai Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Xinjun Hao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Yiren Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Manming Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Zhoubin Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 Wei Yan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 Zhenpeng Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Zhiyong Wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Chenglong Shen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 Yutian Chi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='6 Mengjiao Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='6 Jingnan Guo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2 and Yang Du7 1 School of Earth and Space Sciences/Deep Space Exploration Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' China 2 CAS Center for Excellence in Comparative Planetology/CAS Key Laboratory of Geospace Environment/Mengcheng National Geophysical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' China 3 Space Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Austrian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Graz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Austria 4 Institute of Space Science and Applied Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Harbin Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' China 5 National Astronomical Observatories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' China 6 Institute of Deep Space Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Deep Space Exploration Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' China 7 Shanghai Institute of Satellite Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' China ∗ Corresponding author,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Email: ymwang@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='cn Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Mars Orbiter MAGnetometer (MOMAG) is a scientific instrument onboard the orbiter of China’s first mission for Mars — Tianwen-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' It started to routinely mea- sure the magnetic field from the solar wind to magnetic pile-up region surrounding Mars since November 13, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Here we present its in-flight performance and first science re- sults based on the first one and a half months’ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' By comparing with the magnetic field data in the solar wind from the Mars Atmosphere and Volatile EvolutioN (MAVEN), the magnetic field by MOMAG is at the same level in magnitude, and the same mag- netic structures with the similar variations in three components could be found in MO- MAG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In the first one and a half months, we recognize 158 clear bow shock (BS) crossings from MOMAG data, whose locations statistically match well with the mod- eled average BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We also identify 5 pairs of simultaneous BS crossings of the Tianwen- 1’s orbiter and MAVEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' These BS crossings confirm the global shape of modeled BS as well as the south-north asymmetry of the Martian BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Two presented cases in this paper suggest that the BS is probably more dynamic at flank than near the nose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' So far, MOMAG performs well, and provides accurate magnetic field vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' MOMAG is continuously scanning the magnetic field surrounding Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' These measurements com- plemented by observations from MAVEN will undoubtedly advance our understanding of the plasma environment of Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Introduction Tianwen-1 is the first mission of China to explore and study Mars from its space environment to the surface [Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' It consists of an orbiter, a lander and a rover, called Zhurong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Mars Orbiter MAG- netometer (MOMAG) is one of the scientific instruments onboard the orbiter[Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' It investigates the magnetic field environment of Mars by measuring the local vector magnetic field, and therefore provides some key infor- mation for the understanding of the history and evolution of Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The magnetic field surrounding Mars has two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' One is the dynamic magnetic field resulted from the cou- pling between the solar wind and the Martian ionosphere, and the other is the static crustal magnetic field of Mars itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Since Mars has no global intrinsic magnetic field, the solar wind carrying interplanetary magnetic field directly in- teracts with Martian ionosphere, and forms the bow shock (BS) and induced magnetosphere, which consists of mag- netic pileup region (MPR) and wake region[e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', Bertucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Brain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Between the BS and MPR, there is magnetosheath separated from MPR with the mag- netic pileup boundary (MPB)[e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', Mazelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The magnetic field in these regions influenced by solar wind is highly dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Escape of ions in Martian atmosphere is one of the core science issues of Tianwen-1, and is closely related to the magnetic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' For example, the strong static crustal magnetic field on the southern hemisphere may reach upto a high altitude and reconnect with interplanetary mag- netic field, causing the escape of ions[Brain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2015], just like the behavior of Venus[Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Besides, var- ious waves in the ionosphere may heat particles, causing ion outflow[Ergun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2006], and when these heated ions transport outside the MPB, they will interact with magnetic field carried by solar wind stream to further generate ion cyclotron wave, boosting the escape of the ions[Russell and Blanco-Cano, 2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The escape rate during storm times will be one to two orders higher than that during quite time[Jakosky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2015b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00677v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='EP] 2 Jan 2023 X - 2 WANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 Tianwen-1 orbiter was running on a highly eccentric or- bit with the periapsis of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='08 Mars radii (rm) and the apoapsis of about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='17 rm, and the orbital plane is highly inclined during November and December in 2021, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' During that period, the periapsis was right above the northern pole of Mars, the apoapsis far above the southern pole in the solar wind, and the orbital period was about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='8 hr, with about 50% – 75% of time in so- lar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Thus, MOMAG mainly measured the magnetic field from solar wind to the MPR on the dawn-dusk side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Later, the inclination angle of the orbit will decrease to al- low the orbiter detect the wake region of Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' These data will help us understand the structure and evolution of Mar- tian magnetic field environment and provide clues for ion escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Since the Mars Atmosphere and Volatile EvolutioN (MAVEN, Jakosky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 2015a), which also carries a mag- netometer (MAG, Connerney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 2015), is still working, the successful operation of MOMAG will make us for the first time to study Martian magnetic field environment from two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In this paper, we show and analyze the data during November 13 – December 31, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In Section 2, we will describe the basic information and the current status of MO- MAG, and present some measured magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Then we show the first results of MOMAG about Martian BS with the comparison with MAVEN/MAG data in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In the last section, we summarize the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In-flight Calibration and Performance MOMAG contains two sensors mounted on a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='19 m long boom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The outer sensor accommodates at the top of the boom and the inner sensor is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='9 m away (see Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 2020 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Since the orbiter of Tianwen-1 does not have magnetic cleanliness control, the boom is actually not long enough to avoid the contaminations of the magnetic field from the orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Thus, how to remove the magnetic interference based on the magnetic fields measured by the two separated sensors become pivotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The procedure of the mitigation of the orbiter’s mag- netic field generally includes two steps, which is similar to the procedure applied on the magnetometer of Venus Ex- press[Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Pope et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The first step is to remove the magnetic interference due to the opera- tions of instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Such interferences behave as jumps in the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' If a real discontinuity in the solar wind passes the spacecraft, the amplitudes of the jump at the two sensors should be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' However, since the distances of the two sensors from the instrument are dif- ferent, an artificial jump will show different amplitudes at the two sensors, and therefore could be distinguished from real jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' For these artificial jumps, we use the method of Pope et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' [2011] to remove them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The second step is to remove the static magnetic field of the orbiter and cor- rect the offset of the fluxgate magnetometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' This step is mainly based on the property of Alfv´enic waves, of which the magnetic field almost rotates in a plane without the change of magnitude[Wang and Pan, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We process the raw data of MOMAG to the level 2 (or level C in China’s convention) data for scientific use through these steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Based on the data of the first several months since November 13, 2021, when MOMAG started to formally operate, we iterate the procedure and reach the first ver- sion of the level 2 data, that have been released at the Planet Exploration Program Scientific Data Release Sys- tem (http://202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='98:8081/marsdata/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The data used in this paper and in our forthcoming papers will also be put on the official website of the MOMAG team at University of Science and Technology of China (USTC, http://space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='cn/dreams/tw1_momag/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' A com- plete description of the in-flight calibration procedure as well as the demonstration of the reliability of the calibrated data is given in the separated paper by Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' [2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Figure 2a shows the magnetic field in the Mars-centered Solar Orbital (MSO) system measured by MOMAG dur- ing 01:00 – 09:00 UT on 2021 December 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The orbiter was running in the magnetosheath before 02:55 UT, and at around 03:01:40 UT the orbiter crossed the BS where the amplitude of the magnetic field discontinuity was more than 10 nT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' After about four hours, the orbiter crossed the BS again where the amplitude of the magnetic field dis- continuity was about 16 nT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Around 08:20 UT, the orbiter even crossed the MPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The BS during the second crossing was obviously stronger than that during the first crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The reason is that the BS was compressed during the second crossing, which can be seen from Figure 2d–g that the two BS crossings were on the south, the first cross- ing was outside and further away from the modeled averaged BS [Edberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2008] than the second BS crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' If we look into the details of the first BS crossing as shown in Figure 2h, it could be found that the orbiter crossed out the BS around 02:56:30 UT and crossed in again at 02:57:45 UT before it finally entered the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The magnetic field changes during these preceding crossings suggest that the BS was slightly stronger than the BS at 03:01:40 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' This could also be explained as the compression of the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' During these crossings, the orbiter was moving away from Mars as the indicated by the color-coded orbit in Figure 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The locations of the preceding crossings were closer to Mars than that of the final crossing at 03:01:40 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The magnetic fields in the solar wind stayed around 9 – 10 nT, and were much less fluctuated than those in the magnetosheath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Figure 2b displays the power spectral den- sity of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' It is generated by using a 10-min window and 1-min running step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' It can been seen that the solar wind was indeed quiet except at very low frequency, whereas in the magnetosheath the magnetic fluctuation was enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Behind the bow shock, weak magnetic waves right below proton gyro-frequency appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In the solar wind, a small structure can be found between 05:15 and 06:35 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Though the total magnetic field only slightly en- hanced, the most notable change occurred for By, which decreased from about 4 nT to zero twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' For comparison, Figure 3 shows the MAVEN measure- ments of the magnetic field and solar wind during 04:00 – 07:00 UT on the same day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' MAVEN had a quite dif- ferent orbit, of which the orbital period was shorter than 4 hours (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='3c–g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Within one hour, it crossed the BS twice, but the positions of the crossings were both closer to the shock nose than those of Tianwen-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Since the two crossings stay close to the same modeled BS, suggesting the solar wind conditions during the two crossings are almost the same, the amplitudes of their magnetic field discontinu- ities were similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We also show the detailed BS crossings in Figure 3h and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' No multiple BS crossings happened at MAVEN, probably suggesting the different behavior of BS at different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Though the solar wind region that MAVEN detected was far away from that by Tianwen-1, a similar structure could be found between 05:20 and 05:55 UT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='3a), which cor- responds to the first dip recorded in MOMAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' During that time, the solar wind speed was about 310 km s−1 and the number density of protons was about 7 cm−3 (see the blue and red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Different from MOMAG data, the magnetic field at MAVEN was highly fluctuated with a no- table frequency at the proton gyro frequency (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' This might be because MAVEN closer to Mars than Tianwen-1 when they fly in the solar wind, and the particles escaping from Martian atmosphere more easily interact with the so- lar wind and interplanetary magnetic field to generate such fluctuations at a closer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' However, according to the statistical study of MAVEN data [Ruhunusiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2017], such a pattern seems not to be evident and needs further validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' WANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 X - 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The orbits of Tianwen-1’s orbiter (solid lines) and MAVEN (dashed lines) during November 13 – December 31, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The modeled Martian bow shock and MPB [Edberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2008] are indicated by thick and thin black lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' a 4- 2021-11-16 TW1 2021-11-16 MVN 2021-11-26 TW1 2021-11-26 MVN 2021-12-06 TW1 3 - 2021-12-06 MVN 2021-12-16 TW1 2021-12-16 MVN ryz, MSo (RM) 2021-12-26 TW1 2021-12-26 MVN 2 1 0 2 1 0 1 2 3 4 3 XMSO (RM) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 4 2 b d C 1 2 2 0 (RM) 0 1 0 ZMSO 2 2 2 3 4 4 0 2 2 2 0 0 2 XMSO (RM) XMSO (RM) YMSO (RM)X - 4 WANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The magnetic field measured by MOMAG during 01:00 – 09:00 UT on December 30, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Panel (a) shows the three components of the magnetic field in MSO coordinates with the total magnitude overplotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Two purple arrows mark the crossings of the bow shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Panel (b) shows the power spectral density of the magnetic field fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The green line indicates the proton’s gyro frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Panel (c) shows the height of the Tianwen-1 orbiter, and Panel (d) – (g) display its orbit in the MSO coordinates during the period of interest, in which the two circled dots mark the positions of the bow shock crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Note that Panel (d) is in aberrated MSO coordinates with the modeled BS and MPB indicated by black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Panel (h) and (i) show the two BS crossings, respectively, in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 40 a Bx B朗 30 IBI 20 (lu) 10 Mag 11 0 10 20 5 p gyro 4 10-1 (ZH) 3 Freq 2 10-2 1 10-3 0 10000 C Height (km) 7500 5000 2500 0 01:00 02:00 03:00 04:00 05:00 06:00 07:00 08:00 09:00 Start time 2021-12-30 01:00:00 UT (hr) 2 4 d g e 9 (Rm) (RM) (RM) 0 elapse 5 0 2 ZMSO yMSO ZMSO 2 2 Time 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='4 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='4 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content="5 XMSO' (RM) XMSO (RM) XMSO (RM) yMSo (Rm)30 h Bx By (lu) 20 Bz Mag [BI 10 0 02:59 03:01 03:05 02:55 02:56 02:57 02:58 03:00 03:02 03:03 03:04 Start time 2021-12-30 02:55:00 UT30 ！" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Bx Mag (nT) 20 Bz IBI M 10 M 0 07:00 07:01 07:02 07:03 07:04 07:05 07:06 07:07 07:08 07:09 07:10 Start time 2021-12-30 07:00:00 UTWANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 X - 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The magnetic field and solar wind plasma measured by MAVEN during the same period as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In Panel (c) the solar wind speed and number density of ions are presented with the blue and red lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The arrangements of other panels are the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' a Bx 40 RBI IBI 30 20 (lu) Mag 10 0 10 5 Log PSD (nT2/Hz) p gryo 4 10-1 Freq (Hz) ３ 2 1 10-3 0 40 c 4000 Height 300 (km) 3 Ivl 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 3000 Number density cm s Height ( km 2000 200 uo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='u > 1000 10 100 0 30 04:00 30 04:30 30 05:30 30 06:00 30 06:30 30 05:00 30 07:00 Start time 2021-12-30 04:00:00 UT elapse (hr) 2 2 1 (Rm) 9 (RM) (Rm) (RM) 2 0 ryz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=" MSo' 2 ZMSO yMSO ZMSO 0 1 0 0 1 0 2 0 0 0 1 XMSO' (RM) XMSO (RM) XMSO (RM) YMSO (RM)30 h Bx By (lu) 20 Bz MM VM/A Mag [BI 10 LMAA W 05:00 05:01 05:02 05:03 05:04 05:05 05:06 05:07 05:08 05:09 05:10 Start time 2021-12-30 05:00:00 UT30 Bx BV (lu) 20 Bz Mag [BI 10 W 0 05:55 05:56 05:57 05:58 05:59 06:00 06:01 06:02 06:03 06:04 06:05 Start time 2021-12-30 05:55:00 UTX - 6 WANG ET AL." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The bow shock (BS) crossings during the period of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In Panel (a), the red asterisks mean the BS crossings of Tianwen-1’s orbiter, and the blue dots the BS crossings of MAVEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The modeled average BS [Edberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2008] is displayed by the black line, and the other dashed lines display the BS when the uncertainties of the BS model parameters are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Panel (b) shows the 5 pairs of simultaneous (within 2 minutes) BS crossings of the Tianwen-1’s orbiter and MAVEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Each pair is indicated with the same colored dots, but the dots of Tianwen-1 are enclosed by black circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The BS crossing time of the Tianwen-1’s orbiter is given in the upper-right corner followed by a time interval with the positive value meaning the later BS crossing of MAVEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' All these data are presented in the aberrated MSO coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Bow Shock Crossings Bow shock is one of the notable features in the Mar- tian space environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Its shape may reflect the up- stream solar wind conditions and solar EUV inten- sity and the interaction processes between solar wind and Martian atmosphere [Mazelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Ram- stad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Thus, studying Martian BS is our first choice to show the science re- sults of MOMAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' During 2021 November 13 – Decem- ber 31, we recognize 158 BS crossings from MOMAG data by manually checking the magnetic field strength variation and the fluctuation level which is measured by the standard deviation of magnetic field within one minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In principle, there should be more crossings, but Tianwen-1 mostly crossed the flank of the BS where the characteristic of a shock may be too weak to be rec- ognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' During the same period, we recognize 454 BS crossings from MAVEN/MAG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Figure 4a shows all of the BS crossings in the aber- rated MSO coordinates (MSO coordinates are rotated by 4◦ about the z-axis to reduce the effect of the Mars orbital motion on the solar wind flow direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Since the spatial coverage of the crossings is not wide enough, we do not try to fit these crossings to find the best-fit BS model, but instead to compare with the previously established BS model [Edberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We can find in the figure that the crossings statistically match the model fairly well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Martian BS position and global shape were derived from many single crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Now we can check this based on the joint magnetic field observations from Tianwen-1/MOMAG and MAVEN/MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' By assum- ing that the BS remains unchanged within 2 minutes, we use two crossings of Tianwen-1 and MAVEN within 2 minutes to exam the BS global shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We choose 2 minutes because the upstream solar wind conditions that determine the BS position and shape, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', the fast- mode Mach number and dynamic pressure, are usually stable within this time-scale as revealed by the follow- ing analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Figure 5a shows the characteristic speeds in the solar wind, which are calculated every minute based on the MAVEN/SWIA [Halekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=", 2017] measurements of a TW1 4 MVN 3 2 2 0 1 2 3 1 XMSO' (RM)4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 b 2021-11-19 05:05:40,-30s 2021-12-11 13:18:29, 90s 2021-12-12 07:14:39, -90s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2021-12-16 19:43:35, -90s 2021-12-17 02:44:42, 8s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 (RM) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content="5 ryz, MSo' ( 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 1 2 0 1 2 3 XMSO° (Rm)WANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 X - 7 the solar wind velocity, ion density and temperature and MAVEN/MAG measurements of magnetic field during November 13 – December 31, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The Alfv´en speed, vA, ranges from almost zero to more than 100 km s−1 with the peak around 30 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Since Alfv´en wave propagates along the magnetic field, if we take the direction of magnetic field, that mostly concen- trates around 86◦ with respect to the x-axis in MSO (as indicated by the black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5a), into account, the Alfv´en speed along the x-axis approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The sound speed, vcs, is overall larger than the Alfv´en speed, and is rarely smaller than 30 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The fast-mode magnetoacoustic speed along the x-axis, vf,x, is overall larger than both the Alfv´en speed and the sound speed with its peak around 60 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Since the solar wind propagates along the x-axis and vf,x is the fastest among these characteristic speeds, the fast-mode Mach number in x-axis, Mf,x, is calcu- lated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5b shows the median value of Mf,x within one minute during the period of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We can read from the line that the dynamic range of Mf,x is about 7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', ranging from about 2 to 9 with the peak at about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We further exam the inhomogeneity of Mf,x by calculating the difference between the max- imum and minimum values of Mf,x within a given time scale varying from one minute to 29 minutes as shown by the color-coded thin lines in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Each line presents the distribution of the difference or the range of the Mf,x in a given time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We can see that these distributions extend toward large values with increas- ing time scales, suggesting the enhancement of the in- homogeneity in terms of Mf,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Then we determine the middle value of Mf,x for each distribution, at which the distribution is divided equally, and define the inhomo- geneity as the ratio of the middle value to the dynamic range of Mf,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The dependence of the inhomogeneity on the time scale is plotted in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' If considering that inhomogeneity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1 is an acceptable level for a stale solar wind, we may conclude that the time scale of stable solar wind is about 2 minutes in terms of Mf,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The similar analysis is applied on the solar wind dy- namic pressure, pd, as shown in Figure 5d and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The dynamic pressure also shows a single-peak distribution ranging from about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='01 nPa to nearly 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 nPa with the peak around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='3 nPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The inhomogeneity of pd also increases as the time scale increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' By setting the dynamic range of pd to be 2, we find that the in- homogeneity is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='1 even at the time scale of 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' This suggests that Mf,x is much more dy- namic than pd in the upstream of Martian BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Based on the above analysis, we search the BS- crossing pairs of Tianwen-1 and MAVEN within 2 min- utes in the period of interest, and plot the results in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' A total of 5 pairs are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' A first impres- sion is that the global shape of the BS is slightly more flattened than the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' But this just reflects the south-north asymmetry of the Martian BS[e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', Edberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Dubinin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', 2008], as Tianwen-1 orbiter crossed the southern flank of the BS, while MAVEN crossed the BS at low latitude on the northern hemi- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Figures 6–10 show the 5 pairs of the BS crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Around 05:05 UT on November 19, both spacecraft crossed the BS from the solar wind into magnetosheath, when Tianwen-1 orbiter was far above the southern pole of Mars and MAVEN was close to the BS nose (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The magnetic fields in the solar wind measured before they entered magnetosheath look quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Between 04:56 and 05:00 UT, we can see the large vari- ation patterns in the three components of the magnetic fields without a significant change in the total mag- nitude, which are probably the features of an Alfv´en wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' This featured structure arrived at Tianwen-1 or- biter later than MAVEN by nearly 30 s, which was roughly the time spent by solar wind travelling from MAVEN to Tianwen-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The magnetic fields measured by the two spacecraft after they crossed the BS show different patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' From the first panel of Figure 6, it seems that the Tianwen-1 orbiter crossed the BS three times within 7 minutes, finally returned back to solar wind at 05:11:30 UT, and at about 05:17:20 UT, the orbiter started to cross the BS again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Not like Tianwen-1, MAVEN stayed in the magnetosheath after the crossing except one turning back at around 05:07:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' It is similar to the case shown in Figure 2h and 3h, of which the Tianwen-1 orbiter crossed the BS three times in 7 minutes, but MAVEN had only one clear crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' These phenom- ena suggest that the Martian BS is very dynamic with the time scale even less than one minute, and the BS flank is more dynamic than the nose during this time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Such multiple-crossings in minutes deserve fur- ther study, especially for events with Tianwen-1 orbiter crossing the BS and MAVEN staying in the solar wind to monitor the upstream condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The second pair of the BS crossings is found around 13:19 UT on December 11 as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Both spacecraft were crossing the BS from magnetosheath to the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We can see a sharp jump at 13:18:30 UT in the MOMAG data, and a sharp jump at 13:20:00 UT in the MAVEN/MAG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' In both data, we also can find a large dip in the total magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' It is hard to determine if they are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The third pair was around 07:14 UT on December 12 with one crossing from the solar wind into magnetosheath and the other from magnetosheath into the solar wind (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The fourth pair was around 19:43 UT on December 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Both spacecraft travelled from the magnetosheath into the solar wind (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The last pair is found around 02:45 UT on December 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Both spacecraft also travelled from the magnetosheath into the solar wind (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' If looking at the total strengths of the magnetic fields in the magnetosheath for all the BS crossing pairs, we X - 8 WANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Characteristic speeds and inhomogeneities of solar wind fast Mach number and dynamic pressure based on MAVEN data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Panel (a) shows the distributions of the minutely-averaged Alfv´en speed (blue), sound speed (orange) and the fast-mode magnetoacoustic speed along the x direction in MSO coordinates (green) during November 13 and December 31, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The black line gives the distribution of the absolute value of the angle between the minutely-averaged magnetic field vector and the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Panel (b) shows the distributions of the range of the fast-mode Mach number along the x direction, Mf,x, within the various time scales, and the distribution of the minutely-averaged Mf,x (see the main text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Panel (c) gives the inhomogeneity as a function of time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The definition of inhomogeneity here can be found in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Panel (d) and (e) are for solar dynamic pressure with the same arrangement as Panel (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' may find they are more or less similar no matter how large in distance the two spacecraft are apart, suggest- ing the consistency of the global magnetic structure surrounding Mars at the level of large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Cone angle(°) 0 30 60 90 120 150 180 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='05 VA Vcs Vfast,x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='04 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='03 Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 50 100 150 200 250 300 0 Characteristic speed (km/s)1 min 16 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='07 17 min 2 min 3 min 18 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='15 geneity 4 min 19 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='06: 5 min 20 min 21 min 6 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='10 9 22 min 7 min 8 min 23 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='05 - 9 min 24 min 105 25 min 10 min 26 min 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='04 - 11 min 12 min 27 min Fractic 13 min 28 min 5 10 15 20 25 14 min 29 min 0 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='03 15 min 1-min median Time scale (min) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 - 0 6 8 2 4 10 Mf,xd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='10 1 min 16 min e 17 min 2 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='14 3 min 18 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='08 4 min 19 min 5 min 20 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='06 21 min 6 min 22 min 7 min 8 min 23 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='10 9 min 24 min 10 min 25 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='02 26 min 11 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='08 12 min 27 min ctic 13 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 28 min Fra 14 min 5 10 15 20 25 30 29 min 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='06 15 min 1-min median Time scale (min) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 Pα (nPa)WANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 X - 9 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The simultaneous bow shock crossing around 05:05 UT on November 19, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' From the top to bottom, the panels display the magnetic field measured by MOMAG, the magnetic field measured by MAVEN/MAG, and the solar wind velocity, the number density and temperature of ions measured by MAVEN/SWIA, and the orbits of Tianwen-1’s orbiter and MAVEN viewed from different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The purple arrows in the first two panels indicate the times of the bow shock crossings, and the markers in the panels on the bottom indicate the positions of the crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 20 Bx (lu) By MMi 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Mag Bz M [B| 0 TW1 10 Bx MVN Mag (nT) 20 B Bz W B 0 20 360 Ivl 400 270 MVN v (km s Ciock angie Cone angle 180 - e Angl 300 90 W 0 3 15 Number density Temperature 7 10 IM 5 04:55 05:00 05:05 05:10 05:15 05:20 05:25 Start time 2021-11-19 04:55:00 UT (hr) 4 (RM) 1 (RM) 0 (RM) (RM) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content="25 MVN Iyz, MSo' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 2 0 ZMSO YMSO ZMSO 2 TW1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='25 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content="0 XMSO' (RM) XMSO (RM) XMSO (RM) yMSO (RM)X - 10 WANG ET AL." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The simultaneous bow shock crossing around 13:19 UT on December 11, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 20 Bx (lu) By 10 Mag ( Bz [BI 0 TW1 10 Bx 20 (nT) By MVN Mag ( 10 Bz BI 0 10 360 W MVN v (km s 350 270 Ciock angie (。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=') Cone angle 180 Angl 300 90 250 0 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 Number density (x106 Temperature 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 Tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 MVN 5 >sS>S 13:05 13:10 13:15 13:20 13:25 13:30 13:35 Start time 2021-12-11 13:05:00 UT (hr) (RM) (Wy) OsWz (Rm) D (RM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='25 MVN 0 0 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 YMSO ZMSO TW1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='25 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 0 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content="5 1 XMSO' (RM) XMSO (RM) XMSO (RM) YMSO (RM)WANG ET AL." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 X - 11 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The simultaneous bow shock crossing around 07:14 UT on December 12, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' 20 Bx (lu) 10 BB Mag 0 TW1 10 Bx 20 (nT) By MVN Mag ( 10 Bz [BI 0 w iwwy 10 360 >>> 500 Iv] Ciock angie 270 (。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=') 450 Cone angle 180 Angl 400 90 350 0 3 K Number density Temperature 20 2 10 07:05 07:10 07:20 07:25 07:30 07:15 07:35 Start time 2021-12-12 07:05:00 UT (hr) 2 (RM) (Rm) (RM) (Rm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='25 MVN 0 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 ZMSO ZMSO W1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='25 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content="0 XMSO' (RM) XMSO (RM) XMSO (RM) YMSO (RM)X - 12 WANG ET AL." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The simultaneous bow shock crossing around 19:43 UT on December 16, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=" 20 Bx TW1 Mag (nT) By Bz 10 [BI 0 MAA hM 30 Bx By 20 Bz [BI 10 0 360 500 270 Ciock angie 's LNW 。" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='） (km Cone angle Angle 400 180 MVN V ( >>> 90 300 W 0 Number density Temperature 15 3 M 2 10 1 5 19:30 19:35 19:40 19:45 19:50 19:55 20:00 Start time 2021-12-16 19:30:00 UT (hr) (RM) (RM) (RM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='25 MVN 1 7 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 ZMSO 0 YMSO ZMSO 0 TW1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='25 2 : 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 0 2 0 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='0 0 2 XMSO° (RM) XMSO (RM) XMSO (RM) YMSO (RM)WANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 X - 13 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The simultaneous bow shock crossing around 02:44 UT on December 17, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Bx TW1 Mag (nT) By 10 Bz [B 20 Bx (lu) By 10 Mag Bz IB MVN I 10 360 500 270 Ciock angie (。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=') (km Cone angle Angle ( 400 180 MVN V ( 90 300 0 4 Number density Temperature 10 3 2 5 L <>?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='> 02:30 02:35 02:40 02:45 02:50 02:55 03:00 Start time 2021-12-17 02:30:00 UT (hr) (Rm) (Rm) (RM) (RM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content="25 MVN 1 0 1 2 ryz, Mso' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 ZMSO yMSO ZMSO 0 0 TW1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='25 1 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='00 0 2 2 0 2 2 0 0 XMSO° (RM) XMSO (RM) XMSO (RM) YMSO (RM)X - 14 WANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' : MARS ORBITER MAGNETOMETER OF TIANWEN-1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Summary we have presented the in-flight performance and first results of Tianwen-1/MOMAG with the focus on the most notable structure — Martian BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Based on the first one and a half months’ data, we identified 158 clear BS crossings, whose locations are consistent with the BS model in statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The simultaneous BS crossings of the Tianwen-1 and MAVEN verified the south-north asymmetry of the BS, and also showed the similarity of magnetic field profiles from the two spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The first pair of the simultaneous BS crossings along with the BS crossing case on December 30 suggests that the BS is probably more dynamic at flank than near the nose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' By comparing with the MAVEN observations, we also found similar structures propagating with the solar wind from MAVEN to the Tianwen-1 orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We conclude that MOMAG shows an excellent performance and provides accurate measurements of magnetic field vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Now MOMAG has scanned the magnetic field in the MPR, magnetosheath and solar wind near the dawn-dusk side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' These measurements along with the MAVEN data will help us better understand the plasma environment surrounding Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' We acknowledge the use of the data from the MAG and SWIA onboard MAVEN spacecraft, which are obtained from NASA Planetary Data System (https://pds-ppi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' igpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='edu/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' One may apply for the Tianwen-1/MOMAG data at CNSA Data Release System (http://202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='98: 8081/marsdata/) or can just download the data used in this paper from the official website of the MOMAG team (http: //space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='cn/dreams/tw1_momag/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' The work is sup- port by the NSFC (Grant Nos 42130204 and 42188101) and the Strategic Priority Program of the Chinese Academy of Sciences (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' XDB41000000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' is particularly grateful to the support of the Tencent Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' References Bertucci, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', MGS MAG/ER observations at the mag- netic pileup boundary of mars: Draping enhancement and low frequency waves, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=' Space Res.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} +page_content=', submitted, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfx_mp/content/2301.00677v1.pdf'} diff --git a/kdFPT4oBgHgl3EQf2DV-/vector_store/index.faiss b/kdFPT4oBgHgl3EQf2DV-/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..37017f5f1fdb2d7b6724f91853c57e0b27e8bc44 --- /dev/null +++ b/kdFPT4oBgHgl3EQf2DV-/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:53d167e5221cb4541381b55eeea9b54ad8a4a6ff394f674aee1a1ff6fb1f1b14 +size 4849709 diff --git a/ktAzT4oBgHgl3EQfbvyL/content/tmp_files/2301.01391v1.pdf.txt b/ktAzT4oBgHgl3EQfbvyL/content/tmp_files/2301.01391v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..46d9c0b11e7e31d76b580bd331a4f7c5988d8882 --- /dev/null +++ b/ktAzT4oBgHgl3EQfbvyL/content/tmp_files/2301.01391v1.pdf.txt @@ -0,0 +1,4424 @@ +Amplitude representation of +Landau-Lifshitz equation and its +application to ferromagnetic films. +Gang Li1∗ and Valery Pokrovsky1,2 +January 5, 2023 +1Department of Physics and Astronomy, Texas A&M University, College +Station, TX 77843-4242, USA. +2Landau Institute for Theoretical Physics of Russian Academy of Sciences, +Chernogolovka, 142432, Russian Federation. +∗dgzy03@gmail.com +1 +Introduction +In 1935 Lev Landau and Evgenii Lifshitz set the foundation of static and dy- +namics of weakly anisotropic ferromagnets [1, 2]. They formulated the famous +Landau-Lifshitz equation (LLE) that regulates the motion of the ferromagnet +magnetization in the long-wave low-frequency limit. The purpose of this arti- +cle is to develop a systematic approach to the solution of the LLE in terms of +the magnon wave function ψ (r) and apply it to physical phenomena in a thin +ferromagnetic film. +This problem has a long history. First such approach was proposed by Schlö- +man in 1959 [3] for a bulk ferromagnet. It was developed and improved by Carl +Patton and his coworkers (see references in the review article by Krivosik and +Patton [4]). The applications focused on the ferromagnetic resonance (FMR) +and the spin momentum transfer, i.e., spin currents. +The theoretical study of ferromagnetic films started also in the the middle of +20-th century by the seminal work of Damon and Eshbach [5]. They have found +exact solution of the LLE equation for an infinite ferromagnetic film in which +spins interact only through the dipolar forces. In sufficiently thick films the +evanescent waves propagating in opposite direction at the two surfaces appear. +They create a mechanical torque acting on the film. +Gann [6], De Wames and Wolfram [7], Kalinikos and Slavin [8, 9] extended +the Damon-Eshbach theory to a more general situation in which the spins inter- +act also through the exchange forces. An extension of these exact solutions for +1 +arXiv:2301.01391v1 [cond-mat.mes-hall] 3 Jan 2023 + +the tilted external magnetic field was found by Arias [10]. In the work by the au- +thors, Chen Sun and Thomas Nattermann [11] the solution was extended to the +wide range of the film thickness. It enabled us to follow the transition from the +magnon spectrum with two symmetric minima in thick films to one-minimum +spectra in thin films. +The latter result was inspired by the discovery of the Bose-Einstein conden- +sation of magnons (BECM) at room temperature under permanent pumping +of electromagnetic waves made in 2006 by Demokritov et al[12]. The BECM +was found in the Yttrium Iron Garnet (YIG), a strongly insulating ferrite. For +long-wave excitations all spins in the primitive cells move as a whole. It means +that in this regime the ferrite is indistinguishable from a ferromagnet. +The amplitude representation (AR) is ideally adjusted to describe the con- +densation. The condensate amplitudes ψ± are the Fourier components of the +coordinate wave functions ψ (r) at the values of wave vector k = ±Q corre- +sponding to the two symmetric minima of magnon energy. Since we are mostly +interested in the properties of the condensate and its interaction with excited +magnons, our focus in the study of the (AR) will be different that in already +cited works by Schlöman and Patton. Certainly, some overlapping is unavoid- +able, but we try to minimize it. +This article has also a purpose to represent the modern state of art for the +properties of ferromagnetic films and the pumping-induced BECM in them at +room temperature. Thus, it can be considered as a review on basic principles +and the recent advances in the field. +2 +Hamiltonian formulation of the Landau-Lifshitz +equation and Amplitude representation. +2.1 +Poisson brackets for spins, magnetic moments and +magnetization in discrete and continuous models. +Let us start with a discrete 3d-model of the ferromagnet, in which all spins Sr +are located in the centers of cubic cells of volume v0 labeled by vectors r. The +Poisson brackets for the components of spins are: +{Sk (r) , Sl (r′)} = δr,r′εklmSm (r) , +(1) +where Kronecker symbol δr,r′ is equal to 1 when r = r′ and 0 otherwise; εklm +is absolutely antisymmetric 3d tensor with k, l, m independently taking values +1,2,3 or x, y, z that is equal to +1 if the permutation k, l, m is even and -1 if it +is odd. We use the Einstein convention that the summation must be performed +over repeated indices. +The magnetic moment of a primitive cell is +Mk = γSk, +(2) +where γ = +e +2mc is the classical gyromagnetic ratio. The relation (2) becomes +evident if one remembers that a spin projection, for example Sz, is quantized +2 + +in units ℏ. As a consequence, the magnetic moment projection is quantized +in units of the Bohr’s magneton µB = +eℏ +2mc. Eq. (1) implies that the Poisson +brackets for the components of the magnetic moments are: +{Mk (r) , Ml (r′)} = γδr,r′εklmMm (r) . +(3) +The magnetization is defined as magnetic moment of unit volume. It is expressed +in terms of magnetic moments as M (r) = M(r) +v0 +. Therefore the Poisson brack- +ets for magnetization in the discrete model are: +{Mk (r) , Ml (r′)} = γ +v0 +δr,r′εklmMm (r) . +(4) +In continuous approximation the ratio +δr,r′ +v0 +transits into the Dirac δ-function: +lim +v0→0 +δr,r′ +v0 += δ (r − r′) . +(5) +To prove this statement let us introduce an arbitrary continuous function f (r). +Let us consider a sum over the cites of the discrete model: +v0 +� +r′ +δr,r′ +v0 +f (r′) = f(r). +In continuous limit v0 +� +r′ → +´ +d3r′, which, together with previous equation, +proves eq. (5). +Thus, the Poisson brackets for components of magnetization in continuous +limit are: +{Mk (r) , Ml (r′)} = γδ (r − r′) εklmMm +(6) +It is convenient to rewrite these relations explicitly as; +{Mx (r) , My (r′)} = γδ (r − r′) Mz (r) +(7) +Two other Poisson brackets can be obtained from (7) by the cyclical permutation +of the indices x, y and z. +For further applications it is useful to introduce +complex transverse magnetizations: +M± (r) = Mx (r) ± iMy (r) +(8) +For them eq. (7) implies the following Poisson brackets: +{M+ (r) , M− (r′)} = −2iγδ (r − r′) Mz (r) +{M± (r) , Mz (r′)} = ±iγδ (r − r′) M± (r) +(9) +2.2 +Amplitude representation and Poisson brackets for +the magnon wave function. +Let the spontaneous magnetization and external magnetic field be directed along +z−axis, perpendicular to its direction in the plane of film be y and direction +3 + +Figure 1: The coordinate system for a ferromagnetic film of thickness d: z−axis +is chosen along the common direction of the magnetic field and static magneti- +zation, x−axis is perpendicular to the film, θk is the angle between the magnon +wave vector and magnetic field. +perpendicular to the film x as shown in Fig. 1. The wave function of magnons +ψ (r) is determined by the magnon classical Holstein-Primakoff transformation: +M+ (r) = √µBψ (r) +� +2M − µBψ∗ (r) ψ (r) +M− (r) = √µBψ∗ (r) +� +2M − µBψ∗ (r) ψ (r) +Mz = M − µBψ∗ (r) ψ (r) +, +(10) +where M is the magnitude of magnetization vector that is assumed to be con- +stant. The third equation (10) shows that the physical meaning of the square +of modulus ψ∗ (r) ψ (r) is the density of magnons n (r). Note that the order +of factors in eqs. +(10) is not important. +The second useful remark is that +� +2M − µBψ∗ (r) ψ (r) = √M + Mz. +The equations (9) are compatible with the amplitude representation (10) if +and only if the wave functions satisfy the following permutation relations: +{ψ (r) , ψ∗ (r′)} = − i +ℏδ (r − r′) +{ψ (r) , ψ (r′)} = {ψ∗ (r) , ψ∗ (r′)} = 0 +(11) +Let us prove this theorem for the second equation (9). We will use the algebraic +identity valid for any algebra of operators with defined operations of addition +and non-commutative multiplication: +{AB, C} = A {B, C} C + {A, C}B +(12) +4 + +X +Z +K +Héz +dEmploying this rule and the third equation (10), we find: +{M+, Mz} = √µB +� +ψ (r) √M + Mz (r) , Mz (r′) +� += +� +µB (M + Mz (r)) {ψ (r) , Mz (r′)} = − +� +µ3 +B (M + Mz (r)) {ψ (r) , ψ∗ (r′) ψ (r′)} +Applying again the identity (12) and assuming that {ψ (r) , ψ (r′)} = 0, we +arrive at relation +{M+, Mz} = − +� +µ3 +B (M + Mz (r))ψ (r′) {ψ (r) , ψ∗ (r′)} +The right-hand side of this equation must be equal to iγδ (r − r′) M+ (r) ac- +cording to the second equation (9) . The necessary and sufficient requirement +to satisfy this condition is given by eqs. (11). The validity of the first equation +(9) can be checked by a similar calculation. +2.3 +Landau-Lifshitz Hamiltonian. +The Landau-Lifshitz Hamiltonian HLL for our problem contains several parts: +the exchange interaction Hex, the dipolar interaction Hdip and the Zeeman +interaction HZ. It can also may contain the anisotropy (spin-orbit) energy Han +. First we write them in terms of magnetization: +HLL = Hex + Hdip + HZ + Han, +(13) +where: +Hex = D +2 +ˆ +(∇M)2 dV ≡ D +2 +ˆ +∂iMj∂iMjdV ; +(14) +Hz = −H +ˆ +MzdV +(15) +Hdip = 1 +2 +¨ +(M∇) (M′∇′) +1 +|r − r′|dV dV ′, +(16) +In eq. (16) we omitted for brevity the arguments in functions denoting M = M (r); +M′ = M (r′) ; ∇ = ∇r; ∇′ = ∇r′. When employing the amplitude representa- +tion for the components of magnetization (10), we similarly use abbreviations +ψ ≡ ψ (r), ψ′ ≡ ψ (r′) and ∂± ≡ ∂x ± i∂y, ∂′ +± ≡ ∂x′ ± i∂y′. The exchange +constant D determines the exchange length ℓ = +√ +D that separates the length +range, in which the dipolar interaction dominates l ≫ ℓ, from the range l ≪ ℓ +where exchange interaction dominates. +The LL equation assumes that the magnitude of the magnetization vector +rapidly relaxes to its equilibrium value. Thus, the LL equation describes the +relatively slow motion of the vector M (r, t) on the sphere. The slowness of this +motion in space and time is controlled by two small parameters a/λ and ωτM, +where a is the lattice constant, λ is the wave-length or another characteristic +length of the magnetization motion, ω is its characteristic frequency and τM is +the relaxation time of the magnetization magnitude. All magnetic phenomena +in this limit are dominantly classical since the number of magnons in the volume +5 + +with the linear size of the order of λ is large and the change of this number by +1 produces negligibly small change of magnetization. +In terms of amplitudes the three parts of the Hamiltonian given by equations +(14,15,16) are +Hex = µ2 +Bℓ2 +2 +´ � +∇ |ψ|2�2 +dV + +µBℓ2 +2 +´ ����∇ +� +ψ +� +2M − µB |ψ|2 +����� +2 +dV +(17) +Hz = µBH +ˆ +|ψ|2 dV +(18) +Hdip = 1 +2 +¨ +ˆΩ (r) ˆΩ (r′) dV dV ′ +|r − r′|, +(19) +where +ˆΩ (r) = +� +M − µB |ψ|2� +∂z + +� +µB +� +2M − µB |ψ|2� +2 +(ψ∂− + ψ∗∂+) +(20) +3 +Spectrum and wave functions of magnons. +In this section we consider the approximation of free magnons and find their +spectrum and wave function. For that purpose it is necessary to separate the +part of the total Hamiltonian quadratic in amplitudes ψ, ψ∗ and diagonalize it. +3.1 +Quadratic part of the Hamiltonian. +The Zeeman part of the Hamiltonian HZ given by eq. (18) is naturally quadratic. +The quadratic parts of the exchange and dipolar Hamiltonians are: +H(2) +ex = µBMℓ2 +ˆ +|∇ψ|2 dV +(21) +H(2) +dip = µBM +4 +¨ +(ψ∂− + ψ∗∂+) +� +ψ′∂′ +− + ψ′∗∂′ ++ +� +1 +|r − r′|dV dV ′ +(22) +Note that quadratic parts of the exchange Hamiltonian is local in space and it +conserves the total number of magnons N = +´ +|ψ|2 dV , whereas the quadratic +part of dipolar Hamiltonian is non-local and it violates the conservation of the +magnon number. All three parts of the quadratic Hamiltonian are invariant with +respect to any translation in the film plane. Therefore, it is natural to describe +the motion in plane as a superposition of running plane waves. In other words, +the problem must be partly diagonalized by the Fourier-transformation: +ψ (r) = +1 +√ +A +� +q +χq (x) eiqr, +(23) +6 + +where q = iqyˆy +iqzˆz is the in-plane wave vector; the Fourier-coefficients χq (x) +depend on the transverse-to-plane coordinate x; A is the area of any film cross- +section parallel to its surfaces. The inverse Fourier transformation gives the +amplitude of a magnon with the wave vector q in a general state with the wave +function ψ (r): +χq (x) = +1 +√ +A +¨ +ψ (r) eiqrdydz +(24) +Employing the Poisson brackets for ψ (r) eq. (11), the Poisson brackets for the +amplitudes χq (x) are: +� +χq (x) , χ∗ +q′ (x′) +� += − i +ℏδq,q′δ (x − x′) . +(25) +In terms of the variables χq (x) the three parts of the Hamiltonian are:Q +H(2) +ex = µBMℓ2 � +q +d/2 +ˆ +−d/2 +����� +dχq (x) +dx +���� +2 ++ q2 |χq (x)|2 +� +dx +(26) +H(2) +Z += µBH +� +q +d/2 +ˆ +−d/2 +|χq (x)|2 dx +(27) +H(2) +dip = πµBM � +q +˜ d/2 +−d/2 +� +χq (dx − qy) + χ∗ +−q (dx + qy) +� +× +� +χ′ +−q (dx′ + qy) + χ′∗ +q (dx′ − qy) +� +Gq (x − x′) , +(28) +where we omitted for brevity the arguments x and x′ writing χq instead of +χq (x) and χ′ +q instead of χq (x′) and employed the abbreviation dx ≡ +d +dx. The +symbol Gq (x) stays for for the Green function of the 1d Helmholtz equation: +Gq (x) = e−q|x| +2q +(29) +It obeys the 1d Helmholtz equation with a point source at origin: +� +d2 +x − q2� +Gq (x) = −δ (x) . +(30) +3.2 +Bogoliubov transformation. +The exchange and Zeeman parts of the quadratic Hamiltonian are diagonal in +the variables χq (x), but the dipolar part mixes χq (x) with χ∗ +−q (x). To di- +agonalize the total quadratic Hamiltonian we apply the extended Bogoliubov +transformation introducing for each q an infinite series of variables ηqn associ- +ated with χq (x) and χ∗ +−q (x) by a linear transformation: +ηqn = +d/2 +ˆ +−d/2 +� +uqn (x) χq (x) + vqn (x) χ∗ +−q (x) +� +dx. +(31) +7 + +To be canonical this transformation must produce correct Poisson brackets for +variables ηqn: +� +ηqn, η∗ +q′n′ +� += − i +ℏδq,q′δn,n′ +(32) +This requirement is equivalent to the condition of canonical transformation +in classical mechanics [13] or unitary transformation in quantum mechanics. +Therefore we will also use the word "unitarity" or "unitary" as equivalent to +"canonical". The requirement (32) together with the Bogoliubov transformation +(31) and Poisson brackets for χq (x) (25) implies a series of constraints: +d/2 +ˆ +−d/2 +� +uqn (x) u∗ +qn′ (x) − vqn (x) v∗ +qn′ (x) +� +dx = δn,n′ +(33) +The inverse Bogoliubov transformation determines χq (x) as a linear combina- +tion of ηqn: +χq (x) = +� +n +� +Uqn (x) ηqn + Vqn (x) η∗ +−qn +� +(34) +Replacing the amplitudes ηqn, η∗ +−qnin eq. (34) by their Bogoliubov representa- +tion (31), we arrive at equations relating direct and inverse Bogolyubov trans- +formations: +� +n +� +Uqn (x) uqn (x′) + Vqn (x) v∗ +−qn (x′) +� += δ (x − x′) +� +n +� +Uqn (x) vqn (x′) + Vqn (x) u∗ +−qn (x′) +� += 0 +(35) +On the other hand, the unitarity of the inverse Bogoliubov transformation re- +quires +� +n +� +Uqn (x) U ∗ +qn (x′) − Vqn (x) V ∗ +qn (x′) +� += δ (x − x′) +(36) +Comparing this equation with the first eq. (29), we arrive at conclusion that +Uqn (x) = u∗ +qn (x) and Vqn (x) = −v−qn (x). +Thus, the inverse Bogolyubov +transformation can be rewritten as +χq (x) = +� +n +� +u∗ +qn (x) ηqn − v−qn (x) η∗ +−qn +� +(37) +In addition from the U − V unitarity condition (36) we find the dual unitarity +condition in terms of the initial Bogolyubov coefficients: +� +n +� +u∗ +qn (x) uqn (x′) − v−qn (x) v∗ +−qn (x′) +� += δ (x − x′) +(38) +3.3 +The wave functions and spectrum of magnons. +3.3.1 +Spectrum of magnons. +The magnon amplitudes must satisfy the stationary Schrödinger equation whose +classical analogue is +� +H(2), ηq,n +� += −iωq,nηq,n. +(39) +8 + +The Poisson brackets of the quadratic Hamiltonian and the vector of amplitudes +is a linear anti-Hermitian operator acting on this vector. Thus, the vector of +amplitudes ηq,n is the eigenvector and the frequency of a magnon is the corre- +sponding eigenvalue of the Hermitian operator i +� +H(2), +� +. In this subsection we +express these equations in terms of the Bogoliubov coefficients. Their solutions +in some limiting cases will be found in the next subsection. +In order to write the left part of eq. (39) explicitly, we employ eqs. (26,27,28) +for the three parts of the quadratic Hamiltonian, equation (32) for the Poisson +brackets of the two amplitude vectors and the Bogoliubov transformation (37) +from the amplitudes χq,n to magnon amplitudes ηq,n. In resulting equations +we omit for brevity the subscripts q and n since they are invariant under the +Bogoliubov transformation. Thus, equations (39) can be rewritten as: +� +ω + γ +� +H + Mℓ2 � +q2 − d2 +x +��� +u = −2πγM +�� +q2 +y − d2 +x +� +ζu + (qy − dx)2 ζv +� +; +� +ω − γ +� +H + Mℓ2 � +q2 − d2 +x +��� +v = 2πγM +�� +q2 +y − d2 +x +� +ζv + (qy + dx)2 ζu +� +, +(40) +where we denoted γ = |e|/(2mc) is the classical gyromagnetic constant and +ζu,v (x) = +d/2 +ˆ +−d/2 +G (x − x′) u (x′) +v (x′) dx′. +(41) +The physical meaning of the integral terms in the r.-h. side of eqs. (40) is the +magnetic field h generated by magnon magnetization m. The magnetic field +can be expressed in terms of magnetostatic potential φ as h = −∇φ. If it is +generated by the magnetization m (r), then +φ (r) = −∇ · +ˆ +m (r′) |r − r′|−1 d3x′ +(42) +The coefficients u and v should be identified with the x- and y-components +of magnetization, the operators ±iqy − dx with the complex presentation of +gradient and divergence. Then equation (41) is equivalent to (42) integrated +over y and z. +The reference (40) is a system of two integral-differential equations. However, +they can be transformed in the purely differential linear equations by employing +operator q2 − d2 +x (Laplacian) to both sides of equations (40) and employing eq. +(30) to eliminate the Green function G (x − x′). The application of this operator +to ζu,v (x) transforms these integrals into u (x) and v (x) , respectively. +Thus, we obtain a system ordinary linear differential equations of the fourth +order: +� +ω + γ +� +H + Mℓ2 � +q2 − d2 +x +��� � +q2 − d2 +x +� +u += −2πγM +�� +q2 +y − d2 +x +� +u + (qy − dx)2 v +� +; +� +ω − γ +� +H + Mℓ2 � +q2 − d2 +x +��� � +q2 − d2 +x +� +v += 2πγM +�� +q2 +y − d2 +x +� +v + (qy + dx)2 u +� +. +(43) +9 + +Their solutions must be a superposition of exponents eiκx with κ being a root of +the secular polynomial. To find this polynomial, it is convenient to introduce the +vector k with the components kx = |κ|, ky,z = qy,z whose square if magnitude +is k2 = q2 + κ2. Let us define a simplest solution of the system (43) is: +u (x) = u0eiκx; v (x) = v0eiκx +(44) +Substituting this solution into eq.(43), we obtain a system of two linear homo- +geneous equations for u0, v0. The condition of its solvability is the nullification +of their determinant (secular equation): +ω2k2 = γ2 � +H + Mℓ2k2� +× +�� +H + Mℓ2k2� +k2 + 4πM +� +k2 − k2 +z +�� . +(45) +This equation can be interpreted as dispersion relation for magnons: +ω = γ +� +� +� +�(H + Mℓ2k2) +� +H + Mℓ2k2 + 4πM +� +k2x + k2y +� +k2 +� +(46) +It is valid if a/λ = ka/(2π) ≪ 1. At room temperature the thermal wavelength +λ = ℏ/√2mkBT. For effective mass of magnon for YIG of the order of mag- +nitude m ≈ 3me, λ is about 0.7nm, whereas the lattice constant a = 1.2nm. +Therefore, eq. +(46) is invalid for thermal magnons. +The calculation of the +magnon spectrum at high energies for YIG were given in the seminal article by +Kolokolov, L’vov and Cherepanov [14]. +3.3.2 +Bulk and evanescent waves. +At fixed parameters ℓ, M, H, kz = qz and frequency ω, eq. +(45) is a cubic +equation for the variable k2. Note that its coefficients do not depend not only +on the film thickness d but also on the value ky. Inspection of the coefficients +of the cubic equation shows that the product of three roots is positive, whereas +their sum is negative. Therefore, there are two opportunities: i) one roots k2 is +positive and two others are negative or ii) one root is positive and two others +are complex conjugated with negative real part. Sonin proved [15] that in thick +films d ≫ ℓ and for kl ≪ 1, the opportunity i) is realized. Gang. Li et al. [11] +proved that the opportunity ii) leads to negative ω2 and therefore is forbidden. +For thick films d ≫ ℓ and kz ≪ 1/ℓ and ω2 < γ2H (H + 4πM), the positive +root k2 +1 can be found approximately. +In this case it is possible to retain in +eq. (45) only terms linear in k2 and independent on k2 and neglect the terms +quadratic and cubic in k2. The result is: +k2 +1 = k2 +z +4πγ2HM +γ2H (H + 4πM) − ω2 +(47) +Two others negative solutions k2 = −k2 +1,2 are determined by equation: +k2 +1,2 = +� +2π + H +M ± +� +4π2 + +ω2 +γ2M 2 +� +ℓ−2 +(48) +10 + +When frequency approaches the ferromagnetic resonance value ωF R = γ +� +H (H + 4πM) +to the distance ωF R − ω ≲ +k2 +zℓ2 +√ +1+ 4πH +M +2πγM, the inequality k1ℓ ≪ 1 becomes in- +valid and instead of quadratic the cubic equation must be solved. +At large +frequency ω ≫ ωF R, the exchange energy dominates and ω ≈ γMℓ2k2. It cor- +responds to the region of large wave vectors kℓ ≫ 1. For thick films d ≫ ℓ, the +four wave functions of the type χq (x) ∝ exp +� +−k1,2 +� d +2 ± x +�� +correspond to the +four evanescent waves localized in a layer of the depth ∼ ℓ near the surfaces of +the film x = ±d/2. +3.4 +Self-consistency. +We proved that any propagating in-plane excitation is a superposition of several +transverse modes. The transverse modes may be either superposition of cos kxx +and sin kxx or the evanescent waves. However, the inverse statement that any +such superposition is a solution of the initial equations of motion is wrong. This +happens because the initial equations of motion were integral-differential. The +system of ordinary differential equations was obtained from them by applica- +tion of additional differential operators. This operation introduces additional +solutions of resulting system of equations that are not solutions of the initial +problem. Below we derive the selection rules that separate only solutions of the +initial integral-differential equations (40,41). +Equations for the Bogoliubov transformation functions (40,41) permit real +solution. Therefore the Bogoliubov functions can be searched in the form: +uq,n (x) = an cos kxx + bn sin kxx+ +� +m=1,2 +� +Anm cosh kmx +cosh kmd/2 + Bnm sinh kmx +sinh kmd/2 +� +; +(49) +vq,n (x) = cn cos kxx + dn sin kxx+ +� +m=1,2 +� +Cnm cosh kmx +cosh kmd/2 + Dnm sinh kmx +sinh kmd/2 +� +, +(50) +where all coefficients an, bn, Anm, Bnm,cn, dn, Cnm, Dnm are real numbers. In +further calculations we omit the subscripts n and q since they are fixed. All +evanescent waves exponentially decrease far from boundaries on the scale ∼ ℓ +as exp +� +−km +�� d +2 ± x +��� +. +Substitution of expressions (49,50) to the integral-differential equations (40,41) +leads to appearance of exponential functions that do not belong to the 6 expo- +nents permitted by the secular equation (45). They are produced by the integrals +(41). Their explicit calculation can be reduced to the four basic integrals: +Ic (x) ≡ +´ d/2 +−d/2 +e−q|x−x′| +2q +cos kxx′dx′ += cos kxx +k2 +− e−qd/2 +qk2 +cosh qx f1; +(51) +Is (x) ≡ +´ d/2 +−d/2 +e−q|x−x′| +2q +sin kxx′dx′ += sin kxx +k2 +− e−qd/2 +qk2 +sinh qx f2; +(52) +11 + +Jcm (x) ≡ +´ d/2 +−d/2 +e−q|x−x′| +2q +cosh kmx′dx′ += cosh kmx +q2−k2m − +e−qd/2 +q(q2−k2m) cosh qx g1m; +(53) +Jsm (x) ≡ +´ d/2 +−d/2 +e−q|x−x′| +2q +sinh kmx′dx′ += sinh kmx +q2−k2m − +e−qd/2 +q(q2−k2m) sinh qx g2m, +(54) +where the notations f1,2, g1,2 are used for the following functions: +f1 = q cos kxd +2 +− kx sin kxd +2 ; +(55) +f2 = q sin kxd +2 ++ kx cos kxd +2 ; +(56) +g1m = q cosh kmd +2 ++ km sinh kmd +2 ; +(57) +g2m = q sinh kmd +2 ++ km cosh kmd +2 . +(58) +Employing these results, it is possible to calculate ζu (x) and ζv (x) defined by +eq. (41): +ζu (x) = aIc + bIs + +2 +� +m=1 +� +AmJcm +cosh kmd +2 ++ BmJsm +sinh kmd +2 +� +; +(59) +ζv (x) = cIc + dIs + +2 +� +m=1 +� +CmJcm +cosh kmd +2 ++ DmJsm +sinh kmd +2 +� +. +(60) +The terms with Ic and Is in these equations contain the functions cosh qx and +sinh qx or equivalently exp (±qx). +The wave vector k = q does not satisfy +the secular equation (45). Therefore, they should vanish in the r.-h. side of +eqs. (40). These requirements represent four constraints onto 12 coefficients +a, b, c, d, A1, B1, C1, D1, A2, B2, C2, D2 [11]. Neglecting evanescent waves in the +integrals, we obtain 4 equations for 4 coefficients a, b, c, d at “bulk” waves: +� +q2 +y − q2� +af1 ++ +� +q2 +y + q2� +cf1 ++2qyqdf2 += 0 +� +q2 +y − q2� +bf2 ++2qyqcf1 ++ +� +q2 +y + q2� +df2 += 0 +� +q2 +y + q2� +af1 +−2qyqbf2 ++ +� +q2 +y − q2� +cf1 += 0 +−2qyqaf1 ++ +� +q2 +y + q2� +bf2 ++ +� +q2 +y − q2� +df2 += 0 +, +(61) +The determinant of this system is identically zero . Thus, this system does not +determine quantization of kx. A simple reason why any 4×4 minor of the 4×24 +matrix formed by coefficients at e±qx in each of the mentioned above twelve +coefficients has zero determinant is that all of them obey an inhomogeneous +Helmholtz equation, for example, +d2Ic +dx2 − q2Ic = cos(kxx); d2Jcm +dx2 +− q2Jcm = cosh(kxx). +(62) +12 + +Since the solutions of such equations can include any linear combination of e±qx, +the condition of zero coefficients at these function cannot put any restriction of +the 4 × 24 matrix. It means that any its 4 × 4 minor has zero determinant. +The self-consistency equations are equivalent to the MBC, but they simplify +calculations. +3.5 +Boundary conditions and the quantization of trans- +verse modes. +3.5.1 +Spin boundary conditions. +There are two kinds of boundary conditions: magnetostatic (MBC) associated +with the variation of the magnetic field and induction near the boundary and +the spin boundary conditions (SBC) associated with variation of spin (magneti- +zation) at the boundary. The MBC requires continuity of tangential component +of magnetic field h and the normal component of the induction b = h + 4πm +at two surfaces x = ±d/2 of the film. The MBC are satisfied automatically +if the magnetic potential is related to the magnetization by the equation (42). +Therefore, only the SBC must be taken into account. +Let us consider the simplest possibility that spins on the surfaces are free. +The variation of the exchange energy (14) gives the surface term: +δHex = ℓ2 ´ d/2 +−d/2 dx +˜ ∞ +−∞ dydz∂iδmα · ∂imα = +ℓ2 ˜ ∞ +−∞ dydzδmα∂xmα +��� +d/2 +−d/2 + volume terms +. +(63) +The volume terms contribute exchange terms in equations of motion, whereas +the surface term in this equation implies that on both surfaces magnetization +obeys the spin boundary condition: +∂xm|x=±d/2 = 0. +(64) +The variation of the Zeeman and dipolar Hamiltonians does not give the surface +term since they do not contain derivatives of magnetization. +Returning to the amplitude representation, we identify as before the two +components of magnetization with the Bogolyubov coefficients u and v at fixed +q. Thus, eq. (64) in amplitude representation is: +∂xu|x=±d/2 = ∂xv|x=±d/2 = 0 +(65) +For the thick film and kxℓ ≪ 1, these equations imply that the magnitudes of +coefficients at the evanescent waves Am, Bm, Cm, Dm are less than the magni- +tudes of amplitudes of the bulk waves a, b, c, d by the factor ∼ kxℓ.[15] To see +that, let us put all coefficients except of a, c and A1, C1 equal to zero. Then +equation (65) takes form: +(c − a) kx sin kxd +2 += (A1 + C1) k1 +(66) +13 + +This equation proves the Sonin’s statement since k1 ∼ 1/ℓ. Nevertheless the +evanescent waves allow to satisfy the MBC at fixed amplitudes of the bulk +waves. +Neglecting in equations of motion (40) evanescent waves, we can rewrite +them as: +ˆ +M +� +� +� +� +a +b +c +d +� +� +� +� = 0, +(67) +where the 4 × 4 matrix +ˆ +M is: +ˆ +M = +� +� +� +� +ω − A +0 +B +C +0 +ω − A +−C +B +−B +C +ω + A +0 +−C +−B +0 +ω + A +� +� +� +� , +(68) +and +A = γ +� +H + Mℓ2k2 + +2πM(k2 +x+k2 +y) +k2 +� +B = +2πγM(k2 +x−k2 +y) +k2 +C = 4πγMkxky +k2 +. +(69) +The determinant of the matrix +ˆ +M is +det ˆ +M = +� +ω2 − A2 + B2 + C2�2 . +(70) +It turns into zero at ω = +√ +A2 − B2 − C2 that gives the obtained earlier disper- +sion relation (46). The eigenvalues ±ω of the matrix +ˆ +M are double degenerate. +Therefore, their eigenvectors contain two independent coordinates, for exam- +ple the amplitudes a and b, whereas two others are expressed as their linear +combination as it follows from the equations (67): +c += +B +ω±Aa − +C +ω±Ab +d += +C +ω±Aa + +B +ω±Ab +(71) +Note that the two eigenvectors corresponding to different signs in denominators +are orthogonal at mass shell, i.e., at ω = +√ +A2 − B2 − C2 and any choice of +coordinates a and b. +Let us substitute the amplitudes c and d from eqs. (71) for the sign + into +the first two of self-consistency equations (61). Then we find a system of two +homogeneous equations of the form: +Pa + Qb = 0 +Ra + Sb = 0 , +(72) +14 + +where +P = +� +q2 +y − q2 + (q2 +y+q2)B +ω+A +� +f1 − 2qyqC +ω+A f2 +Q = (q2 +y+q2)C +ω+A +f1 + 2qyqB +ω+A f2 +R = −(q2 +y+q2)C +ω+A +f1 + 2qyqC +ω+A f1 +S = +� +q2 +y − q2 + (q2 +y+q2)B +ω+A +� +f2 + 2qyqC +ω+A f1 +(73) +The determinant of the system (72) PS − QR must be zero. It determines the +quantization of kx. Equation PS − QR = 0 gives: +f 2 +1 − f 2 +2 = 2Γf1f2; Γ = +� +q2 +y − q2� +ω + +� +q2 +y + q2� +B +2qyqB +. +(74) +From this equation we find: +f1 +f2 += Λ ≡ Γ ± +� +Γ2 + 1. +(75) +Note that the change of sign in front of square root turns Λ into −1/Λ. Em- +ploying equations (55,56), we represent the quantization condition in a more +explicit form: +tan kxd +2 += q − Λkx +Λq + kx +. +(76) +The change Λ → −1/Λ transforms the fraction q−Λkx +Λq+kx into inverse value with +opposite sign, i.e., − Λq+kx +q−Λkx . For the waves propagating along spontaneous mag- +netization (ky = 0), the quantization condition becomes +tan kxd +2 += q +kx +or tan kxd +2 += −kx +q +(77) +The first of them was first found by Damon and Eshbach [5] for purely dipo- +lar interaction and reproduced by Sonin.[15] It corresponds to the pure cosine +solution (b = 0). The second sign at ky = 0 corresponds to the pure sine solu- +tion (a = 0).[11] For general direction of propagation in-plane the two different +signs in front of square root in eq. (75) correspond to two different branches of +discrete solutions. We denote them by discrete index ν accepting two values ±. +3.5.2 +Quantization of transverse wave vectors. Parallel propagation. +Equations (77) have a discrete set of solutions for kxn in the intervals +� +πn +d , π(n+1/2) +d +� +for the cosine and in the intervals +� +π(n+1/2) +d +, π(n+1) +d +� +for the sine transverse mag- +netization, where n is any non-negative integer. It is clearly seen from Fig. 2. +In the limit qd ≫ 1 the approximate analytical solution is possible for n ≪ qd. +15 + +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0.0 +0.5 +1.0 +1.5 +2.0 +kz ( +MD +) +kxn ( +MD +) +Figure 2: Plots of the dependence of quantized transverse wave vectors kxn on +kz in units +� +H +MD for d = 10 in units +� +MD +H +. Black and red curves correspond +to even and odd transverse modes, respectively +In this case kx ≪ q so the ratio q +kx ≫ 1 for the first series of quantized kx. +Therefore, kxd +2 +in the first equation (77) must be close to +� +n + 1 +2 +� +π and +k(+) +xn ≈ (2n + 1) π +d +� +1 − 2 +qd +� +(78) +Here we used the index + as notation of the first series (even transverse distri- +bution of magnetization). For large n and qd ≫ 1 the approximate equation for +the quantized values of the first series is: +k(+) +xn ≈ 2nπ +d ++ 2 +d arctan qd +2nπ +(79) +It accurately matches the result (78) for 1 ≪ n ≪ qd. +For the second series the quantized transverse wave vectors for qd ≫ 1 and +n ≪ qd are +k(−) +xn ≈ 2nπ +d +� +1 − 2 +qd +� +(80) +and for n ≫ 1 +k(−) +xn ≈ (2n + 1) π +d ++ 2 +d arctan qd +2nπ +(81) +3.5.3 +Wave vectors and effective masses at minimum energy. +Two energy minima ±Q are located on z−axis and correspond to minimal value +n = 0 and symmetric branch of the transverse momentum quantization, i.e. +kx ≈ π +d . Let us minimize explicitly the energy or frequency eq. (46). For a +thick film d ≫ ℓ, the energy is ε = ℏω (q, kx). It is more convenient to minimize +16 + +the square of energy +ε2 (q, kx) = µ2 +B +� +H2 + 2HMℓ2k2 + 4πHM +� +k2 +x + k2 +y +� +k2z +� +. +(82) +We first minimize square of energy over qy putting qy = 0 and in the square of +total momentum k2 = k2 +x + q2 +y + q2 +z neglect k2 +x. Taking derivative over qz from +ε2 (qz, 0, kx) at kx = π +d , we get: +2ε ∂ε +∂qz += 4µ2 +BHM +� +ℓ2qz − 2π3 +q3zd2 +� +. +(83) +At minimum energy the derivative +∂ε +∂qz = 0. From this requirement we find, +that two minima are located at qz = ±Q, where +Q = +� +2π3�1/4 +√ +ℓd +. +(84) +This result was obtained by E. Sonin.[15] +The main value of the mass tensor mz in z direction relates to the second +derivative ∂2ε +∂q2z for qz = ±Q as mz = ℏ2/ ∂2ε +∂q2z +��� +qz=Q. By differentiation of eq. (83) +and putting qz = Q, εmin = µBH, we find: +mz = +ℏ2 +8µBMℓ2 +(85) +To find my, we need to take the second derivative of ε2 (q, kx) given by eq. +(82) over qy at qy = 0, qz = Q neglecting kx. The searched effective mass is +my = ℏ2/ ∂2ε +∂q2y +��� +qy=0. An elementary calcualtion gives: +my = +ℏ2Q2 +8πµBM +(86) +The mass my is much less than mz: their ratio is my/mz = ℓ/ (πd) ≪ 1. +For the film of YIG 5μm thick Q ≈ 6.44 × 105cm−1 , mz = 7.37 × 10−27g; +my = 1.78 × 10−29g. +3.5.4 +Quantization of transverse wave vector: arbitrary direction of +propagation. +Despite of rather involved structure of quantization condition (76) its solution +can be written explicitly in the limit d ≫ ℓ, and qd ≫ 1. The roots of this +equation are kxνn, where n = 0, 1, 2... is the number of quantized value kx, +ν = ± stays for even or odd transverse distribution of magnetization. +The +explicit analytical expression for these roots in the asymptotic region and large +n ≫ 1 is +kxνn = 2nπ +d ++ 2 +d arctan qd − 2πnΛνn +qdΛνn + 2πn. +(87) +17 + +To find parameters Λνn = Γ+ν +√ +Γ2 + 1 it is necessary to replace kx by 2πn/d in +the equations (75) for Λ and (74) in all functions containing kx in its arguments. +Equation (87) has precision 1/qd and is valid for 1 ≪ n ≪ qd. In the entire this +region the difference between the quantized values of kx with the same number +in the two branches is +kx+n − kx−n = π +d +(88) +The ratio of amplitudes in this range of variables is +bνn +aνn += −qA +ω Λν − C +ω +(89) +At fixed direction of in-plane propagation given by the angle θ between the +wave vector and direction of the spontaneous magnetization M, the frequency +as function of the wave vector magnitude has minimum at +q0 = 2√πχ3/4√ +cos θ +� +2 + χ sin2 θ +�1/4 +� +kxνn +ℓ +, +(90) +where χ = 4πM +H . From this equation and strong inequality qℓ ≪ 1 it follows +that q0 ≫ kxνn ≈ 2π +d n. +3.5.5 +Motion of energy minimum vs. kx. +At very large n ≫ d +ℓ the value k2 becomes so large that the exchange interactions +dominates and the frequency of a magnon becomes equal to ω = γMℓ2k2. Then +the minimum energy occurs at q = 0. It means that the position of minimum +of frequency q0 first grows with kx and reaches its maximum at some specific +kx1 ∼ 1/ℓ. At further growth of kx the position of frequency minimum q0 (kx) +decreases and reaches zero at another specific value of kx = kx2. At further +growth of n it remains zero. Theory gives exact analytical answers for all these +values, namely: +k2 +x1 = 1 +3k2 +1 + 2 + χ +12π tan2 θk4 +1ℓ2, +(91) +where +k2 +1 = +H +6Mℓ2 +��� +2 + χ sin2 θ +�2 + 6χ cos θ − 2 − χ sin2 θ +� +. +(92) +The maximal value of q0 is given by +q2 +0 max = k2 +1 + k2 +x1. +(93) +Finally the value of k2 +x at which the minimum of frequency merges with maxi- +mum located at q = 0 is +k2 +x2 = +H +4Mℓ2 +��� +2 + χ sin2 θ +�2 + 8χ cos θ − 2 − χ sin2 θ +� +. +(94) +18 + +The position of maximum k0 (kx) for kxℓ ≳ 1 is given by +k2 +0 (kx) = +H +Mℓ2 +2 + χ sin2 θ +2 +w (ξ) , +(95) +where w (ξ) is the solution of a cubic equation: +w3 + w2 = ξ +(96) +and +ξ = +χ2 cos2 θk2 +xℓ2 +π +� +2 + χ sin2 θ +�3 +(97) +Details of these calculations can be found in the Appendix[motion of minima]. +In the analysis of this subsection we followed the work [11]. +3.6 +Comparison with other calculations and experiment. +The results of numerical calculations of quantized spectra eq. (82) with quan- +tized kxn for propagation perpendicular and parallel to magnetization and d = +18.2 in units +� +MD +H , χ = 2.5 are shown in Fig 3(a) and 3(b), spectra of the first +transverse modes for a number of different directions of propagation specified +by the angle θ = arctan ky +kz are shown in Fig. 3(c). +The spectra for parallel and perpendicular propagation (Fig. 3(a) and 3(b)) +agree very well with the numerical calculations of the work [16] based on diag- +onalization of a large matrix. We also discovered an excellent agreement with +similar calculations of the same work made for the YIG film with a thickness of +5 µm. +Figure 4 shows a comparison of the theoretical spectrum with the experiment +[17, 18]. +Brillouin scattering spectroscopy was used in the experiment. +Its +precision is not sufficient for resolution of excited states. A dramatic increase +in precision was achieved by an experimental group led by J. Ketterson [19]. +His method makes use of direct microwave excitation of magnons via a specially +designed antenna. +It is made up of periodically repeated emitters that are +powered by an adjustable frequency generator. The excited magnon wave-length +coincides with the distance between emitters λ. The magnon frequency at this +wave vector kz = (2π)/λ is a frequency at which the resonance adsorption of +microwave radiation reaches maximum. The increased resolution allowed for the +observation of multiple magnon modes (up to nine). This is the first time that +different transverse magnon modes have been experimentally observed. Figure +5 shows a comparison of theoretical spectrum with experimental results [19]. +The agreement between theory and experiment is excellent. +3.7 +Thin films. +In what follows till the end of this section we use +� +M/Hℓ as unit of length +and (γH)−1 as unit of time. +In this part we discuss the case of thin films. +19 + +0.0 +0.1 +0.2 +0.3 +0.4 +1.8 +2.0 +2.2 +2.4 +ky ( +MD +) +ω (γℋ) +(a) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.0 +1.5 +2.0 +2.5 +kz ( +MD +) +ω (γℋ) +(b) +0.0 +0.1 +0.2 +0.3 +0.4 +1.2 +1.4 +1.6 +1.8 +2.0 +k +ℋ +MD +ω γℋ +θ 0 +θ π/6 +θ=π/4 +θ=π/3 +θ=π/2 +(c) +Figure 3: Results of numerical calculations for the case d = 18.2 in units +� +MD +H +and χ = 2.5. (a) The spectra of first four quantized modes for direction of +propagation perpendicular to magnetization. (b) Spectra of the first four modes +for direction of propagation parallel to magnetization. (c) Spectra of the first +transverse modes for θ = 0, π +6 , π +4 , π +3 , π +2 . Black solid curves correspond to our +numerical calculations, red dashed line is the Damon-Eshbach surface mode, +circles are numerical calculations by Kreisel et al.. [16]. These figures agree +with the figures from [11]. +0.00 +0.05 +0.10 +0.15 +0.20 +1.40 +1.42 +1.44 +1.46 +1.48 +ky ( +MD +) +ω (γℋ) +(a) +0.00 +0.02 +0.04 +0.06 +0.08 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +kz ( +MD +) +ω (γℋ) +(b) +0.005 0.010 0.015 0.020 0.025 0.030 +0.9 +1.0 +1.1 +1.2 +1.3 +ky ( +MD +) +ω (γℋ) +(c) +Figure 4: Comparison of theoretical spectrum with experiments. In experiments +the Brillouin light scattering spectroscopy was used.(a) Comparison with A. A. +Serga et al.[17] d = 5 µm, H=1750 Oe . (b) Comparison with V. E. Demidov et +al.[18] d = 5.1 µm, H=1000 Oe for direction of propagation parallel to magneti- +zation. (c) Comparison with V. E. Demidov et al.[18] d = 5.1 µm, H=1000 Oe. +for fixed kz = 3.4 × 104cm−1. These figures agree with the figures from [11]. +20 + +0.02 +0.04 +0.06 +0.08 +0.10 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +kz ( +MD +) +ω (γℋ) +Figure 5: Comparison of theoretical spectrum with experiment. Solid curves +are our calculations of the first 15 transverse modes for the YIG film of thick- +ness 5µm, 4πM= 1940 Oe and H= 1960 Oe . Circles on them are frequencies +measured by J. Lim et al. [19] at three fixed wavelengths for different transverse +mode. This figure agrees with the figure from [11]. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +20 +40 +60 +80 +100 +ky ( +MD +) +ω (γℋ) +(a) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +20 +40 +60 +80 +100 +kz ( +MD +) +ω (γℋ) +(b) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +1.7 +1.8 +1.9 +2.0 +2.1 +2.2 +k +ℋ +MD +ω γℋ +θ 0 +θ π/6 +θ=π/4 +θ=π/3 +θ=π/2 +(c) +Figure 6: Results of numerical calculations for a thin film d = 1 in units +� +MD +H +and χ = 2. +(a) The spectra of first four quantized modes for direction of +propagation perpendicular to magnetization. (b) Spectra of the first four modes +for direction of propagation parallel to magnetization. (c) Spectra of the first +transverse modes for θ = 0, π +6 , π +4 , π +3 , π +2 . These figures agree with the figures from +[11]. +21 + +0 +2 +4 +6 +8 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +d ( +MD ) +kz ( +MD +) +(a) +0 +2 +4 +6 +8 +10 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +d ( +MD ) +ω (γℋ) +(b) +0 +1 +2 +3 +4 +5 +6 +0.1 +0.2 +0.3 +0.4 +0.5 +d ( +MD ) +kxn ( +MD +) +kz=0.1 +kz=0.2 +kz=0.3 +kz=0.4 +(c) +Figure 7: Results of numerical calculations for the case χ = 2.5 and θ = 0. (a) +Position of minima for the lowest mode vs d for thin films. (b) The value of +frequency in minimum for the lowest mode vs d for thin films. (c) kxn for the +lowest mode vs d at fixed kz = 0.1, 0.2, 0.3, 0.4. Black solid curves correspond to +our numerical calculations, circles are numerical calculations by Kreisel et al.. +[16].These figures agree with the figures from [11]. +If the film’s thickness is of the order of one or less (ℓ in dimensional units), +it is regarded as thin. The experimental realization of ultrathin films of YIG +with d ≪ 1 looks very improbable since the typical value of ℓ (in YIG) is +a few tens of nanometers. +It may be accomplished in thin, monolayer-thick +ferromagnetic materials. Transverse modes with high n in thin films with d ∼ 1 +have kxn ≈ πn/d ≫ 1 in the exchange dominance area. Thus, only a few modes +with the lowest frequencies are of theoretical and experimental relevance. In +these modes, evanescent waves penetrate to the film at a depth of the same +order of magnitude as its thickness. They therefore play an equally essential +role in spectral characteristics and TDM as the oscillating wave. +A compact analytic expression has been found only for frequency as function +of the wave vector (see eq. (46)). +Fig. 6 shows examples of spectra in thin films that are qualitatively similar +to spectra in thick films. Each mode determined by numbers ν, n at not very +big n has a frequency minimum at some k∥ ̸= 0, but it does not follow equation +∂ω2 +∂k2 +∥ = 0 since kxn also depends on k∥. Fig. 6(a) and Fig. 6(b) show that at +d = 1, the energy of transverse excitation weakly depends on kz, a feature that +could be expected for ultrathin films. +The graphs of position of minima and the value of frequency in minimum +for the lowest mode vs d for thin films are shown in Fig.7. In the same figures +7(a) and 7(b), we compared our results with calculations of the same values by +Kreisel et al. [16]. Finally, the graphs of kxn for the lowest mode vs d at fixed +k∥ and θ = 0 are shown in Fig. 7(c). An example of TDM for lowest mode and +first excited mode in thin films is shown in Fig. 8. +All ground state spectra cross at the point k∥ = 0, ω ≈ √1 + χ ( +√ +3 ≈ 1.73 +for χ = 2), exactly the same result as for the thick film. This is manifestation of +a general property of films with arbitrary thickness: at k∥ = 0, the transverse +wave vector of the lowest transverse mode is also equal to zero. The frequency +of the lowest mode equals to ω0 = √1 + χ (ferromagnetic resonance frequency). +22 + +-0.4 +-0.2 +0.0 +0.2 +0.4 +1.5 +2.0 +2.5 +3.0 +x ( +MD ) +TDM +my +mx +(a) +-0.4 +-0.2 +0.0 +0.2 +0.4 +-1.0 +-0.5 +0.0 +0.5 +1.0 +x ( +MD ) +TDM +my +mx +(b) +Figure 8: For the case χ = 2 and θ = 0 (a) TDM for the lowest mode at k∥ = 0.1 +and a1x = 1. (b) TDM for the first excited mode at k∥ = 0.1 and b1x = 1. These +figures agree with the figures from [11]. +We consider first the limiting case of ultrathin films d → 0 when θ = 0. +It will be shown that only wave vectors of the lowest transverse mode with +ν = −, n = 0 remains finite in this limit. All excited transverse state with other +ν or n have wave vectors that go to infinity as 1/d. We just take into account +the simplest scenario of waves propagating along magnetization and magnetic +field in order to simplify calculations. The transverse mode then has a definite +parity. +In such a case, the non-zero amplitudes are ai for even modes and bi for +odd modes. For finite wave vectors ki in the taken limit, sin kixd/2 ≈ kixd/2 +and cos kixd/2 ≈ 1 are appropriate values. This fact simplifies the SBC (64) +and self-consistency equations (61). The second simplification results from the +fact that the relationship between the x and y components of the vectors ai and +bi is reduced to aiy = +ω +1+k2 +i aix and biy = +ω +1+k2 +i bix, respectively. Here we denote +three kernels of cubic equation for k2 (45) as k2 +1, k2 +2, k2 +3 and corresponding vector +amplitudes at sin(kixx) and cos(kixx) as ai, bi. Let us remind that k2 +1 > 0, +whereas k2 +2, k2 +3 < 0. After all these simplifications, the quantization of an even +mode is described by the system of three equations with three independent +amplitudes aix : +� +� +� +� +� +�3 +i=1 k2 +ixaix += 0 +�3 +i=1 +k2 +ix +1+k2 +i aix += 0 +�3 +i=1 +aix +k2 +i += 0 +(98) +Zeros of determinant of this system determine quantized values of k2 +xn. In order +to transform this determinant into an explicit function of kxn one should employ +the relations k2 +1 = k2 +xn + k2 +z, +k2 +2,3 = −1 − χ +2 − k2 +1 +2 ± +�� +1 + χ +2 + k2 +1 +2 +�2 +− χk2z +k2 +1 +(99) +23 + +approximation +kxn +0 +2 +4 +6 +8 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +kz ( +MD +) +kxn ( +MD +) +Figure 9: Plot of kxn at d → 0 and approximation to it when χ = 2 and θ = 0. +This figure agrees with the figure from [11]. +and k2 +ix = k2 +i − k2 +z. The only positive root of this equation at small kz ≪ 1 is +kxn ≈ +� +χ +2 + χ +�1/4 � +kz +(100) +At large kz, kxn asymptotically approaches a constant value kxn ≈ +� +χ/2. +Both these asymptotic values agree very well with numerical calculations of the +dependence of kxn on kz at d → 0 (see Fig.9). The fact that kxn = 0 at kz = 0 +is confirmed by the asymptotic behavior of kxn at small kz. As a result, both in +the limit of small d and the limit of large d, the value of frequency at k∥ = 0 is +√1 + χ. On Fig. 10, the plots of kxn vs. kz at d = 1 and d = 0 are compared. +We can now demonstrate the general proposition that, regardless of thick- +ness, the frequency of the lowest mode at k∥ = 0 equals √1 + χ . Set ky = 0 +and consider kz ≪ 1/d2. We will show that the same equation (100) deter- +mines the first quantized value kxn, but the arguments must be modified. In +order to prove the result (100), let us assume that the initial quantized value +of kxn obeys the strong inequalities kz ≪ kxn ≪ 1. Then eq. (99) implies that +k2 +2x ≈ −χk2 +z/ +� +(2 + χ) k2 +xn +� +has small magnitude, whereas k2 +3x ≈ −2 − χ has +the magnitude of the order of unity. Let us first consider the SBC (64) that in +considered situation take form +k2 +xna1x + k2 +2xa2x − +� +2 + χ2 sinh √2 + χd/2 +d +a3x = 0 +k2 +xna1x + k2 +2xa2x + 2√2 + χ sinh √2 + χd/2 +(1 + χ)d +a3x = 0 +(101) +These equations imply a3x = 0. Then they become identical and define the +ratio a2x/a1x = −k2 +xn/k2 +2x. Next consider the self-consistency equations that in +the same limit have a form: +a1x +k2 +1 ++ a2x +k2 +2 += 0 +24 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +kz ( +MD +) +kxn ( +MD +) +d +0 +d=1 +Figure 10: kxn vs. kz for the lowest mode at d → 0 and d = 1 at χ = 2 and +θ = 0.This figure agrees with the figure from [11]. +Using the previously found ratio a1x/a2x, we again obtain eq. (100) for this more +general situation. It shows that in the limit kz → 0, the limit of ratio k2 +z/k2 +xn +is also zero and limiting value of ω is √1 + χ independently on thickness. Note +that in the limit k∥ = 0 the magnetization in the lowest spin-wave mode does +not depend on transverse coordinate. +Although thin films are more sensitive to the exact form of the SBC than +thick films, changing forms of these requirements have no effect on the symmetry +or general features of solutions. An important problem is how the wave vector +kzmin corresponding to the minimum of energy changes with thickness. For thick +films it behaves as 1/ +√ +d [15] and grows when film becomes thinner. However, +in the case of ultrathin films, it decreases linearly with thickness. +It means that the wave vector kzmin as function of d has a maximum. Ac- +cording to numerical calculations shown in Fig. 7(a) for χ = 2.5 the maximum +is located at d ≈ 6, and the maximum value of kzmin is around 0.3. For d = 5µm +and χ = 2, kzmin is around 0.02. Thus, by decreasing thickness from 5µm to +15−30 nm, the wave vector kzmin may be modified by a factor of roughly 15. The +size of any soliton-like formation constructed of magnons that may be utilized +for information transfer without dissipation or with very little dissipation has +an upper limit determined by the minimal wavelength of a magnon, according +to [20]. +4 +Interaction of magnons. +Previously we considered only quadratic in amplitudes part of the Hamilto- +nian. Here we take into account higher order contributions, i.e, we consider the +magnon interaction. The expansion will be limited by the terms of the third and +the fourth order. The expansion must be applied only to the exchange (17) and +dipolar (19) Hamiltonians since the Zeeman Hamiltonian is purely quadratic. +25 + +4.1 +Third order terms. +Let us first write out the 3rd order terms of the Hamiltonian, which come solely +from the dipolar part: +Hd3 = −µB +√2µBM +2 +¨ � +|ψ|2 + 1 +4 |ψ′|2 +� +∂z +� +ψ′∂′ +− + ψ′∗∂′ ++ +� dV dV ′ +|r − r′|. +(102) +In terms of the Fourier transforms defined by eq. (23) and employing the identity +1 +|r − r′| = 4π +A +� +q +eiq(r−r′)Gq (x − x′) , +(103) +where the 1d Green function is defined by eq. (29), we find: +Hd3 = − 2πµB +√2µBM +√ +A +˜ ∞ +−∞ dxdx′ +� +q1,q2,q3,q +� +χq1χ∗ +q2δq1−q2+qδq3−q ++ 1 +4χ′ +q1χ′∗ +q2δqδq1−q2+q3−q +� +iqz× +� +χ′ +q3 (dx′ − qy) + χ′∗ +−q3 (dx′ + qy) +� +Gq(x − x′) +(104) +The second term in the sum contains the factor δq that makes qy = qz = 0. +Thus, the square bracket in this equation is equal to +� +χ′ +q3 + χ′∗ +−q3 +� +dx′. Acting to +Gq(x−x′), the operator dx′ transforms it into qsign (x − x ′) Gq(x − x ′) = +sign(x−x ′) +2 +. +Thus, the second term in the sum is zero. The Kronecker δ−symbols in the first +term imply that q = q3 = q2 − q1. Thus, the dipolar Hamiltonian of the third +order is simplified to +Hd3 = − 2πµB +√2µBM +√ +A +˜ ∞ +−∞ dxdx′ +� +q1,q2 χq1χ∗ +q2i (q2z − q1z) +� +χ′ +q2−q1 (dx′ − q2y + q1y) ++χ′∗ +q1−q2 (dx′ + q2y − q1y) +� +G|q1−q2| (x − x′) +(105) +4.1.1 +Third order non-linearity in terms of quantized magnon am- +plitudes. +In this section we perform the Bogoliubov transformation (37) from transverse +modes χq (x) to the quantized amplitudes of magnons ηq,n, η∗ +q,n. After some +algebra we arrive at a cubic form for these amplitudes limited by the requirement +of the momentum conservation (translational invariance): +Hd3 = − 2πµB +√2µBM +√ +A +� +q1n1;q2n2;q3n3 δq1−q2+q3 +� +I(+++) +d3 +ηq1n1η−q2n2ηq3n3 + I(++−) +d3 +ηq1n1η−q2n2η∗ +−q3n3 +I(+−+) +d3 +ηq1n1η∗ +q2n2ηq3n3 + I(−++) +d3 +η∗ +−q1n1η−q2n2ηq3n3 + c.c +� , +(106) +where the eight coefficients I(ρστ) +d3 +with ρ, σ, τ taking values +, − are matrix +elements of the three transverse modes: the first is u∗ +q1n1 (x) for ρ = + and +26 + +u−q1n1 for ρ = −; the second is v∗ +−q2n2 (x) for σ = + and vq2n2 for σ = −; the +third is given by +iq3z +� +u∗ +q3n3 (x′) (dx′ − q3y) − v∗ +q3n3 (x′) (dx′ + q3y) +� +Gq3 (x − x′) +for τ = + and +iq3z +� +u−q3n3 (x′) (dx′ + q3y) − v−q3n3 (x′) (dx′ − q3y) +� +Gq3 (x − x′) +for τ = −. +The matrix element is the double integral over x and x′ from the products of +any set of these three modes. +For the reader convenience we place below explicit expressions for the integrals +I(ρστ) +d3 +with all three indices + and with two + and one −: +I(+++) +d3 += −iq3z +˜ +dxdx′u∗ +q1n1v∗ +−q2n2× +� +u′∗ +q3n3 (dx′ − q3y) − v′∗ +q3n3 (dx′ + q3y) +� +Gq3(x − x′) +I(++−) +d3 += −iq3z +˜ +dxdx′u∗ +q1n1v∗ +−q2n2× +� +u′ +−q3n3 (dx′ + q3y) − v′ +−q3n3 (dx′ − q3y) +� +Gq3(x − x′) +I(+−+) +d3 += iq3z +˜ +dxdx′u∗ +q1n1vq2n2× +� +u′∗ +q3n3 (dx′ − q3y) − v′∗ +q3n3 (dx′ + q3y) +� +Gq3(x − x′) +I(−++) +d3 += iq3z +˜ +dxdx′u−q1n1v∗ +−q2n2× +� +u′∗ +q3n3 (dx′ − q3y) − v′∗ +q3n3 (dx′ + q3y) +� +Gq3(x − x′) +. +(107) +In order to obtain the Hamiltonian Hd3 (106) and coefficients I(σρτ) +d3 +we have +used the fact that some terms (e.g. the term with η∗ +−q1n1η∗ +q2n2η∗ +−q3n3) can be +expressed as complex conjugates of others (e.g. the term with ηq1n1η−q2n2ηq3n3) +by permutation of the summation indices q1 ↔ q2 that implies q3 → −q3 . +Later we will use this kind of relations when calculating 4th-order terms. Note +also that the three terms involving one complex conjugated function in eq. (106) +can also be received each from other by renaming the summation indices. Thus, +these three sums are identical. On the other hand two last of them are complex +conjugates each to other. Therefore, all these sums are real. +4.1.2 +Cherenkov radiation of a low energy magnon by the high en- +ergy magnons. +In the theory of BECM the life-time of the condensate magnons is dominantly +determined by their merging with a high energy magnon and by the inverse +process of the Cherenkov radiation of the condensate magnon by a high energy +magnons. +Here we consider a more general problem when the high energy +magnon emits or absorbs a low energy magnon. The high-energy magnon is +assumed to have the exchange dominated dispersion ωq,kx = γℓ2k2, whereas +the low-energy magnon dispersion is given by eq. +(46). +In the Bogoliubov +coefficients uqn the coefficients a, b dominate for ν = +, c, d dominate for ν = −, +whereas vqn = 0 . For low-energy magnons generally the coefficients a, b, c, d +27 + +are of the same order of magnitude. They are defined by eqs. (49,50). For thick +films in the integrals (107) defining the matrix elements of the Cherenkov or +inverse Cherenkov process, the terms corresponding to evanescent waves can be +neglected. +4.2 +Fourth order terms. +Here we consider the 4th order terms of the Hamiltonian. In terms of general +magnon wave function ψ (r) they are: +H4 = +Hex4 + Hd4 +Hex4 = +µ2 +Bℓ2 +2 +´ � +− |ψ|2 |∇ψ|2 + 1 +2 +� +∇ +� +|ψ|2��2� +dV, +Hd4 = +µ2 +B +2 +˜ � +|ψ|2 |ψ′|2 ∂z∂′ +z − 1 +4 |ψ|2 (ψ∂− + ψ∗∂+) +� +ψ′∂′ +− + ψ′∗∂′ ++ +�� +dV dV ′ +|r−r′| +(108) +4.2.1 +Fourth order Hamiltonian in terms of magnon amplitudes χq (r). +Employing Fourier transformation to the wave vector representation (23), we +find the following expressions for Hex4 and Hd4: +Hex4 = µ2 +Bℓ2 +2A2 +´ � +q1,q2,q3,q4 +� +−χq1χ∗ +q2 +� +dxχq3dxχ∗ +q4 + q3q4χq3χ∗ +q4 +� ++ 1 +2dx(χq1χ∗ +q2)dx(χq3χ∗ +q4) + 1 +2(q1 − q2)(q3 − q4)χq1χ∗ +q2χq3χ∗ +q4 +� +ei(q1−q2+q3−q4)rdV += µ2 +Bℓ2 +4A +´ � +q1,q2,q3,q4 +� +dxχq1χ∗ +q2dxχq3χ∗ +q4 + χq1dxχ∗ +q2χq3dxχ∗ +q4 +−(q2 +1 + q2 +2)χq1χ∗ +q2χq3χ∗ +q4 +� +δq1−q2+q3−q4dx += µ2 +Bℓ2 +4A +´ � +q1,q2,q3,q4 +� +dxχq1χ∗ +q2dxχq3χ∗ +q4 + χq1dxχ∗ +q2χq3dxχ∗ +q4 +− 1 +2(q2 +1 + q2 +2 + q2 +3 + q2 +4)χq1χ∗ +q2χq3χ∗ +q4 +� +δq1−q2+q3−q4dx +(109) +Hd4 = +2πµ2 +B +A3 +˜ � +q1,q2,q3,q4,q +� +q2 +zχq1χ∗ +q2χ′ +q3χ′∗ +q4ei[(q1−q2)r+(q3−q4)r′+q(r−r′)]− +1 +4χq1χ∗ +q2 +� +χq3 (dx + qy) + χ∗ +−q3 (dx − qy) +� � +χ′ +q4 (dx′ − qy) + χ′∗ +−q4 (dx′ + qy) +� +×ei[(q1−q2)r+q3r+q4r′+q(r−r′)]� +Gq(x − x′)dV dV ′ +(110) +After integration over y, z and y′, z′ the 4th-order dipolar Hamiltonian trans- +forms into the sum over momenta and integral over transverse coordinates: +Hd4 = +2πµ2 +B +A +˜ � +q1,q2,q3,q4,q +� +q2 +zχq1χ∗ +q2χ′ +q3χ′∗ +q4δq1−q2+qδq3−q4−q +− 1 +4χq1χ∗ +q2 +� +χq3 (dx + qy) + χ∗ +−q3 (dx − qy) +� � +χ′ +q4 (dx′ − qy) + χ′∗ +−q4 (dx′ + qy) +� +×δq1−q2+q3+qδq4−q} Gq(x − x′)dxdx′. +(111) +In these calculation we used the symmetry with respect to permutations of +running momenta participating in the sum and the relation between Fourier +component of 1/ |r − r′| and one-dimensional Green function Gq (x − x′) (see +eq. (29)). +28 + +4.2.2 +Fourth order Hamiltonian in terms of the magnon amplitudes +ηqνn. +Employing the Bogoliubov transformation (38), we represent the 4-th order +Hamiltonian in terms of the homogeneous fourth order polynomials of the form +(the subscripts qi, ni in the coefficients I4 are omitted for brevity): +H4 = +� +qinkρl(i,k,l=1...4) +I(ρ1ρ2ρ3ρ4) +4 +� +� +4 +� +j=1 +η(ρj) +qjnj +� +� δq1+q2+q3+q4, +(112) +where η(+) +qn = ηqn; η(−) +qn = η∗ +qn. It is obvious that the matrix I4 can be made +invariant under permutation of four its composite indices γj = (ρjqjnj) ; j = +1, 2, 3, 4 since the product in eq. (112) is invariant under such permutation. +Therefore, it is more reasonable to denote the matrix elements of the matrix I4 +as (I4)γ1γ2γ3γ4. The table of coefficients (I4)γ1γ2γ3γ4 is given in the Appendix +[Hamiltonian of the 4-th order]. +4.2.3 +Interaction of condensate magnons in thick films. +Here we show the results of calculations of the interaction between condensate +of magnons that have momenta either Q = Qˆz or −Q. +When the conden- +sate exists, the chemical potential µ is equal to the minimal magnon energy +∆. Therefore the wave functions of the condensates ψ±Q do not depend on +time (we remind that the time dependence of the wave function is given by +exp +� +− i(∆−µ)t +ℏ +� +). Further for brevity we denote the wave functions of the two +condensates as ψ± and present them in terms of the densities of condensates n± +and their time-independent phases φ± as +ψ± = √n±eiφ±f (x) , +(113) +where f (x) = +√ +2 cos πx +d is the transverse wave function corresponding to the +ground state of a magnon. The total wave function is +ψ (r) = ψ+eiQr + ψ−e−iQr = +�√n+ei(Qz+φ+) + √n−ei(−Qz+φ−)� +f (x) . +(114) +Introducing notation n = n+ + n− for the total density of condensate and +Φ (z) = 2Qz + φ+ − φ− for the phase difference of the two condensates, we find +the square of modulus of the wave function: +|ψ (r)|2 = +� +n + 2√n+n− cos Φ (z) +� +f 2 (x) . +(115) +The square of gradient of the wave function is +|∇ψ|2 = Q2 � +n − 2√n+n− cos Φ (z) +� +f 2 (x) ++ +� +n + 2√n+n− cos Φ (z) +� � +df +dx +�2 +. +(116) +29 + +The fourth order exchange Hamiltonian contains two terms − |ψ|2 |∇ψ|2 and +1 +2 +� +∇ |ψ|2�2 +. Assuming that densities of condensates n± and their phases φ± +vary in plane on the distances much larger than period of density oscillation +L = 2π/Q, the density of interaction energy of condensates is equal to the +exact value of interaction energy averaged over period of oscillation L inte- +grated over the transverse coordinate x. For thick films the terms in ∇ψ and +∇ |ψ|2containing derivatives df +dx can be neglected in comparison with the terms +containing derivatives over z or equivalently the value Q since Qd ≫ 1. Per- +forming simple operations of averaging and integration for exchange interaction +we find: +Hex4 +V += −3µ2 +Bℓ2 +16 +Q2 � +n2 − 6n+n− +� +(117) +Analyzing in similar way the interaction energy generated by dipolar Hamilto- +nian of the 4-th order, we should find the average of the integrand in the third +equation (108). To make it, we will use the identity: +1 +|r − r′| = 1 +π +∞ +¨ +−∞ +dqydqzeiq∥(r∥−r′ +∥)Gq∥ (x − x′) , +(118) +where the subscript ∥ at a vector means that it is parallel to the surfaces of the +film, i.e., they have only y and z−components; we remind that the 1-dimensional +Green function of the Helmholtz equation Gq (x) is defined by eq. (29). The +proof of the identity (118) is given in the Appendix [1/r-G-identity]. Thus, the +dipolar Hamiltonian of the 4-th order can be rewritten as follows: +Hd4 = µ2 +B +2π +˝ +dV dV ′d2q∥eiq∥(r∥−r′ +∥) +� +|ψ|2 |ψ′|2 ∂z∂′ +z − 1 +8 +� +|ψ|2 + |ψ′|2� +× +(ψ∂− + ψ∗∂+) +� +ψ′∂′ +− + ψ′∗∂′ ++ +�� +Gq∥ (x − x′) +(119) +Note that we symmetrized the integrand over the variables r and r′. Except +of the exponential function eiq∥(r∥−r′ +∥) the integrand does not depend of y and +y′. +Therefore the integration over y′ gives 2πδ (qy). +The partial derivatives +∂± = ∂x ± iqy become equal each to other and equal to ∂x. The magnitude +of derivatives ∂z, ∂z′ is equal to Q, whereas the magnitude of the derivatives +∂x, ∂x′ is equal to 2π/d. For thick films Q ≫ 1/d, therefore, the first term in +the square brackets of this equation dominates. In this approximation we find +Hd4 = µ2 +B +´ +dV +´ d/2 +−d/2 dx′ ´ ∞ +−∞ dz′ ´ ∞ +−∞ dqz +� +n + 2√n+n− cos Φ (z) +� � +n + 2√n+n− cos Φ (z′) +� +[f (x) f (x′)]2 q2 +zeiqz(z−z′)G|qz| (x − x′) . +(120) +Since the integrand does not depend on y, the integration over this variable +gives the linear size of sample Ly. +Let us make change of variables Z = +z+z′ +2 , ζ = z − z′. +The Jacobian of this transformation is 1. +The only term +30 + +in the product of two square brackets in eq. (120) that together with expo- +nential factor eiqz(z−z′) gives non-zero average is 4n+n− cos Φ (z) cos Φ (z′) = +2n+n− [cos (Φ (z) + Φ (z′)) + cos (Φ (z) − Φ (z′))]. From these two terms only +the second gives nonzero average over z: +∞ +ˆ +−∞ +dζeiqzζ2 cos (2Qζ) = 2π [δ (qz − 2Q) + δ (qz + 2Q)] +(121) +This result allows us to perform also integration over qz. Besides of that the +integrand does not depend on Z and integration over this variable gives the +linear size Lz. These integrations strongly simplify the expression for Hd4: +Hd4 = 4πµ2 +BLyLzQn+n− +d/2 +¨ +−d/2 +f 2 (x) f 2 (x′) e−2Q|x−x′|dxdx′ +(122) +The calculation of the double integral in eq. (122) is elementary and gives: +˜ d/2 +−d/2 f 2 (x) f 2 (x′) e−2Q|x−x′|dxdx′ += − +−3d5Q5−5π2d3Q3+π4(−2dQ−e−2dQ+1) +2Q2(d2Q2+π2)2 +, +(123) +In the limit of thick film Qd ≫ 1 the leading term is equal to 3d/2Q. This +dependence of the integral in (123) on parameters as d/Q could be predicted +without detailed calculation since the exponent e−Q|x−x′| cut in the square of +integration a band of the width ∼ 1/Q along the diagonal, whereas the average +value of f 2 is 1. However, strong fluctuations of f 2 from 0 to 1 with period 1/8 +of the diagonal requires explicit calculation to get exact numerical coefficient at +the leading term: +Hd4 +V += 6πµ2 +Bn+n−. +(124) +Thus, we have found the density of interaction energy between condensates of +different minima (the inter-minima interaction). It can be written as +U4int = Bn+n− +(125) +with B = 6πµ2 +B > 0. It is repulsion. Note that the terms of the same form +in the exchange interaction energy (117) has coefficient B which differs from +dipolar value by a factor ∼ Q2ℓ2 ∼ ℓ/d ≪ 1 that can be neglected. +Another term that enters Hex4/V but is absent in Hd4/V is interaction of +the condensate magnons within one minimum +U4inn = A +2 +� +n2 ++ + n2 +− +� +(126) +with A = − 3 +8µ2 +BQ2ℓ2. Thus, the interaction within one minimum is attraction. +The magnitude |A| is much smaller than B: |A| /B = π1/2 +27/2 +ℓ +d. For YIG film 5μm +thick at room temperature |A| /B = 0.012. +31 + +4.2.4 +Quasi-equilibrium state. +In the experiment by Demokritov et al. [12] the low energy magnons in the YIG +film were generated by a microstrip resonator. A photon of frequency ωres emit- +ted by the resonator decays into two magnons with practically opposite momenta +and frequency ωp = ωres/2 (in classical electrodynamics this process is called +parametric resonance or parametric pumping). The resonator frequency is cho- +sen to be less than 4∆/ℏ, where ∆ ≈ 2µBH is the minimal energy of magnons +(gap in the spectrum). Then the decays of pumped magnons are forbidden, +whereas their collisions with other low energy magnons remain possible. These +collisions establish the equilibrium. The relaxation time τr is just the time be- +tween collisions. An important role is played by the processes of the Cherenkov +radiation of a low-energy magnon by a thermal magnon and inverse process of +the absorption of the low-energy magnon by a thermal magnon. These processes +determine the lifetime of low-energy magnons τl. In YIG at room temperature +τr ≪ τl. It means that during the relaxation the number of magnons is con- +served and they go to equilibrium with the finite chemical potential µ. The +role of pumping is to restore the stationary number of magnons in exchange of +absorbed ones. We will call such a stationary state quasi-equilibrium. +Let us consider the balance of magnons following Bun’kov and Volovik.[21] +The occupation number of a low-energy magnon with energy ε in the quasi- +equilibrium state is n (ε) = +T +ε−µ. The occupation number of the magnon with +the same energy in equilibrium without pumping is n0 (ε) = +T +ε . +The total +density npm (T, µ) of pumped magnons is +npm (T, µ) = +∞ +ˆ +0 +[n (ε) − n0 (ε)] ¯g (ε) dε, +(127) +where ¯g (ε) is the magnon density of state per unit volume. It can be rewritten +as +npm (T, µ) = +∞ +ˆ +0 +Tµ +ε (ε − µ) ¯g (ε) dε. +(128) +The density of magnons pumped per unit time is determined by the pumped +power W per unit volume as +2W +ℏωres . In a stationary state it must be equal to +the density of pumped magnons that disappear per unit time npm +τl . Thus, the +established density of pumped magnons is +npm = 2W +ℏωres +τl. +(129) +Replacing npm by the integral in the r.-h. side of eq. (128), we obtain equation +relating the chemical potential µ to the pumped power W. This equation implies +that µ grows monotonically with W growing. At a critical value of the pumped +32 + +power +W (c) = ℏωres +2τl +∞ +ˆ +∆ +T∆ +ε (ε − ∆) ¯g (ε) dε, +(130) +chemical potential reaches its maximum possible value µmax = ∆ and the density +of pumped magnons reaches its critical value +n(c) +pm = +∞ +ˆ +∆ +T∆ +ε (ε − ∆) ¯g (ε) dε. +(131) +Chemical potential cannot grow more since at µ > ∆, the occupation number +of magnons with energy between ∆ and µ would be negative that is nonsense. +Therefore, at W > W (c) the chemical potential remains unchanged µ = ∆. The +excessive magnons go to the state with minimal energy ∆ and form the BEC. +The condensate density is +nc = 2 +� +W − W (c)� +ℏωres +τl. +(132) +All these calculations assumed that the integrals are converging. +There are +two possible sources of divergence: large energies ε → ∞ and ε close to ∆ for +W ≥ W (c). For large ε the exchange interaction dominates, the magnon energy +is quadratic function of momentum and ¯g (ε) ∝ √ε,whereas the denominator +of integrand in eq. +(128) asymptotically approaches ε2. +Thus, the integral +converges at ε → ∞. This result physically means that the pumped magnons +after relaxation remain in the range of low energy ∼ ∆. Paradoxically their +energy escapes into the range ι ∼ T. Indeed, the pumped energy is +Epm = +∞ +ˆ +0 +Tµ +ε (ε − µ)ε¯g (ε) dε. +(133) +This integral diverges at ε → ∞. It happens because we applied low-energy +Rayleigh-Jeans approximation n (ε) = +T +ε−µ, n0 (ε) = T +ε for the occupation num- +bers of magnons, which at high energy must be replaced by the Planck-Bose- +Einstein distribution n (ε) = +� +exp ε−µ +T +− 1 +�−1 , n0 (ε) = +� +exp ε +T − 1 +�−1. Thus, +the integral (133) is cut-off at ε ∼ T. +Neglecting µ in denominator of in- +tegrand, we find the rough estimate of the pumped energy per unit volume +µT 3/2/ +� +(µBM)3/2 ℓ3� +that corresponds to the change of the magnons temper- +ature by δT ≈ ∆/kB. For YIG film in external magnetic field H = 600Oe and +at room temperature, the resulting increase of temperature is about 0.04K. +The convergence at the points of minimum energy ϵ = ∆ follows from the +fact that, in the continuous limit, they are isolated points in 3-dimensional space. +Therefore, the density of states near each minimum goes to zero as +√ +ϵ − ∆. +33 + +4.2.5 +Spontaneous violation of the reflection symmetry in the quasi- +equilibrium state. +In the state of quasi-equilibrium its energy (more accurately its Helmholtz free +energy) must be minimum. At fixed temperature and volume, the free energy +has minimum when the occupation numbers obey the Bose-Einstein law and +excessive magnons occupy the state with minimal energy ∆. In ferromagnetic +films there are two such states. Therefore, the ground state of the ideal magnon +gas is highly degenerate: the condensate energy Eid = V nc∆ depends only on +the total number of magnons in condensate Nc = V nc = N+ + N− and does +not depend on how these magnons are distributed between two minima. This +Nc + 1−fold degeneration is lifted by magnons interaction.[20] +As it was derived in the subsection,4.2.3 the 4-th order interaction density +of energy is +U4 = A +2 +� +n2 ++ + n2 +− +� ++ Bn+n−, +(134) +with A < 0 and B > 0 for thick films. The interaction energy U4 has minimum +equal to U4 = − A +2 n2 either at n+ = n, n− = 0 or at n+ = 0, n− = n. In +both cases the symmetry with respect to reflection in the plane z = 0 combined +with the time reversal is violated. Unfortunately such a most asymmetric state +contradicts to the experiment and to a more sophisticated theory. +Let us start with the experiment. In 2012 in the work by P. Novik-Boltyk +et al. [22] the Münster experimental team led by S. Demokritov discovered a +stripe interference structure of the magnetization Mz in the YIG sample (see +the interference picture in Fig. 11.) It can be interpreted as the measurement +of +|ψ|2 = +��√n+eiQz+φ+ + √n−e−iQz+φ−�� = n + 2√n+n− cos (2Qz + φ+ − φ−) . +This equation clearly shows that the interference picture can be observed +only if both n+ and n− are not zero. In order to explain this result, F. Li, W. +Saslow and V. Pokrovsky[23] proposed to consider the additional term in the +4-th order interaction Hamiltonian of purely dipolar origin of the form +C +2 +�� +ψ∗ ++ψ2 ++ψ− + c.c. +� ++ (+ ↔ −) +� +(135) +where the abbreviation c.c. stays for complex conjugate, C is a real constants +whose magnitude in terms of parameters is of the same order as |A|, however +the numerical constant in C is by a factor 1/2π3 ≈ 0.016 smaller. This term +is contained in the earlier neglected terms of the 4-th order dipolar interac- +tion containing derivatives over x. The real processes associated with this term +would be decay of one condensate magnon in three and inverse process of merg- +ing three condensate magnon in one. All such processes are forbidden by the +energy conservation. However, they determine additional (anomalous) 4-order +interaction energy: +H4an +V += Cn√n+n− cos (φ+ + φ−) +(136) +34 + +Figure 11: Measurement of the BLS intensity. Dashed circles indicate the posi- +tions of two defects causing an appearance of two vortices of positive circulation +in different components of the condensate. The vortices show themselves as forks +in the interference pattern.Reprinted by permission from Macmillan Publishers +Ltd: Scientific Reports [22], Copyright 2012. +Note that this energy depends on a different combination of phases φ+ + φ− +than the Goldstone phase φ+ − φ− whose variation does not change energy. +The minimum energy is reached at φ+ + φ− = π or 0 depending on the sign +of the coefficient C. On the line C = 0 the transition from 0− to π−phase or +vice versa proceeds. In both these phases the minimum anomalous interaction +energy is negative: +min +�H4an +V +� += − |C| n√n+n−. +(137) +Thus, the total 4−th order interaction energy acquires the form: +U4 ≡ H4 +V += A +2 +� +n2 ++ + n2 +− +� ++ Bn+n− − |C| n√n+n− +(138) +Its minimization at a fixed n gives: +n +√n+n− += 2 (B − A) +|C| +. +(139) +Let us denote R = B−A +|C| + +�� +B−A +|C| +�2 +− 1 and Θ = +|C| +2(B−A)R. The value R is +very big, whereas the value Θ ≈ +1 +4R2 is very small. The two solutions of this +equation are either +n+ = (1 − Θ) n; +n− = Θn, +(140) +35 + +5.0 +4.5 +4.0 +1.0 +0.9 +3.5 +0.8 +0.7 +3.0 +0.6 +y(μm) +0.5 +2.5 +0.4 +2.0 +0.3 +0.2 +1.5 +0.1 +0.0 +1.0 +0.5 +0.0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +z(μm)or n+ and n− interchange. In each solution one of two condensate densities is +much larger than another, but the smaller one turns into zero only if C = 0. +The total interaction energy in this phase is +U4 = n2 � +A +2 + |C| +2R (1 − Θ) − |C| +� +Θ (1 − Θ) +� +≈ n2 � +A +2 − C2 +4B +� +< 0 +(141) +4.2.6 +Instability of homogeneous asymmetric phase. +We have found that the homogeneous phase with the violated reflection sym- +metry has negative interaction energy proportional to n2. It means that the +interaction energy decreases when the volume occupied by the condensate de- +creases. In the weakly non-ideal attractive Bose-gas of N particles with the +coupling constant g < 0 and mass m of particle this tendency leads to the me- +chanical instability of the gas and its collapse at a critical value of number of +particles Nc . At this value the isothermal compressibility κT = − 1 +V +� ∂V +∂P +� +T is +zero and at N > Nc becomes negative. Due to quantum uncertainty, the kinetic +energy per particle can be written as K/N = +ℏ2 +2mV 2/3 , whereas the interaction +energy is U = gN 2 +2V . Thus, the total energy is +E (N, V ) = N +ℏ2 +2mV 2/3 + gN 2 +2V . +(142) +The pressure is +P = −∂E +∂V = +ℏ2N +3mV 5/3 + gN 2 +2V 2 +(143) +and the compressibility is +κT = +5ℏ2N +9mV 5/3 + gN 2 +V 2 +(144) +Equation κT = 0 determines the critical number of particles Nc = − 5ℏ2V 1/3 +9mg +. +At N > Nc, the compressibility is negative and the gas becomes mechanically +unstable. It starts to contract. Since this process proceeds simultaneously in +the total volume occupied by the gas, the process will stop when the volume +wil be divided into N/Nc cells each containing Nc particles and isolated each +from other. The volume of such a cell is v = V Nc/N, therefore the critical +number in a cell is different than the critical number in the entire volume. It +should be found from equation Nc = +5ℏ2 +9m|g| +� V Nc +N +�1/3. It is convenient to express +the coupling constant g in terms of the Born scattering length as as g = ℏ2 +m as. +Then Nc = 53/2 +27 +1 +n1/6|as|1/2 , where n = N/V is the average density of particles. +For a weakly interacting Bose gas n1/3 |as| ≪ 1. +Therefore Nc ≫ 1. +The +collapse was observed in cooled gases of alkali atoms 7Li [24] and 85Rb [25]. At +finite temperature the pressure from excitations must be included. It changes +the critical values for starting the collapse, but the collapse persists. Gases of +36 + +cooled attracting alkali atoms after the collapse flew out the magnetic or laser +trap. Our calculations relate to the Bose gas of quasiparticles that cannot avoid +the system in which they exist like excitons in semiconductors or spin waves in +magnets. +Theoretical predictions for starting parameters of collapse in weakly attract- +ing Bose gas at finite temperature were made by Mueller and Baym.[26] Dynamic +approach to the same problem was developed by Pitaevskii. [27] +For the magnon condensation in a ferromagnetic film, the problem is effec- +tively two-dimensional. It is because the minimum of energy corresponds to +the transverse standing wave, period of which fits between surfaces of the film. +The effective masses are strongly anisotropic (see subsection 3.5.3). The curve +of constant kinetic energy is the ellipsis +ℏ2k2 +y +2my + ℏ2k2 +z +2mz = K. Therefore we ex- +pect that the collapsed magnon condensate will be limited by an ellipsis with +semi-axes Ry, Rz whose ratio is Ry/Rz = +� +mz/my. Then the kinetic energy of +collapsed condensate can be estimated as +K = N +� +ℏ2 +2myR2y ++ +ℏ2 +2mzR2z +� += +Nℏ2 +√mymzRyRz +, +(145) +whereas the condensate potential energy is +U = +gN 2 +2πRyRzd. +(146) +The pressure at zero temperature is +P = N +V 2 +� +πℏ2d +√mymz ++ g +2N +� +, +(147) +where V = πRyRzd is the volume of the condensate cloud. The compressibility +of the magnon gas in the film differs from the pressure only by numerical factor +2. Thus, the pressure and compressibility simultaneously become zero when N +reaches a critical value +Nc = − +2πℏ2d +√mymzg = 2πd +|as| . +(148) +The film will be divided into N/Nc almost isolated cells each containing Nc +magnons. Let the cell be a rectangle with the sides Ry, Rz. If A is the area of the +sample, then the area of a cell is RyRz = ANc +N +. From this equation and require- +ment Ry/Rz = +� +mz/my we find Ry = +� +mz +my +�1/4 � +ANc +N ; Rz = +� +my +mz +�1/4 � +ANc +N . +According to eq. (148), this result can be rewritten as +Ry = +�mz +my +� � +2π +n |as|; Rz = +�my +mz +�1/4 � +2π +n |as|, +(149) +where n = N/(Ad) is the average density of magnons. The collapse destroys the +homogeneous coherent condensate transforming it into a set of isolated islands. +37 + +For YIG with n = 1018cm−3 we find Nc ≈ 1.14 × 105, Ry ≈ 1.16 × +10−6cm; Rz ≈ 0.58 × 10−7cm. There is no experimental evidence of the cell +structure in YIG films. +4.3 +New experiments and our new theoretical ideas about +slow inter-minima relaxation and laser effects. +Two recent articles by the Münster University experimental team led by S.O. +Demokritov [28, 29] revealed several important facts about the Bose-Einstein +condensation of magnons (BECM) under permanent pumping first discovered in +2006 [12]. Existing theories of this phenomenon predict an attractive interaction +between magnons [30, 31, 23] and a strong spontaneous violation of the reflection +symmetry [23]. However these theories implicitly assumed that all relaxation +processes were fast compared to the lifetime of magnons, whereas one of them, +the relaxation between two energy minima, is slow. +We predict the properties of the stationary state of the magnon gas with +condensate, that is far from equilibrium with respect to variables responsible +for inter-minima coherence. The momentum-flip relaxation time is no less than +1 hour, which exceeds even the time of the experiment without considering +the lifetime. It means that the equilibrium between condensates in different +minima is never reached. As a result, the condensates’ stationary state is far +from equilibrium. In this regard, it is analogous to the laser stationary state, +and, like the laser, the magnon condensate state can produce coherent magnon +radiation [32, 33]. +The very slow inter-minima relaxation implies that the appearance of the +stationary condensate in a ferromagnetic film is a dynamic phase transition. +Since the inter-minima equilibrium is not established, the pumping, which is +symmetric with respect to the two minima, creates equal numbers of magnons +in the two condensates n+ = n− = nc/2. Therefore, the inter-minima repul- +sion energy Bn+n− = Bn2 +c/4 strongly exceeds the magnitude of in-minimum +attraction |A| +2 (n2 ++ + n2 +−) = |A|n2 +c/4. This consideration explains why experi- +menters observe repulsion of magnons in the stationary state with the conden- +sate. With a somewhat more sophisticated point of view, the mirror symmetry +of the pumping does not necessarily lead to the same symmetry of the conden- +sate. In principle, dynamic violation of mirror symmetry is possible. But in +this case, there is no reason why it should be strong. This issue requires further +theoretical investigation. +It is difficult to avoid a slight asymmetry of the device in real-world experi- +ments, which favors a slightly asymmetric stationary state. Such a device asym- +metry could explain the asymmetry observed in the experiment II by Borisenko +et al. If the asymmetry is relatively small, then in eq. (139) the term Bn+n− +is dominant and positive, but it completely conceals the possibility of dynamic +spontaneous violation of the reflection symmetry. +Because the inter-minima equilibrium is not established, the consistent the- +ory of the stationary state with condensate necessitates solving the Boltzmann +38 + +kinetic equation for magnons and the Gross-Pitaevskii equations for the two +condensates. It can be accomplished using either a variational technique based +on the idea of maximal entropy production or by solving a problem with proper +initial conditions that asymptotically approaches a stationary state. +This kinetic approach may help to bridge another gap between the existing +theories [21, 20] and experiment [32, 33]. The theory proofs that the pumped +magnons are accumulated in the low-energy region assuming the temperature +of accumulated magnons to be the same as initial temperature of the system +(room temperature). The temperature of low energy magnons is approximately +three times higher, according to experimental data. +If the temperature is a +slow-varying function of energy (momentum) that saturates to the system’s +room temperature at some intermediate energy between µBH and the room +temperature, the controversy may be resolved. +Acknowledgments. +We are thankful to T. Nattermann, W.M. Saslow, Fuxiang Li, Chen Sun to- +gether with whom were obtained many results mentioned in this article. Our +gratefulness is due to S.O. Demokritov and participants of his experimental team +V. E. Demidov, I. Borisenko, B. Divinskii, P. Novik-Boltyk for many useful dis- +cussions of the experimental results and cooperation. We thank J. Ketterson +and J. Lim for explanation of their experiment and discussion of its results. +We are indebted to B. Hillebrands, A. Serga and D. Bozhko for discussion of +their experiments. Many theoretical problems were discussed with A.N. Slavin, +V.S. L’vov and G. E. Volovik, who also informed us on a vast literature on the +subjecr. Our thanks to them. We remember thankfully the discussion with +deceased L.P. Pitaevskii on the instability of attractive Bose condensate. +Appendix 1. Motion of minima. +The dependence of frequency on wave vector is determined by equation (82) of +the main text. For the reader’s convenience we reproduce it: +ω2 = µ2 +BH2 � +1 + k2� � +1 + k2 + χ − χk2 +z +k2 +� +(150) +Here k2 = k2 +∥ + k2 +x, where kx is a positive quantized transverse component of +wave vector. Generally to find minimum of frequency for a given mode with fixed +quantum numbers and direction of propagation, it is necessary to take in account +the dependence of quantized kx on k∥. This dependence can be neglected in thick +films with d ≫ 1. Indeed according to the main text, quantized values of kx are +equal to kx,ν,n = 2πn +d +µν,n. Here µν,n = 2 +d arctan fν,n +� +k∥ +� +, where fν,n +� +k∥ +� +is a +smooth function. According to this definition, µν,n varies in the limits +� +− π +d , π +d +� +when k∥ changes at least by 1/ +√ +d. Therefore, the derivative dkx +dk∥ ≲ +1 +√ +d ≪ 1 and +the values k∥ and kx can be considered as independent. In this approximation +39 + +the value of parallel wave vector k∥0 at which frequency has minimum can be +found from equation: +∂ω2 +∂ +� +k2 +∥ +� = 2k2 + 2 + χ sin2 θ − χk2 +x cos2 θ +k4 += 0 +(151) +At small kx i.e. at n ≪ d/2π, the value k2 satisfying eq. (151) is also small and +equal to +k2 +0 ≈ k2 +∥0 ≈ +� +χ +2 + χ sin2 θkx cos θ +(152) +It is however much larger than k2 +x. +The value of frequency in minimum is +ωmin ≈ +� +1 + χ sin2 θ. +The equation for k2 +0 valid in the range of larger kx +comparable with 1 can be found by the following scaling transformation: +k2 +0 = 2 + χ sin2 θ +2 +w (ξ) ; ξ = +4χk2 +x cos2 θ +� +2 + χ sin2 θ +�3 , +(153) +where function w (ξ) obeys cubic equation: +w3 + w2 = ξ +(154) +At small ξ, this equation gives the result (152). This equation shows that at +small kx, the wave vector corresponding to minimal frequency k∥0 grows with +kx. To study the motion of minimum in a broader interval of kx it is useful +to look at the derivative +dk2 +∥0 +d(k2x). According to eq. (151), it can be expressed as +follows: +dk2 +∥0 +d(k2x) = − +∂2ω2 +∂ +� +k2 +∥ +� +∂(k2x) +∂2ω2 +� +∂ +� +k2 +∥ +��2 +(155) +From this equation it follows that maximal value of k∥0 can be found from +equation: +∂2ω2 +∂ +� +k2 +∥ +� +∂ (k2x) += 2 − χcos2 θ +k4 ++ 2χk2 +x cos2 θ +k6 += 0 +(156) +It is cubic equation for k2. It must be solved together with equation of frequency +minimum (151). Eliminating k2 +x from these two equations, we arrive at a closed +equation for k2: +6k6 + 2 +� +2 + χ sin2 θ +� +k4 − χ cos2 θk2 = 0 +(157) +Dividing this equation by k2 ̸= 0, we obtain a quadratic equation for k2, whose +solution reads: +k2 +m = +�� +2 + χ sin2 θ +�2 + 6χ cos2 θ − +� +2 + χ sin2 θ +� +6 +(158) +40 + +The value of k2 +x corresponding to maximal value of k∥0 can be found by elimi- +nating k6 from eqs. (151,157). It reads: +� +k2 +x +� +m = +1 +3χ cos2 θ +�� +2 + χ sin2 θ +� +k4 +m + χ cos2 θk2 +m +� +(159) +The maximal value of k2 +∥0 is equal to +� +k2 +∥0 +� +max = k2 +m − +� +k2 +x +� +m = 2 +3k2 +m − +� +2 + χ sin2 θ +� +k4 +m +3χ cos2 θ +At further increase of kx, the position of minimum k∥0 decreases and finally +becomes zero. +At this point, k2 = k2 +x and eq. +(151) turns into quadratic +equation for k2 +x. Its solution reads: +� +k2 +x +� +f = +�� +2 + χ sin2 θ +�2 + 8χ cos2 θ − +� +2 + χ sin2 θ +� +4 +At this value of kx, minimum merges with a local maximum at k∥ = 0. At larger +values of kx, the only minimum of frequency is at k∥ = 0. +Appendix 2. Hamiltonian of the 4-th order. +. According to the subsection 4.2.2, the 4-th order Hamiltonian is: +H4 = +4 +� +i,k,l=1 +� +qinkρl +I(ρ1ρ2ρ3ρ4) +4q1n1,q2n2,q3n3,q4n4 +� +� +4 +� +j=1 +η(ρj) +qjnj +� +� δq1+q2+q3+q4, +(160) +where η(+) +qn = ηqn, η(−) +qn = η∗ +−qn and upper indices ρl (l = 1, 2, 3, 4) take values ++, − independently each from others. In terms of complex indices γi = (ρiqini) +the Hamiltonian H4 can be rewritten as +H4 = +� +γi +Iγ1γ2γ3γ4ηγ1ηγ2ηγ3ηγ4δq1+q2+q3+q4 +(161) +Since the product of four ηj is symmetric at any permutation P of four j, +it is possible to replace the initial coefficients Iγ1γ2γ3γ4 by the symmetrized +coefficients +Is +γ1γ2γ3γ4 = 1 +24 +� +P +IγP 1γP 2γP 3γP 4, +(162) +where Pj means the number appearing on j−th place at permutation P. For +example, for the permutation 1, 2, 3, 4 → 4, 3, 2, 1 one finds P1 = 4, P2 = 3, +P3 = 2, P4 = 1. +Let us now analyze what are constraints for symmetrized coefficients fol- +lowing from the fact that the energy is real. To make notations more compact +41 + +further we omit the subscript 4 and round brackets in upper part of initial +coefficients. Then eq. (161) turns into +H4 = +� +qknlρm +Isρ1ρ2ρ3ρ4 +q1n1q2n2q3n3,q4n4 +4 +� +j=1 +ηρj +qjnj. +(163) +Since η− +qjnj = +� +η+ +−qjnj +�∗ +, the energy is real if the following relations are satis- +fied: +Is−ρ1−ρ2−ρ3−ρ4 +q1n1q2n2q3n3,q4n4 = +� +Isρ1ρ2ρ3ρ4 +−q1n1−q2n2−q3n3,−q4n4 +�∗ +(164) +However, the initial non-symmetrized coefficients Iρ1ρ2ρ3ρ4 +q1n1q2n2q3n3,q4n4 calcu- +lated according to the rules formulated in the subsection 4.2.2. do not obey +these relationships. Nevertheless, not all of them are independent. In this Ap- +pendix we derive the integral presentation for independent coefficients and find +relations that allow to find the rest of them. +At fixed values qi, ni; i = 1, 2, 3, 4, there are 24 = 16 different combinations +of ρj = ± that defines coefficients Iρ1ρ2ρ3ρ4 +q1n1q2n2q3n3,q4n4. Each of them contains +contributions from exchange Ie and dipolar Id interactions, in total 32 coeffi- +cients. In each of them ρj take the same value + or − more than once. It +allows to make the partial symmetrization over repeating indices. For a further +compactification of notations we denote the pair j ≡ qjnj and j = −qjnj; +j = 1, 2, 3, 4. Then the resulting relationships for exchange coefficients are: +I−−−− +e1234 += +� +I++++ +e4321 +�∗ +(165) +I−−−+ +e1234 += +� +I++−+ +e2143 +�∗ += +� +I−+++ +e4321 +�∗ +(166) +I+−−− +e1234 += +� +I+−++ +e2143 +�∗ += +� +I+++− +e4321 +�∗ +(167) +I−−+− +e1234 += +� +I+−++ +e4321 +�∗ += +� +I+++− +e2143 +�∗ +(168) +I−+−− +e1234 += +� +I++−+ +e4321 +�∗ += +� +I−+++ +e2143 +�∗ +(169) +I−−++ +e1234 += I++−− +e3412 += I+−−+ +e3214 += I−++− +e1432 +(170) +Altogether there are 10 equations for 16 exchange coefficients. Thus, only 6 of +them are independent. This 6 coefficients can be chosen as: +I++++ +e1234 += µ2 +Bℓ2 +4A +´ d/2 +−d/2 [(dxu∗ +1) v∗ +2 (dxu∗ +3) v∗ +4 + u∗ +1 (dxv∗ +2) u∗ +3 (dxv∗ +4) +− 1 +2 +��4 +j=1 q2 +j +� +u∗ +1v∗ +2u∗ +3v∗ +4 +� +dx; +(171) +I+++− +e1234 += − µ2 +Bℓ2 +4A +´ d/2 +−d/2 [(dxu∗ +1) v∗ +2 (dxu∗ +3) u4 + u∗ +1 (dxv∗ +2) u∗ +3 (dxu4) +− 1 +2 +��4 +j=1 q2 +j +� +u∗ +1v∗ +2u∗ +3u4 +� +dx; +(172) +42 + +I−+++ +e1234 += − µ2 +Bℓ2 +4A +´ d/2 +−d/2 [(dxv1) v∗ +2 (dxu∗ +3) v∗ +4 + v1 (dxv∗ +2) u∗ +3 (dxv∗ +4) +− 1 +2 +��4 +j=1 q2 +j +� +v1v∗ +2u∗ +3v∗ +4 +� +dx; +(173) +I++−− +e1234 += µ2 +Bℓ2 +4A +´ d/2 +−d/2 [(dxu∗ +1) v∗ +2 (dxv3) u4 + u∗ +1 (dxv∗ +2) v3 (dxu4) +− 1 +2 +��4 +j=1 q2 +j +� +u∗ +1v∗ +2v3u4 +� +dx; +(174) +I+−+− +e1234 += µ2 +Bℓ2 +4A +´ d/2 +−d/2 [(dxu∗ +1) u2 (dxu∗ +3) u4 + u∗ +1 (dxu2) u∗ +3 (dxu4) +− 1 +2 +��4 +j=1 q2 +j +� +u∗ +1u2u∗ +3u4 +� +dx; +(175) +I−+−+ +e1234 += µ2 +Bℓ2 +4A +´ d/2 +−d/2 [(dxv1) v∗ +2 (dxv3) v∗ +4 + v1 (dxv∗ +2) v3 (dxv∗ +4) +− 1 +2 +��4 +j=1 q2 +j +� +v1v∗ +2v3v∗ +4 +� +dx. +(176) +There only 8 relationships for the dipolar part of 4-th order Hamiltonian: +I−−−− +d1,2,3,4 = +� +I++++ +d2,1,3,4 +�∗ +(177) +I−+−− +d1,2,3,4 = +� +I−+++ +d2,1,3,4 +�∗ +(178) +I+−−− +d1,2,3,4 = +� +I+−++ +d2,1,3,4 +�∗ +(179) +I−−+− +d1,2,3,4 = +� +I++−+ +d2,1,3,4 +�∗ +(180) +I−−−+ +d1,2,3,4 = +� +I+++− +d2,1,3,4 +�∗ +(181) +I−−++ +d1,2,3,4 = +� +I++−− +d2,1,3,4 +�∗ +(182) +I+−−+ +d1,2,3,4 = +� +I+−+− +d2,1,3,4 +�∗ +(183) +I−++− +d1,2,3,4 = +� +I−+−+ +d2,1,3,4 +�∗ +(184) +Thus, only 8 of them are independent. This 8 coefficients can be chosen as: +I++++ +d1,2,3,4 = πµ2 +B +2A +˜ +dxdx′× +� +(q1z + q2z)2 (u∗ +1v∗ +2 + v∗ +1u∗ +2) (u′∗ +3 v′∗ +4 + v′∗ +3 u′∗ +4 ) G|q1+q2| (x − x′) +−u∗ +1v∗ +2u∗ +3u′∗ +4 (dx + q4y)2 G|q4| (x − x′) ++u∗ +1v∗ +2u∗ +3v′∗ +4 +� +d2 +x − q2 +4y +� +G|q4| (x − x′) ++u∗ +1v∗ +2v∗ +3u′∗ +4 +� +d2 +x − q2 +4y +� +Gq4 (x − x′) +−u∗ +1v∗ +2v∗ +3v′∗ +4 (dx − q4y)2 Gq4 (x − x′) +� +(185) +43 + +I−+++ +d1,2,3,4 = − πµ2 +B +2A +˜ +dxdx′× +� +(q1z + q2z)2 (v1v∗ +2 + v∗ +1v2) (u′∗ +3 v′∗ +4 + v′∗ +3 u′∗ +4 ) G|q1+q2| (x − x′) +−v1v∗ +2u∗ +3u′∗ +4 (dx + q4y)2 G|q4| (x − x′) ++v1v∗ +2u∗ +3v′∗ +4 +� +d2 +x − q2 +4y +� +G|q4| (x − x′) ++v1v∗ +2v∗ +3u′∗ +4 +� +d2 +x − q2 +4y +� +Gq4 (x − x′) +−v1v∗ +2v∗ +3v′∗ +4 (dx − q4y)2 Gq4 (x − x′) +� +(186) +I+−++ +d1,2,3,4 = − πµ2 +B +2A +˜ +dxdx′× +� +(q1z + q2z)2 (u∗ +1u2 + v1u∗ +2) (u′∗ +3 v′∗ +4 + v′∗ +3 u′∗ +4 ) G|q1+q2| (x − x′) +−u∗ +1u2u∗ +3u′∗ +4 (dx + q4y)2 G|q4| (x − x′) ++u∗ +1u2u∗ +3v′∗ +4 +� +d2 +x − q2 +4y +� +G|q4| (x − x′) ++u∗ +1u2v∗ +3u′∗ +4 +� +d2 +x − q2 +4y +� +Gq4 (x − x′) +−u∗ +1u2v∗ +3v′∗ +4 (dx − q4y)2 Gq4 (x − x′) +� +(187) +I++−+ +d1,2,3,4 = − πµ2 +B +2A +˜ +dxdx′× +� +(q1z + q2z)2 (u∗ +1v∗ +2 + v∗ +1u∗ +2) +� +v′ +3v′∗ +4 + v′∗ +3 v′ +4 +� +G|q1+q2| (x − x′) +−u∗ +1v∗ +2v3u′∗ +4 (dx + q4y)2 G|q4| (x − x′) ++u∗ +1v∗ +2v3v′∗ +4 +� +d2 +x − q2 +4y +� +G|q4| (x − x′) ++u∗ +1v∗ +2u3u′∗ +4 +� +d2 +x − q2 +4y +� +Gq4 (x − x′) +−u∗ +1v∗ +2u3v′∗ +4 (dx − q4y)2 Gq4 (x − x′) +� +(188) +I+++− +d1,2,3,4 = − πµ2 +B +2A +˜ +dxdx′× +� +(q1z + q2z)2 (u∗ +1v∗ +2 + v∗ +1u∗ +2) +� +u′∗ +3 u′ +4 + u′ +3u′∗ +4 +� +G|q1+q2| (x − x′) +−u∗ +1v∗ +2u∗ +3v′ +4 (dx + q4y)2 G|q4| (x − x′) ++u∗ +1v∗ +2u∗ +3u′ +4 +� +d2 +x − q2 +4y +� +G|q4| (x − x′) ++u∗ +1v∗ +2v∗ +3v′ +4 +� +d2 +x − q2 +4y +� +Gq4 (x − x′) +−u∗ +1v∗ +2v∗ +3u′ +4 (dx − q4y)2 Gq4 (x − x′) +� +(189) +I++−− +d1,2,3,4 = πµ2 +B +2A +˜ +dxdx′× +� +(q1z + q2z)2 (u∗ +1v∗ +2 + v∗ +1u∗ +2) +� +v′ +3u′ +4 + u′ +3v′ +4 +� +G|q1+q2| (x − x′) +−u∗ +1v∗ +2v3v′ +4 (dx + q4y)2 G|q4| (x − x′) ++u∗ +1v∗ +2v3u′ +4 +� +d2 +x − q2 +4y +� +G|q4| (x − x′) ++u∗ +1v∗ +2u3v′ +4 +� +d2 +x − q2 +4y +� +Gq4 (x − x′) +−u∗ +1v∗ +2u3u′ +4 (dx − q4y)2 Gq4 (x − x′) +� +(190) +44 + +I+−+− +d1,2,3,4 = πµ2 +B +2A +˜ +dxdx′× +� +(q1z + q2z)2 (u∗ +1u2 + u1u∗ +2) +� +u′∗ +3 u′ +4 + u′ +3u′∗ +4 +� +G|q1+q2| (x − x′) +−u∗ +1u2u∗ +3v′ +4 (dx + q4y)2 G|q4| (x − x′) ++u∗ +1u2u∗ +3u′ +4 +� +d2 +x − q2 +4y +� +G|q4| (x − x′) ++u∗ +1u2v∗ +3v′ +4 +� +d2 +x − q2 +4y +� +Gq4 (x − x′) +−u∗ +1u2v∗ +3u′ +4 (dx − q4y)2 Gq4 (x − x′) +� +(191) +I−+−+ +d1,2,3,4 = πµ2 +B +2A +˜ +dxdx′× +� +(q1z + q2z)2 (v1v∗ +2 + v∗ +1v2) +� +v′ +3v′∗ +4 + v′∗ +3 v′ +4 +� +G|q1+q2| (x − x′) +−v1v∗ +2v3u′∗ +4 (dx + q4y)2 G|q4| (x − x′) ++v1v∗ +2v3v′∗ +4 +� +d2 +x − q2 +4y +� +G|q4| (x − x′) ++v1v∗ +2u3u′∗ +4 +� +d2 +x − q2 +4y +� +Gq4 (x − x′) +−v1v∗ +2u3v′∗ +4 (dx − q4y)2 Gq4 (x − x′) +� +(192) +All the integrals participating in Id can be calculated explicitly since the in- +tegrand is the product of sines, cosines and exponential function of |x − x′|. +However, the large number of different combinations of sines and cosines and +the necessity to use different exponents depending on the sign of x − x′ makes +real calculation sufficiently tiresome to charge a computer with this task. For +the coefficients Ie the calculations are much simpler since they include only sines +and cosines and integrals over one variable x. However, 6 independent coeffi- +cients Ie contain about 30 different integrals, so that charging computer with +this task is again justified. +Appendix 3. 1/r-G-identity . +From the Fourier transfromation of +1 +|r−r′| we have +1 +|r − r′| += +1 +(2π)3 +∞ +˚ +−∞ +dqeiqr 4π +q2 += +1 +(2π)3 +∞ +˚ +−∞ +dqeiq∥(r∥−r′ +∥)+iqx(x−x′) +4π +q2 +∥ + q2x += +1 +(2π)3 +∞ +¨ +−∞ +dqydqzeiq∥(r∥−r′ +∥) +∞ +ˆ +−∞ +dqxeiqx(x−x′) +4π +q2 +∥ + q2x +45 + +Since +´ ∞ +−∞ dqxeiqx(x−x′) +4π +q2 +∥+q2x = 4π2 +q∥ e−q∥|x−x′| = 8π2Gq∥ Then we get the 1/r- +G-identity +1 +|r − r′| = 1 +π +∞ +¨ +−∞ +dqydqzeiq∥(r∥−r′ +∥)Gq∥ (x − x′) +(193) +References +[1] L.D. Landau and E.M. Lifshitz, Phys. Zs. Sowiet. 8, 153, 1935. +[2] L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media, +Elsevier, 2nd Edition, 1984, Ch. 5. +[3] E. Schlöman, Phys. Rev. 116, 828 (1959). +[4] Pavol Krivosik and Carl E. Patton, Phys. Rev. B 82, 184428 (2010). +[5] R.W. Damon and J.R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961). +[6] V.V. Gann, Sov. Phys. Solid State 8, 2537. +[7] T. Wolfram and R.R. De Wames, Phys. Rev. Lett. 24, 1489 (1970). +[8] B.A. Kalinikos, IEEE Proc. H 127, 4 (1980). +[9] B.A. Kalinikos and A.N. Slavin, J. Solid State Phys. 19, 7013 (1986). +[10] R.E. Arias, Phys. Rev. B 94, 134408 (2016). +[11] Gang.Li, Chen Sun, T. Nattermann and V.L. Pokrovsky, Phys. Rev. B 98, +014436 (2018). +[12] S.O. Demokritov, V.E. Demidov, O. Dzyapko, G.A. Melkov, A.A. Serga, +B. Hillebrands, and A.N. Slavin, Nature (London) 443, 430 (2006) +[13] H. Goldstein, C.P. Poole and J. Safko, Classical Mechanics, 3d edition, +Pearson Education, 2011. +[14] I.V. Kolokolov, V.S. L’vov and V.B. Cherepanov, Zh. Eksp. Theor. Fiz. 84, +1043 (1983) [Sov. Phys. JETP 57, 605 (1983). +[15] E.B. Sonin, Phys. Rev. B 95, 144432 (2017). +[16] A. Kreisel, F.Sauli, L. Bartosch, and P. Kopietz, Eur. Phys. J B 71, 59 +(2009). +[17] A. A. Serga, C. W. Sandweg, V. I. Vasyuchka, M. B. Jungfleisch, B. Hille- +brands, A. Kreisel, P. Kopietz, and M. P. Kostylev. Phys. Rev. B 86, 134403 +(2012) +46 + +[18] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov, and A. N. +Slavin, Phys. Rev. Lett. 100, 047205 (2008). +[19] J. Lim, W. Bang, J. Trossman, A. Kreisel, M.B. Jüngfleisch, A. Hoffmann, +C.S. Tsal and J.B. Ketterson, Study of micron scale dispersion of spin +waves in Yttrium Iron Garnet film. Absrract of presentation at March APS +Meeting 2018, Los Angeles. +[20] Chen Sun, Thomas Nattermann and Valery L Pokrovsky, J. Phys. D: Appl. +Phys. 50, 143002 (2017). +[21] Y.M. Bunkov and G.E. Volovik, J. Low Temp. Phys. 150, 135 (2008). +[22] P. Novik-Boltyk, O. Dzyapko, V.E. Demidov, N.G. Berloff and S.O. +Demokritov, Sci. Rep. 2, 482 (2012). +[23] Fuxiang Li, Wayne M. Saslow, and Valery L. Pokrovsky Sci Rep 3, 1372 +(2013) +[24] C.C. Bradley, C.A. Sackett, and R.G. Hulet, Phys. Rev. Lett. 78, 985, +(1997); C.C. Bradley, C.A. Sackett, and R.G. Hulet, Phys. Rev. A 55, +3951, (1997). +[25] J. L. Roberts, N. R. Claussen, S. L. Cornish, E. A. Donley, E. A. Cornell, +and C. E. Wieman Phys. Rev. Lett. 86, 4211 (2001). +[26] Eric J. Mueller and Gordon Baym, Phys. Rev. A 62, 053605 (2000). +[27] L.P. Pitaevskii, Physics Letters A 221, 14 (1996). +[28] Borisenko, I., Divinskiy, B., Demidov, V.et al. Direct evidence of spatial +stability of Bose-Einstein condensate of magnons. Nat Commun 11, 1691 +(2020). +[29] Borisenko, I.V., Demidov, V.E., Pokrovsky, V.L. et al. Spatial separation +of degenerate components of magnon Bose-Einstein condensate by using a +local acceleration potential. Sci Rep 10, 14881 (2020). +[30] I. S. Tupitsyn, P. C. E. Stamp, and A. L. Burin. Stability of Bose-Einstein +Condensates of Hot Magnons in Yttrium Iron Garnet Films. Phys. Rev. +Lett. 100, 257202 (2008). +[31] S.M. Rezende. Theory of coherence in Bose-Einstein condensation phenom- +ena in a microwave-driven interacting magnon gas. Phys. Rev. B 79, 174411 +(2009). +[32] Divinskiy, B., Merbouche, H., Demidov, V.E. et al. Evidence for spin cur- +rent driven Bose-Einstein condensation of magnons. Nat Commun 12, 6541 +(2021). +47 + +[33] Noack, Timo B. and Vasyuchka, Vitaliy I. and Pomyalov, Anna and L’vov, +Victor S. and Serga, Alexander A. and Hillebrands, Burkard, Evolution of +room-temperature magnon gas: Toward a coherent Bose-Einstein conden- +sate, Phys. Rev. B 104, L100410 (2021). +48 + diff --git a/ktAzT4oBgHgl3EQfbvyL/content/tmp_files/load_file.txt b/ktAzT4oBgHgl3EQfbvyL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5d8e120f8c70a928bcffc7b5a7bcc5f4d60efc6e --- /dev/null +++ b/ktAzT4oBgHgl3EQfbvyL/content/tmp_files/load_file.txt @@ -0,0 +1,1573 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf,len=1572 +page_content='Amplitude representation of Landau-Lifshitz equation and its application to ferromagnetic films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Gang Li1∗ and Valery Pokrovsky1,2 January 5, 2023 1Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 2Landau Institute for Theoretical Physics of Russian Academy of Sciences, Chernogolovka, 142432, Russian Federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' ∗dgzy03@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='com 1 Introduction In 1935 Lev Landau and Evgenii Lifshitz set the foundation of static and dy- namics of weakly anisotropic ferromagnets [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' They formulated the famous Landau-Lifshitz equation (LLE) that regulates the motion of the ferromagnet magnetization in the long-wave low-frequency limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The purpose of this arti- cle is to develop a systematic approach to the solution of the LLE in terms of the magnon wave function ψ (r) and apply it to physical phenomena in a thin ferromagnetic film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This problem has a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' First such approach was proposed by Schlö- man in 1959 [3] for a bulk ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It was developed and improved by Carl Patton and his coworkers (see references in the review article by Krivosik and Patton [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The applications focused on the ferromagnetic resonance (FMR) and the spin momentum transfer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=', spin currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The theoretical study of ferromagnetic films started also in the the middle of 20-th century by the seminal work of Damon and Eshbach [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' They have found exact solution of the LLE equation for an infinite ferromagnetic film in which spins interact only through the dipolar forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In sufficiently thick films the evanescent waves propagating in opposite direction at the two surfaces appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' They create a mechanical torque acting on the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Gann [6], De Wames and Wolfram [7], Kalinikos and Slavin [8, 9] extended the Damon-Eshbach theory to a more general situation in which the spins inter- act also through the exchange forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' An extension of these exact solutions for 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='01391v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='mes-hall] 3 Jan 2023 the tilted external magnetic field was found by Arias [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the work by the au- thors, Chen Sun and Thomas Nattermann [11] the solution was extended to the wide range of the film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It enabled us to follow the transition from the magnon spectrum with two symmetric minima in thick films to one-minimum spectra in thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The latter result was inspired by the discovery of the Bose-Einstein conden- sation of magnons (BECM) at room temperature under permanent pumping of electromagnetic waves made in 2006 by Demokritov et al[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The BECM was found in the Yttrium Iron Garnet (YIG), a strongly insulating ferrite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For long-wave excitations all spins in the primitive cells move as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It means that in this regime the ferrite is indistinguishable from a ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The amplitude representation (AR) is ideally adjusted to describe the con- densation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The condensate amplitudes ψ± are the Fourier components of the coordinate wave functions ψ (r) at the values of wave vector k = ±Q corre- sponding to the two symmetric minima of magnon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Since we are mostly interested in the properties of the condensate and its interaction with excited magnons, our focus in the study of the (AR) will be different that in already cited works by Schlöman and Patton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Certainly, some overlapping is unavoid- able, but we try to minimize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This article has also a purpose to represent the modern state of art for the properties of ferromagnetic films and the pumping-induced BECM in them at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, it can be considered as a review on basic principles and the recent advances in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 2 Hamiltonian formulation of the Landau-Lifshitz equation and Amplitude representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 Poisson brackets for spins, magnetic moments and magnetization in discrete and continuous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us start with a discrete 3d-model of the ferromagnet, in which all spins Sr are located in the centers of cubic cells of volume v0 labeled by vectors r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The Poisson brackets for the components of spins are: {Sk (r) , Sl (r′)} = δr,r′εklmSm (r) , (1) where Kronecker symbol δr,r′ is equal to 1 when r = r′ and 0 otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' εklm is absolutely antisymmetric 3d tensor with k, l, m independently taking values 1,2,3 or x, y, z that is equal to +1 if the permutation k, l, m is even and -1 if it is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We use the Einstein convention that the summation must be performed over repeated indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The magnetic moment of a primitive cell is Mk = γSk, (2) where γ = e 2mc is the classical gyromagnetic ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The relation (2) becomes evident if one remembers that a spin projection, for example Sz, is quantized 2 in units ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' As a consequence, the magnetic moment projection is quantized in units of the Bohr’s magneton µB = eℏ 2mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (1) implies that the Poisson brackets for the components of the magnetic moments are: {Mk (r) , Ml (r′)} = γδr,r′εklmMm (r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (3) The magnetization is defined as magnetic moment of unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It is expressed in terms of magnetic moments as M (r) = M(r) v0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore the Poisson brack- ets for magnetization in the discrete model are: {Mk (r) , Ml (r′)} = γ v0 δr,r′εklmMm (r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (4) In continuous approximation the ratio δr,r′ v0 transits into the Dirac δ-function: lim v0→0 δr,r′ v0 = δ (r − r′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (5) To prove this statement let us introduce an arbitrary continuous function f (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us consider a sum over the cites of the discrete model: v0 � r′ δr,r′ v0 f (r′) = f(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In continuous limit v0 � r′ → ´ d3r′, which, together with previous equation, proves eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the Poisson brackets for components of magnetization in continuous limit are: {Mk (r) , Ml (r′)} = γδ (r − r′) εklmMm (6) It is convenient to rewrite these relations explicitly as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' {Mx (r) , My (r′)} = γδ (r − r′) Mz (r) (7) Two other Poisson brackets can be obtained from (7) by the cyclical permutation of the indices x, y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For further applications it is useful to introduce complex transverse magnetizations: M± (r) = Mx (r) ± iMy (r) (8) For them eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (7) implies the following Poisson brackets: {M+ (r) , M− (r′)} = −2iγδ (r − r′) Mz (r) {M± (r) , Mz (r′)} = ±iγδ (r − r′) M± (r) (9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 Amplitude representation and Poisson brackets for the magnon wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let the spontaneous magnetization and external magnetic field be directed along z−axis, perpendicular to its direction in the plane of film be y and direction 3 Figure 1: The coordinate system for a ferromagnetic film of thickness d: z−axis is chosen along the common direction of the magnetic field and static magneti- zation, x−axis is perpendicular to the film, θk is the angle between the magnon wave vector and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' perpendicular to the film x as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The wave function of magnons ψ (r) is determined by the magnon classical Holstein-Primakoff transformation: M+ (r) = √µBψ (r) � 2M − µBψ∗ (r) ψ (r) M− (r) = √µBψ∗ (r) � 2M − µBψ∗ (r) ψ (r) Mz = M − µBψ∗ (r) ψ (r) , (10) where M is the magnitude of magnetization vector that is assumed to be con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The third equation (10) shows that the physical meaning of the square of modulus ψ∗ (r) ψ (r) is the density of magnons n (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Note that the order of factors in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (10) is not important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The second useful remark is that � 2M − µBψ∗ (r) ψ (r) = √M + Mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The equations (9) are compatible with the amplitude representation (10) if and only if the wave functions satisfy the following permutation relations: {ψ (r) , ψ∗ (r′)} = − i ℏδ (r − r′) {ψ (r) , ψ (r′)} = {ψ∗ (r) , ψ∗ (r′)} = 0 (11) Let us prove this theorem for the second equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We will use the algebraic identity valid for any algebra of operators with defined operations of addition and non-commutative multiplication: {AB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' C} = A {B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' C} C + {A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' C}B (12) 4 X Z K Héz dEmploying this rule and the third equation (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' we find: {M+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Mz} = √µB � ψ (r) √M + Mz (r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Mz (r′) � = � µB (M + Mz (r)) {ψ (r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Mz (r′)} = − � µ3 B (M + Mz (r)) {ψ (r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' ψ∗ (r′) ψ (r′)} Applying again the identity (12) and assuming that {ψ (r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' ψ (r′)} = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' we arrive at relation {M+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Mz} = − � µ3 B (M + Mz (r))ψ (r′) {ψ (r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' ψ∗ (r′)} The right-hand side of this equation must be equal to iγδ (r − r′) M+ (r) ac- cording to the second equation (9) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The necessary and sufficient requirement to satisfy this condition is given by eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The validity of the first equation (9) can be checked by a similar calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 Landau-Lifshitz Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The Landau-Lifshitz Hamiltonian HLL for our problem contains several parts: the exchange interaction Hex, the dipolar interaction Hdip and the Zeeman interaction HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It can also may contain the anisotropy (spin-orbit) energy Han .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' First we write them in terms of magnetization: HLL = Hex + Hdip + HZ + Han, (13) where: Hex = D 2 ˆ (∇M)2 dV ≡ D 2 ˆ ∂iMj∂iMjdV ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (14) Hz = −H ˆ MzdV (15) Hdip = 1 2 ¨ (M∇) (M′∇′) 1 |r − r′|dV dV ′, (16) In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (16) we omitted for brevity the arguments in functions denoting M = M (r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' M′ = M (r′) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' ∇ = ∇r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' ∇′ = ∇r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' When employing the amplitude representa- tion for the components of magnetization (10), we similarly use abbreviations ψ ≡ ψ (r), ψ′ ≡ ψ (r′) and ∂± ≡ ∂x ± i∂y, ∂′ ± ≡ ∂x′ ± i∂y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The exchange constant D determines the exchange length ℓ = √ D that separates the length range, in which the dipolar interaction dominates l ≫ ℓ, from the range l ≪ ℓ where exchange interaction dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The LL equation assumes that the magnitude of the magnetization vector rapidly relaxes to its equilibrium value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the LL equation describes the relatively slow motion of the vector M (r, t) on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The slowness of this motion in space and time is controlled by two small parameters a/λ and ωτM, where a is the lattice constant, λ is the wave-length or another characteristic length of the magnetization motion, ω is its characteristic frequency and τM is the relaxation time of the magnetization magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' All magnetic phenomena in this limit are dominantly classical since the number of magnons in the volume 5 with the linear size of the order of λ is large and the change of this number by 1 produces negligibly small change of magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In terms of amplitudes the three parts of the Hamiltonian given by equations (14,15,16) are Hex = µ2 Bℓ2 2 ´ � ∇ |ψ|2�2 dV + µBℓ2 2 ´ ����∇ � ψ � 2M − µB |ψ|2 ����� 2 dV (17) Hz = µBH ˆ |ψ|2 dV (18) Hdip = 1 2 ¨ ˆΩ (r) ˆΩ (r′) dV dV ′ |r − r′|, (19) where ˆΩ (r) = � M − µB |ψ|2� ∂z + � µB � 2M − µB |ψ|2� 2 (ψ∂− + ψ∗∂+) (20) 3 Spectrum and wave functions of magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In this section we consider the approximation of free magnons and find their spectrum and wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For that purpose it is necessary to separate the part of the total Hamiltonian quadratic in amplitudes ψ, ψ∗ and diagonalize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 Quadratic part of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The Zeeman part of the Hamiltonian HZ given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (18) is naturally quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The quadratic parts of the exchange and dipolar Hamiltonians are: H(2) ex = µBMℓ2 ˆ |∇ψ|2 dV (21) H(2) dip = µBM 4 ¨ (ψ∂− + ψ∗∂+) � ψ′∂′ − + ψ′∗∂′ + � 1 |r − r′|dV dV ′ (22) Note that quadratic parts of the exchange Hamiltonian is local in space and it conserves the total number of magnons N = ´ |ψ|2 dV , whereas the quadratic part of dipolar Hamiltonian is non-local and it violates the conservation of the magnon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' All three parts of the quadratic Hamiltonian are invariant with respect to any translation in the film plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, it is natural to describe the motion in plane as a superposition of running plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In other words, the problem must be partly diagonalized by the Fourier-transformation: ψ (r) = 1 √ A � q χq (x) eiqr, (23) 6 where q = iqyˆy +iqzˆz is the in-plane wave vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' the Fourier-coefficients χq (x) depend on the transverse-to-plane coordinate x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' A is the area of any film cross- section parallel to its surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The inverse Fourier transformation gives the amplitude of a magnon with the wave vector q in a general state with the wave function ψ (r): χq (x) = 1 √ A ¨ ψ (r) eiqrdydz (24) Employing the Poisson brackets for ψ (r) eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (11), the Poisson brackets for the amplitudes χq (x) are: � χq (x) , χ∗ q′ (x′) � = − i ℏδq,q′δ (x − x′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (25) In terms of the variables χq (x) the three parts of the Hamiltonian are:Q H(2) ex = µBMℓ2 � q d/2 ˆ −d/2 ����� dχq (x) dx ���� 2 + q2 |χq (x)|2 � dx (26) H(2) Z = µBH � q d/2 ˆ −d/2 |χq (x)|2 dx (27) H(2) dip = πµBM � q ˜ d/2 −d/2 � χq (dx − qy) + χ∗ −q (dx + qy) � × � χ′ −q (dx′ + qy) + χ′∗ q (dx′ − qy) � Gq (x − x′) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (28) where we omitted for brevity the arguments x and x′ writing χq instead of χq (x) and χ′ q instead of χq (x′) and employed the abbreviation dx ≡ d dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The symbol Gq (x) stays for for the Green function of the 1d Helmholtz equation: Gq (x) = e−q|x| 2q (29) It obeys the 1d Helmholtz equation with a point source at origin: � d2 x − q2� Gq (x) = −δ (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 Bogoliubov transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The exchange and Zeeman parts of the quadratic Hamiltonian are diagonal in the variables χq (x), but the dipolar part mixes χq (x) with χ∗ −q (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' To di- agonalize the total quadratic Hamiltonian we apply the extended Bogoliubov transformation introducing for each q an infinite series of variables ηqn associ- ated with χq (x) and χ∗ −q (x) by a linear transformation: ηqn = d/2 ˆ −d/2 � uqn (x) χq (x) + vqn (x) χ∗ −q (x) � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (31) 7 To be canonical this transformation must produce correct Poisson brackets for variables ηqn: � ηqn, η∗ q′n′ � = − i ℏδq,q′δn,n′ (32) This requirement is equivalent to the condition of canonical transformation in classical mechanics [13] or unitary transformation in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore we will also use the word "unitarity" or "unitary" as equivalent to "canonical".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The requirement (32) together with the Bogoliubov transformation (31) and Poisson brackets for χq (x) (25) implies a series of constraints: d/2 ˆ −d/2 � uqn (x) u∗ qn′ (x) − vqn (x) v∗ qn′ (x) � dx = δn,n′ (33) The inverse Bogoliubov transformation determines χq (x) as a linear combina- tion of ηqn: χq (x) = � n � Uqn (x) ηqn + Vqn (x) η∗ −qn � (34) Replacing the amplitudes ηqn, η∗ −qnin eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (34) by their Bogoliubov representa- tion (31), we arrive at equations relating direct and inverse Bogolyubov trans- formations: � n � Uqn (x) uqn (x′) + Vqn (x) v∗ −qn (x′) � = δ (x − x′) � n � Uqn (x) vqn (x′) + Vqn (x) u∗ −qn (x′) � = 0 (35) On the other hand, the unitarity of the inverse Bogoliubov transformation re- quires � n � Uqn (x) U ∗ qn (x′) − Vqn (x) V ∗ qn (x′) � = δ (x − x′) (36) Comparing this equation with the first eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (29), we arrive at conclusion that Uqn (x) = u∗ qn (x) and Vqn (x) = −v−qn (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the inverse Bogolyubov transformation can be rewritten as χq (x) = � n � u∗ qn (x) ηqn − v−qn (x) η∗ −qn � (37) In addition from the U − V unitarity condition (36) we find the dual unitarity condition in terms of the initial Bogolyubov coefficients: � n � u∗ qn (x) uqn (x′) − v−qn (x) v∗ −qn (x′) � = δ (x − x′) (38) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 The wave functions and spectrum of magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 Spectrum of magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The magnon amplitudes must satisfy the stationary Schrödinger equation whose classical analogue is � H(2), ηq,n � = −iωq,nηq,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (39) 8 The Poisson brackets of the quadratic Hamiltonian and the vector of amplitudes is a linear anti-Hermitian operator acting on this vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the vector of amplitudes ηq,n is the eigenvector and the frequency of a magnon is the corre- sponding eigenvalue of the Hermitian operator i � H(2), � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In this subsection we express these equations in terms of the Bogoliubov coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Their solutions in some limiting cases will be found in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In order to write the left part of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (39) explicitly, we employ eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (26,27,28) for the three parts of the quadratic Hamiltonian, equation (32) for the Poisson brackets of the two amplitude vectors and the Bogoliubov transformation (37) from the amplitudes χq,n to magnon amplitudes ηq,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In resulting equations we omit for brevity the subscripts q and n since they are invariant under the Bogoliubov transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, equations (39) can be rewritten as: � ω + γ � H + Mℓ2 � q2 − d2 x ��� u = −2πγM �� q2 y − d2 x � ζu + (qy − dx)2 ζv � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' � ω − γ � H + Mℓ2 � q2 − d2 x ��� v = 2πγM �� q2 y − d2 x � ζv + (qy + dx)2 ζu � , (40) where we denoted γ = |e|/(2mc) is the classical gyromagnetic constant and ζu,v (x) = d/2 ˆ −d/2 G (x − x′) u (x′) v (x′) dx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (41) The physical meaning of the integral terms in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' side of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (40) is the magnetic field h generated by magnon magnetization m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The magnetic field can be expressed in terms of magnetostatic potential φ as h = −∇φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' If it is generated by the magnetization m (r), then φ (r) = −∇ · ˆ m (r′) |r − r′|−1 d3x′ (42) The coefficients u and v should be identified with the x- and y-components of magnetization, the operators ±iqy − dx with the complex presentation of gradient and divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then equation (41) is equivalent to (42) integrated over y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The reference (40) is a system of two integral-differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' However, they can be transformed in the purely differential linear equations by employing operator q2 − d2 x (Laplacian) to both sides of equations (40) and employing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (30) to eliminate the Green function G (x − x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The application of this operator to ζu,v (x) transforms these integrals into u (x) and v (x) , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, we obtain a system ordinary linear differential equations of the fourth order: � ω + γ � H + Mℓ2 � q2 − d2 x ��� � q2 − d2 x � u = −2πγM �� q2 y − d2 x � u + (qy − dx)2 v � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' � ω − γ � H + Mℓ2 � q2 − d2 x ��� � q2 − d2 x � v = 2πγM �� q2 y − d2 x � v + (qy + dx)2 u � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (43) 9 Their solutions must be a superposition of exponents eiκx with κ being a root of the secular polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' To find this polynomial, it is convenient to introduce the vector k with the components kx = |κ|, ky,z = qy,z whose square if magnitude is k2 = q2 + κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us define a simplest solution of the system (43) is: u (x) = u0eiκx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' v (x) = v0eiκx (44) Substituting this solution into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (43), we obtain a system of two linear homo- geneous equations for u0, v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The condition of its solvability is the nullification of their determinant (secular equation): ω2k2 = γ2 � H + Mℓ2k2� × �� H + Mℓ2k2� k2 + 4πM � k2 − k2 z �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (45) This equation can be interpreted as dispersion relation for magnons: ω = γ � � � �(H + Mℓ2k2) � H + Mℓ2k2 + 4πM � k2x + k2y � k2 � (46) It is valid if a/λ = ka/(2π) ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At room temperature the thermal wavelength λ = ℏ/√2mkBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For effective mass of magnon for YIG of the order of mag- nitude m ≈ 3me, λ is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='7nm, whereas the lattice constant a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (46) is invalid for thermal magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The calculation of the magnon spectrum at high energies for YIG were given in the seminal article by Kolokolov, L’vov and Cherepanov [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 Bulk and evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At fixed parameters ℓ, M, H, kz = qz and frequency ω, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (45) is a cubic equation for the variable k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Note that its coefficients do not depend not only on the film thickness d but also on the value ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Inspection of the coefficients of the cubic equation shows that the product of three roots is positive, whereas their sum is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, there are two opportunities: i) one roots k2 is positive and two others are negative or ii) one root is positive and two others are complex conjugated with negative real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Sonin proved [15] that in thick films d ≫ ℓ and for kl ≪ 1, the opportunity i) is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Gang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [11] proved that the opportunity ii) leads to negative ω2 and therefore is forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For thick films d ≫ ℓ and kz ≪ 1/ℓ and ω2 < γ2H (H + 4πM), the positive root k2 1 can be found approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In this case it is possible to retain in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (45) only terms linear in k2 and independent on k2 and neglect the terms quadratic and cubic in k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The result is: k2 1 = k2 z 4πγ2HM γ2H (H + 4πM) − ω2 (47) Two others negative solutions k2 = −k2 1,2 are determined by equation: k2 1,2 = � 2π + H M ± � 4π2 + ω2 γ2M 2 � ℓ−2 (48) 10 When frequency approaches the ferromagnetic resonance value ωF R = γ � H (H + 4πM) to the distance ωF R − ω ≲ k2 zℓ2 √ 1+ 4πH M 2πγM, the inequality k1ℓ ≪ 1 becomes in- valid and instead of quadratic the cubic equation must be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At large frequency ω ≫ ωF R, the exchange energy dominates and ω ≈ γMℓ2k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It cor- responds to the region of large wave vectors kℓ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For thick films d ≫ ℓ, the four wave functions of the type χq (x) ∝ exp � −k1,2 � d 2 ± x �� correspond to the four evanescent waves localized in a layer of the depth ∼ ℓ near the surfaces of the film x = ±d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We proved that any propagating in-plane excitation is a superposition of several transverse modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The transverse modes may be either superposition of cos kxx and sin kxx or the evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' However, the inverse statement that any such superposition is a solution of the initial equations of motion is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This happens because the initial equations of motion were integral-differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The system of ordinary differential equations was obtained from them by applica- tion of additional differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This operation introduces additional solutions of resulting system of equations that are not solutions of the initial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Below we derive the selection rules that separate only solutions of the initial integral-differential equations (40,41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Equations for the Bogoliubov transformation functions (40,41) permit real solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore the Bogoliubov functions can be searched in the form: uq,n (x) = an cos kxx + bn sin kxx+ � m=1,2 � Anm cosh kmx cosh kmd/2 + Bnm sinh kmx sinh kmd/2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (49) vq,n (x) = cn cos kxx + dn sin kxx+ � m=1,2 � Cnm cosh kmx cosh kmd/2 + Dnm sinh kmx sinh kmd/2 � , (50) where all coefficients an, bn, Anm, Bnm,cn, dn, Cnm, Dnm are real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In further calculations we omit the subscripts n and q since they are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' All evanescent waves exponentially decrease far from boundaries on the scale ∼ ℓ as exp � −km �� d 2 ± x ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Substitution of expressions (49,50) to the integral-differential equations (40,41) leads to appearance of exponential functions that do not belong to the 6 expo- nents permitted by the secular equation (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' They are produced by the integrals (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Their explicit calculation can be reduced to the four basic integrals: Ic (x) ≡ ´ d/2 −d/2 e−q|x−x′| 2q cos kxx′dx′ = cos kxx k2 − e−qd/2 qk2 cosh qx f1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (51) Is (x) ≡ ´ d/2 −d/2 e−q|x−x′| 2q sin kxx′dx′ = sin kxx k2 − e−qd/2 qk2 sinh qx f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (52) 11 Jcm (x) ≡ ´ d/2 −d/2 e−q|x−x′| 2q cosh kmx′dx′ = cosh kmx q2−k2m − e−qd/2 q(q2−k2m) cosh qx g1m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (53) Jsm (x) ≡ ´ d/2 −d/2 e−q|x−x′| 2q sinh kmx′dx′ = sinh kmx q2−k2m − e−qd/2 q(q2−k2m) sinh qx g2m, (54) where the notations f1,2, g1,2 are used for the following functions: f1 = q cos kxd 2 − kx sin kxd 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (55) f2 = q sin kxd 2 + kx cos kxd 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (56) g1m = q cosh kmd 2 + km sinh kmd 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (57) g2m = q sinh kmd 2 + km cosh kmd 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (58) Employing these results, it is possible to calculate ζu (x) and ζv (x) defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (41): ζu (x) = aIc + bIs + 2 � m=1 � AmJcm cosh kmd 2 + BmJsm sinh kmd 2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (59) ζv (x) = cIc + dIs + 2 � m=1 � CmJcm cosh kmd 2 + DmJsm sinh kmd 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (60) The terms with Ic and Is in these equations contain the functions cosh qx and sinh qx or equivalently exp (±qx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The wave vector k = q does not satisfy the secular equation (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, they should vanish in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' side of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' These requirements represent four constraints onto 12 coefficients a, b, c, d, A1, B1, C1, D1, A2, B2, C2, D2 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Neglecting evanescent waves in the integrals, we obtain 4 equations for 4 coefficients a, b, c, d at “bulk” waves: � q2 y − q2� af1 + � q2 y + q2� cf1 +2qyqdf2 = 0 � q2 y − q2� bf2 +2qyqcf1 + � q2 y + q2� df2 = 0 � q2 y + q2� af1 −2qyqbf2 + � q2 y − q2� cf1 = 0 −2qyqaf1 + � q2 y + q2� bf2 + � q2 y − q2� df2 = 0 , (61) The determinant of this system is identically zero .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, this system does not determine quantization of kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' A simple reason why any 4×4 minor of the 4×24 matrix formed by coefficients at e±qx in each of the mentioned above twelve coefficients has zero determinant is that all of them obey an inhomogeneous Helmholtz equation, for example, d2Ic dx2 − q2Ic = cos(kxx);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' d2Jcm dx2 − q2Jcm = cosh(kxx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (62) 12 Since the solutions of such equations can include any linear combination of e±qx, the condition of zero coefficients at these function cannot put any restriction of the 4 × 24 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It means that any its 4 × 4 minor has zero determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The self-consistency equations are equivalent to the MBC, but they simplify calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 Boundary conditions and the quantization of trans- verse modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 Spin boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' There are two kinds of boundary conditions: magnetostatic (MBC) associated with the variation of the magnetic field and induction near the boundary and the spin boundary conditions (SBC) associated with variation of spin (magneti- zation) at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The MBC requires continuity of tangential component of magnetic field h and the normal component of the induction b = h + 4πm at two surfaces x = ±d/2 of the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The MBC are satisfied automatically if the magnetic potential is related to the magnetization by the equation (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, only the SBC must be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us consider the simplest possibility that spins on the surfaces are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The variation of the exchange energy (14) gives the surface term: δHex = ℓ2 ´ d/2 −d/2 dx ˜ ∞ −∞ dydz∂iδmα · ∂imα = ℓ2 ˜ ∞ −∞ dydzδmα∂xmα ��� d/2 −d/2 + volume terms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (63) The volume terms contribute exchange terms in equations of motion, whereas the surface term in this equation implies that on both surfaces magnetization obeys the spin boundary condition: ∂xm|x=±d/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (64) The variation of the Zeeman and dipolar Hamiltonians does not give the surface term since they do not contain derivatives of magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Returning to the amplitude representation, we identify as before the two components of magnetization with the Bogolyubov coefficients u and v at fixed q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (64) in amplitude representation is: ∂xu|x=±d/2 = ∂xv|x=±d/2 = 0 (65) For the thick film and kxℓ ≪ 1, these equations imply that the magnitudes of coefficients at the evanescent waves Am, Bm, Cm, Dm are less than the magni- tudes of amplitudes of the bulk waves a, b, c, d by the factor ∼ kxℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [15] To see that, let us put all coefficients except of a, c and A1, C1 equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then equation (65) takes form: (c − a) kx sin kxd 2 = (A1 + C1) k1 (66) 13 This equation proves the Sonin’s statement since k1 ∼ 1/ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Nevertheless the evanescent waves allow to satisfy the MBC at fixed amplitudes of the bulk waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Neglecting in equations of motion (40) evanescent waves, we can rewrite them as: ˆ M � � � � a b c d � � � � = 0, (67) where the 4 × 4 matrix ˆ M is: ˆ M = � � � � ω − A 0 B C 0 ω − A −C B −B C ω + A 0 −C −B 0 ω + A � � � � , (68) and A = γ � H + Mℓ2k2 + 2πM(k2 x+k2 y) k2 � B = 2πγM(k2 x−k2 y) k2 C = 4πγMkxky k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (69) The determinant of the matrix ˆ M is det ˆ M = � ω2 − A2 + B2 + C2�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (70) It turns into zero at ω = √ A2 − B2 − C2 that gives the obtained earlier disper- sion relation (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The eigenvalues ±ω of the matrix ˆ M are double degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, their eigenvectors contain two independent coordinates, for exam- ple the amplitudes a and b, whereas two others are expressed as their linear combination as it follows from the equations (67): c = B ω±Aa − C ω±Ab d = C ω±Aa + B ω±Ab (71) Note that the two eigenvectors corresponding to different signs in denominators are orthogonal at mass shell, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=', at ω = √ A2 − B2 − C2 and any choice of coordinates a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us substitute the amplitudes c and d from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (71) for the sign + into the first two of self-consistency equations (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then we find a system of two homogeneous equations of the form: Pa + Qb = 0 Ra + Sb = 0 , (72) 14 where P = � q2 y − q2 + (q2 y+q2)B ω+A � f1 − 2qyqC ω+A f2 Q = (q2 y+q2)C ω+A f1 + 2qyqB ω+A f2 R = −(q2 y+q2)C ω+A f1 + 2qyqC ω+A f1 S = � q2 y − q2 + (q2 y+q2)B ω+A � f2 + 2qyqC ω+A f1 (73) The determinant of the system (72) PS − QR must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It determines the quantization of kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Equation PS − QR = 0 gives: f 2 1 − f 2 2 = 2Γf1f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Γ = � q2 y − q2� ω + � q2 y + q2� B 2qyqB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (74) From this equation we find: f1 f2 = Λ ≡ Γ ± � Γ2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (75) Note that the change of sign in front of square root turns Λ into −1/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Em- ploying equations (55,56), we represent the quantization condition in a more explicit form: tan kxd 2 = q − Λkx Λq + kx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (76) The change Λ → −1/Λ transforms the fraction q−Λkx Λq+kx into inverse value with opposite sign, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=', − Λq+kx q−Λkx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For the waves propagating along spontaneous mag- netization (ky = 0), the quantization condition becomes tan kxd 2 = q kx or tan kxd 2 = −kx q (77) The first of them was first found by Damon and Eshbach [5] for purely dipo- lar interaction and reproduced by Sonin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [15] It corresponds to the pure cosine solution (b = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The second sign at ky = 0 corresponds to the pure sine solu- tion (a = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [11] For general direction of propagation in-plane the two different signs in front of square root in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (75) correspond to two different branches of discrete solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We denote them by discrete index ν accepting two values ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 Quantization of transverse wave vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Parallel propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Equations (77) have a discrete set of solutions for kxn in the intervals � πn d , π(n+1/2) d � for the cosine and in the intervals � π(n+1/2) d , π(n+1) d � for the sine transverse mag- netization, where n is any non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It is clearly seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the limit qd ≫ 1 the approximate analytical solution is possible for n ≪ qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 kz ( MD ) kxn ( MD ) Figure 2: Plots of the dependence of quantized transverse wave vectors kxn on kz in units � H MD for d = 10 in units � MD H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Black and red curves correspond to even and odd transverse modes, respectively In this case kx ≪ q so the ratio q kx ≫ 1 for the first series of quantized kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, kxd 2 in the first equation (77) must be close to � n + 1 2 � π and k(+) xn ≈ (2n + 1) π d � 1 − 2 qd � (78) Here we used the index + as notation of the first series (even transverse distri- bution of magnetization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For large n and qd ≫ 1 the approximate equation for the quantized values of the first series is: k(+) xn ≈ 2nπ d + 2 d arctan qd 2nπ (79) It accurately matches the result (78) for 1 ≪ n ≪ qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For the second series the quantized transverse wave vectors for qd ≫ 1 and n ≪ qd are k(−) xn ≈ 2nπ d � 1 − 2 qd � (80) and for n ≫ 1 k(−) xn ≈ (2n + 1) π d + 2 d arctan qd 2nπ (81) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 Wave vectors and effective masses at minimum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Two energy minima ±Q are located on z−axis and correspond to minimal value n = 0 and symmetric branch of the transverse momentum quantization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' kx ≈ π d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us minimize explicitly the energy or frequency eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For a thick film d ≫ ℓ, the energy is ε = ℏω (q, kx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It is more convenient to minimize 16 the square of energy ε2 (q, kx) = µ2 B � H2 + 2HMℓ2k2 + 4πHM � k2 x + k2 y � k2z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (82) We first minimize square of energy over qy putting qy = 0 and in the square of total momentum k2 = k2 x + q2 y + q2 z neglect k2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Taking derivative over qz from ε2 (qz, 0, kx) at kx = π d , we get: 2ε ∂ε ∂qz = 4µ2 BHM � ℓ2qz − 2π3 q3zd2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (83) At minimum energy the derivative ∂ε ∂qz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' From this requirement we find, that two minima are located at qz = ±Q, where Q = � 2π3�1/4 √ ℓd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (84) This result was obtained by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Sonin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [15] The main value of the mass tensor mz in z direction relates to the second derivative ∂2ε ∂q2z for qz = ±Q as mz = ℏ2/ ∂2ε ∂q2z ��� qz=Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' By differentiation of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (83) and putting qz = Q, εmin = µBH, we find: mz = ℏ2 8µBMℓ2 (85) To find my, we need to take the second derivative of ε2 (q, kx) given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (82) over qy at qy = 0, qz = Q neglecting kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The searched effective mass is my = ℏ2/ ∂2ε ∂q2y ��� qy=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' An elementary calcualtion gives: my = ℏ2Q2 8πµBM (86) The mass my is much less than mz: their ratio is my/mz = ℓ/ (πd) ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For the film of YIG 5μm thick Q ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='44 × 105cm−1 , mz = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='37 × 10−27g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' my = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='78 × 10−29g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 Quantization of transverse wave vector: arbitrary direction of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Despite of rather involved structure of quantization condition (76) its solution can be written explicitly in the limit d ≫ ℓ, and qd ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The roots of this equation are kxνn, where n = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' is the number of quantized value kx, ν = ± stays for even or odd transverse distribution of magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The explicit analytical expression for these roots in the asymptotic region and large n ≫ 1 is kxνn = 2nπ d + 2 d arctan qd − 2πnΛνn qdΛνn + 2πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (87) 17 To find parameters Λνn = Γ+ν √ Γ2 + 1 it is necessary to replace kx by 2πn/d in the equations (75) for Λ and (74) in all functions containing kx in its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Equation (87) has precision 1/qd and is valid for 1 ≪ n ≪ qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the entire this region the difference between the quantized values of kx with the same number in the two branches is kx+n − kx−n = π d (88) The ratio of amplitudes in this range of variables is bνn aνn = −qA ω Λν − C ω (89) At fixed direction of in-plane propagation given by the angle θ between the wave vector and direction of the spontaneous magnetization M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' the frequency as function of the wave vector magnitude has minimum at q0 = 2√πχ3/4√ cos θ � 2 + χ sin2 θ �1/4 � kxνn ℓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (90) where χ = 4πM H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' From this equation and strong inequality qℓ ≪ 1 it follows that q0 ≫ kxνn ≈ 2π d n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 Motion of energy minimum vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At very large n ≫ d ℓ the value k2 becomes so large that the exchange interactions dominates and the frequency of a magnon becomes equal to ω = γMℓ2k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then the minimum energy occurs at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It means that the position of minimum of frequency q0 first grows with kx and reaches its maximum at some specific kx1 ∼ 1/ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At further growth of kx the position of frequency minimum q0 (kx) decreases and reaches zero at another specific value of kx = kx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At further growth of n it remains zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Theory gives exact analytical answers for all these values, namely: k2 x1 = 1 3k2 1 + 2 + χ 12π tan2 θk4 1ℓ2, (91) where k2 1 = H 6Mℓ2 ��� 2 + χ sin2 θ �2 + 6χ cos θ − 2 − χ sin2 θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (92) The maximal value of q0 is given by q2 0 max = k2 1 + k2 x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (93) Finally the value of k2 x at which the minimum of frequency merges with maxi- mum located at q = 0 is k2 x2 = H 4Mℓ2 ��� 2 + χ sin2 θ �2 + 8χ cos θ − 2 − χ sin2 θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (94) 18 The position of maximum k0 (kx) for kxℓ ≳ 1 is given by k2 0 (kx) = H Mℓ2 2 + χ sin2 θ 2 w (ξ) , (95) where w (ξ) is the solution of a cubic equation: w3 + w2 = ξ (96) and ξ = χ2 cos2 θk2 xℓ2 π � 2 + χ sin2 θ �3 (97) Details of these calculations can be found in the Appendix[motion of minima].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the analysis of this subsection we followed the work [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 Comparison with other calculations and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The results of numerical calculations of quantized spectra eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (82) with quan- tized kxn for propagation perpendicular and parallel to magnetization and d = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 in units � MD H , χ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 are shown in Fig 3(a) and 3(b), spectra of the first transverse modes for a number of different directions of propagation specified by the angle θ = arctan ky kz are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The spectra for parallel and perpendicular propagation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3(a) and 3(b)) agree very well with the numerical calculations of the work [16] based on diag- onalization of a large matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We also discovered an excellent agreement with similar calculations of the same work made for the YIG film with a thickness of 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Figure 4 shows a comparison of the theoretical spectrum with the experiment [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Brillouin scattering spectroscopy was used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Its precision is not sufficient for resolution of excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' A dramatic increase in precision was achieved by an experimental group led by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Ketterson [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' His method makes use of direct microwave excitation of magnons via a specially designed antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It is made up of periodically repeated emitters that are powered by an adjustable frequency generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The excited magnon wave-length coincides with the distance between emitters λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The magnon frequency at this wave vector kz = (2π)/λ is a frequency at which the resonance adsorption of microwave radiation reaches maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The increased resolution allowed for the observation of multiple magnon modes (up to nine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This is the first time that different transverse magnon modes have been experimentally observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Figure 5 shows a comparison of theoretical spectrum with experimental results [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The agreement between theory and experiment is excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='7 Thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In what follows till the end of this section we use � M/Hℓ as unit of length and (γH)−1 as unit of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In this part we discuss the case of thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 ky ( MD ) ω (γℋ) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 kz ( MD ) ω (γℋ) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 k\uf605 ℋ MD ω γℋ θ 0 θ π/6 θ=π/4 θ=π/3 θ=π/2 (c) Figure 3: Results of numerical calculations for the case d = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 in units � MD H and χ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (a) The spectra of first four quantized modes for direction of propagation perpendicular to magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (b) Spectra of the first four modes for direction of propagation parallel to magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (c) Spectra of the first transverse modes for θ = 0, π 6 , π 4 , π 3 , π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Black solid curves correspond to our numerical calculations, red dashed line is the Damon-Eshbach surface mode, circles are numerical calculations by Kreisel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='. [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' These figures agree with the figures from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='48 ky ( MD ) ω (γℋ) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 kz ( MD ) ω (γℋ) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 ky ( MD ) ω (γℋ) (c) Figure 4: Comparison of theoretical spectrum with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In experiments the Brillouin light scattering spectroscopy was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (a) Comparison with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Serga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [17] d = 5 µm, H=1750 Oe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (b) Comparison with V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Demidov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [18] d = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 µm, H=1000 Oe for direction of propagation parallel to magneti- zation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (c) Comparison with V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Demidov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [18] d = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 µm, H=1000 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' for fixed kz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 × 104cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' These figures agree with the figures from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 kz ( MD ) ω (γℋ) Figure 5: Comparison of theoretical spectrum with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Solid curves are our calculations of the first 15 transverse modes for the YIG film of thick- ness 5µm, 4πM= 1940 Oe and H= 1960 Oe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Circles on them are frequencies measured by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [19] at three fixed wavelengths for different transverse mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This figure agrees with the figure from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 0 20 40 60 80 100 ky ( MD ) ω (γℋ) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 0 20 40 60 80 100 kz ( MD ) ω (γℋ) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 k\uf605 ℋ MD ω γℋ θ 0 θ π/6 θ=π/4 θ=π/3 θ=π/2 (c) Figure 6: Results of numerical calculations for a thin film d = 1 in units � MD H and χ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (a) The spectra of first four quantized modes for direction of propagation perpendicular to magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (b) Spectra of the first four modes for direction of propagation parallel to magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (c) Spectra of the first transverse modes for θ = 0, π 6 , π 4 , π 3 , π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' These figures agree with the figures from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 21 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 d ( MD ) kz ( MD ) (a) 0 2 4 6 8 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='9 d ( MD ) ω (γℋ) (b) 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 d ( MD ) kxn ( MD ) kz=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 kz=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 kz=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 kz=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 (c) Figure 7: Results of numerical calculations for the case χ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 and θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (a) Position of minima for the lowest mode vs d for thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (b) The value of frequency in minimum for the lowest mode vs d for thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (c) kxn for the lowest mode vs d at fixed kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Black solid curves correspond to our numerical calculations, circles are numerical calculations by Kreisel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='. [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='These figures agree with the figures from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' If the film’s thickness is of the order of one or less (ℓ in dimensional units), it is regarded as thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The experimental realization of ultrathin films of YIG with d ≪ 1 looks very improbable since the typical value of ℓ (in YIG) is a few tens of nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It may be accomplished in thin, monolayer-thick ferromagnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Transverse modes with high n in thin films with d ∼ 1 have kxn ≈ πn/d ≫ 1 in the exchange dominance area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, only a few modes with the lowest frequencies are of theoretical and experimental relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In these modes, evanescent waves penetrate to the film at a depth of the same order of magnitude as its thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' They therefore play an equally essential role in spectral characteristics and TDM as the oscillating wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' A compact analytic expression has been found only for frequency as function of the wave vector (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (46)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 6 shows examples of spectra in thin films that are qualitatively similar to spectra in thick films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Each mode determined by numbers ν, n at not very big n has a frequency minimum at some k∥ ̸= 0, but it does not follow equation ∂ω2 ∂k2 ∥ = 0 since kxn also depends on k∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 6(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 6(b) show that at d = 1, the energy of transverse excitation weakly depends on kz, a feature that could be expected for ultrathin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The graphs of position of minima and the value of frequency in minimum for the lowest mode vs d for thin films are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the same figures 7(a) and 7(b), we compared our results with calculations of the same values by Kreisel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Finally, the graphs of kxn for the lowest mode vs d at fixed k∥ and θ = 0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 7(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' An example of TDM for lowest mode and first excited mode in thin films is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' All ground state spectra cross at the point k∥ = 0, ω ≈ √1 + χ ( √ 3 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='73 for χ = 2), exactly the same result as for the thick film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This is manifestation of a general property of films with arbitrary thickness: at k∥ = 0, the transverse wave vector of the lowest transverse mode is also equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The frequency of the lowest mode equals to ω0 = √1 + χ (ferromagnetic resonance frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 x ( MD ) TDM my mx (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 x ( MD ) TDM my mx (b) Figure 8: For the case χ = 2 and θ = 0 (a) TDM for the lowest mode at k∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 and a1x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (b) TDM for the first excited mode at k∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 and b1x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' These figures agree with the figures from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We consider first the limiting case of ultrathin films d → 0 when θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It will be shown that only wave vectors of the lowest transverse mode with ν = −, n = 0 remains finite in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' All excited transverse state with other ν or n have wave vectors that go to infinity as 1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We just take into account the simplest scenario of waves propagating along magnetization and magnetic field in order to simplify calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The transverse mode then has a definite parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In such a case, the non-zero amplitudes are ai for even modes and bi for odd modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For finite wave vectors ki in the taken limit, sin kixd/2 ≈ kixd/2 and cos kixd/2 ≈ 1 are appropriate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This fact simplifies the SBC (64) and self-consistency equations (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The second simplification results from the fact that the relationship between the x and y components of the vectors ai and bi is reduced to aiy = ω 1+k2 i aix and biy = ω 1+k2 i bix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Here we denote three kernels of cubic equation for k2 (45) as k2 1, k2 2, k2 3 and corresponding vector amplitudes at sin(kixx) and cos(kixx) as ai, bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us remind that k2 1 > 0, whereas k2 2, k2 3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' After all these simplifications, the quantization of an even mode is described by the system of three equations with three independent amplitudes aix : � � � � � �3 i=1 k2 ixaix = 0 �3 i=1 k2 ix 1+k2 i aix = 0 �3 i=1 aix k2 i = 0 (98) Zeros of determinant of this system determine quantized values of k2 xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In order to transform this determinant into an explicit function of kxn one should employ the relations k2 1 = k2 xn + k2 z, k2 2,3 = −1 − χ 2 − k2 1 2 ± �� 1 + χ 2 + k2 1 2 �2 − χk2z k2 1 (99) 23 approximation kxn 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 kz ( MD ) kxn ( MD ) Figure 9: Plot of kxn at d → 0 and approximation to it when χ = 2 and θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This figure agrees with the figure from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' and k2 ix = k2 i − k2 z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The only positive root of this equation at small kz ≪ 1 is kxn ≈ � χ 2 + χ �1/4 � kz (100) At large kz, kxn asymptotically approaches a constant value kxn ≈ � χ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Both these asymptotic values agree very well with numerical calculations of the dependence of kxn on kz at d → 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The fact that kxn = 0 at kz = 0 is confirmed by the asymptotic behavior of kxn at small kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' As a result, both in the limit of small d and the limit of large d, the value of frequency at k∥ = 0 is √1 + χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' On Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 10, the plots of kxn vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' kz at d = 1 and d = 0 are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We can now demonstrate the general proposition that, regardless of thick- ness, the frequency of the lowest mode at k∥ = 0 equals √1 + χ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Set ky = 0 and consider kz ≪ 1/d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We will show that the same equation (100) deter- mines the first quantized value kxn, but the arguments must be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In order to prove the result (100), let us assume that the initial quantized value of kxn obeys the strong inequalities kz ≪ kxn ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (99) implies that k2 2x ≈ −χk2 z/ � (2 + χ) k2 xn � has small magnitude, whereas k2 3x ≈ −2 − χ has the magnitude of the order of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us first consider the SBC (64) that in considered situation take form k2 xna1x + k2 2xa2x − � 2 + χ2 sinh √2 + χd/2 d a3x = 0 k2 xna1x + k2 2xa2x + 2√2 + χ sinh √2 + χd/2 (1 + χ)d a3x = 0 (101) These equations imply a3x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then they become identical and define the ratio a2x/a1x = −k2 xn/k2 2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Next consider the self-consistency equations that in the same limit have a form: a1x k2 1 + a2x k2 2 = 0 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='8 kz ( MD ) kxn ( MD ) d 0 d=1 Figure 10: kxn vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' kz for the lowest mode at d → 0 and d = 1 at χ = 2 and θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='This figure agrees with the figure from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Using the previously found ratio a1x/a2x, we again obtain eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (100) for this more general situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It shows that in the limit kz → 0, the limit of ratio k2 z/k2 xn is also zero and limiting value of ω is √1 + χ independently on thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Note that in the limit k∥ = 0 the magnetization in the lowest spin-wave mode does not depend on transverse coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Although thin films are more sensitive to the exact form of the SBC than thick films, changing forms of these requirements have no effect on the symmetry or general features of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' An important problem is how the wave vector kzmin corresponding to the minimum of energy changes with thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For thick films it behaves as 1/ √ d [15] and grows when film becomes thinner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' However, in the case of ultrathin films, it decreases linearly with thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It means that the wave vector kzmin as function of d has a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Ac- cording to numerical calculations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 7(a) for χ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 the maximum is located at d ≈ 6, and the maximum value of kzmin is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For d = 5µm and χ = 2, kzmin is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, by decreasing thickness from 5µm to 15−30 nm, the wave vector kzmin may be modified by a factor of roughly 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The size of any soliton-like formation constructed of magnons that may be utilized for information transfer without dissipation or with very little dissipation has an upper limit determined by the minimal wavelength of a magnon, according to [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 4 Interaction of magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Previously we considered only quadratic in amplitudes part of the Hamilto- nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Here we take into account higher order contributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e, we consider the magnon interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The expansion will be limited by the terms of the third and the fourth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The expansion must be applied only to the exchange (17) and dipolar (19) Hamiltonians since the Zeeman Hamiltonian is purely quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 Third order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us first write out the 3rd order terms of the Hamiltonian, which come solely from the dipolar part: Hd3 = −µB √2µBM 2 ¨ � |ψ|2 + 1 4 |ψ′|2 � ∂z � ψ′∂′ − + ψ′∗∂′ + � dV dV ′ |r − r′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (102) In terms of the Fourier transforms defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (23) and employing the identity 1 |r − r′| = 4π A � q eiq(r−r′)Gq (x − x′) , (103) where the 1d Green function is defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (29), we find: Hd3 = − 2πµB √2µBM √ A ˜ ∞ −∞ dxdx′ � q1,q2,q3,q � χq1χ∗ q2δq1−q2+qδq3−q + 1 4χ′ q1χ′∗ q2δqδq1−q2+q3−q � iqz× � χ′ q3 (dx′ − qy) + χ′∗ −q3 (dx′ + qy) � Gq(x − x′) (104) The second term in the sum contains the factor δq that makes qy = qz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the square bracket in this equation is equal to � χ′ q3 + χ′∗ −q3 � dx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Acting to Gq(x−x′), the operator dx′ transforms it into qsign (x − x ′) Gq(x − x ′) = sign(x−x ′) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the second term in the sum is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The Kronecker δ−symbols in the first term imply that q = q3 = q2 − q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the dipolar Hamiltonian of the third order is simplified to Hd3 = − 2πµB √2µBM √ A ˜ ∞ −∞ dxdx′ � q1,q2 χq1χ∗ q2i (q2z − q1z) � χ′ q2−q1 (dx′ − q2y + q1y) +χ′∗ q1−q2 (dx′ + q2y − q1y) � G|q1−q2| (x − x′) (105) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 Third order non-linearity in terms of quantized magnon am- plitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In this section we perform the Bogoliubov transformation (37) from transverse modes χq (x) to the quantized amplitudes of magnons ηq,n, η∗ q,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' After some algebra we arrive at a cubic form for these amplitudes limited by the requirement of the momentum conservation (translational invariance): Hd3 = − 2πµB √2µBM √ A � q1n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q2n2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3n3 δq1−q2+q3 � I(+++) d3 ηq1n1η−q2n2ηq3n3 + I(++−) d3 ηq1n1η−q2n2η∗ −q3n3 I(+−+) d3 ηq1n1η∗ q2n2ηq3n3 + I(−++) d3 η∗ −q1n1η−q2n2ηq3n3 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='c � , (106) where the eight coefficients I(ρστ) d3 with ρ, σ, τ taking values +, − are matrix elements of the three transverse modes: the first is u∗ q1n1 (x) for ρ = + and 26 u−q1n1 for ρ = −;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' the second is v∗ −q2n2 (x) for σ = + and vq2n2 for σ = −;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' the third is given by iq3z � u∗ q3n3 (x′) (dx′ − q3y) − v∗ q3n3 (x′) (dx′ + q3y) � Gq3 (x − x′) for τ = + and iq3z � u−q3n3 (x′) (dx′ + q3y) − v−q3n3 (x′) (dx′ − q3y) � Gq3 (x − x′) for τ = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The matrix element is the double integral over x and x′ from the products of any set of these three modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='For the reader convenience we place below explicit expressions for the integrals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I(ρστ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='d3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='with all three indices + and with two + and one −: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I(+++) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='d3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= −iq3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='dxdx′u∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q1n1v∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='−q2n2× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='u′∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3n3 (dx′ − q3y) − v′∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3n3 (dx′ + q3y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='Gq3(x − x′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I(++−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='d3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= −iq3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='dxdx′u∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q1n1v∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='−q2n2× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='u′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='−q3n3 (dx′ + q3y) − v′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='−q3n3 (dx′ − q3y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='Gq3(x − x′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I(+−+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='d3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= iq3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='dxdx′u∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q1n1vq2n2× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='u′∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3n3 (dx′ − q3y) − v′∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3n3 (dx′ + q3y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='Gq3(x − x′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I(−++) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='d3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= iq3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='dxdx′u−q1n1v∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='−q2n2× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='u′∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3n3 (dx′ − q3y) − v′∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3n3 (dx′ + q3y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='Gq3(x − x′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (107) In order to obtain the Hamiltonian Hd3 (106) and coefficients I(σρτ) d3 we have used the fact that some terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' the term with η∗ −q1n1η∗ q2n2η∗ −q3n3) can be expressed as complex conjugates of others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' the term with ηq1n1η−q2n2ηq3n3) by permutation of the summation indices q1 ↔ q2 that implies q3 → −q3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Later we will use this kind of relations when calculating 4th-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Note also that the three terms involving one complex conjugated function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (106) can also be received each from other by renaming the summation indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, these three sums are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' On the other hand two last of them are complex conjugates each to other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, all these sums are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 Cherenkov radiation of a low energy magnon by the high en- ergy magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the theory of BECM the life-time of the condensate magnons is dominantly determined by their merging with a high energy magnon and by the inverse process of the Cherenkov radiation of the condensate magnon by a high energy magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Here we consider a more general problem when the high energy magnon emits or absorbs a low energy magnon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The high-energy magnon is assumed to have the exchange dominated dispersion ωq,kx = γℓ2k2, whereas the low-energy magnon dispersion is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the Bogoliubov coefficients uqn the coefficients a, b dominate for ν = +, c, d dominate for ν = −, whereas vqn = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For low-energy magnons generally the coefficients a, b, c, d 27 are of the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' They are defined by eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (49,50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For thick films in the integrals (107) defining the matrix elements of the Cherenkov or inverse Cherenkov process, the terms corresponding to evanescent waves can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 Fourth order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Here we consider the 4th order terms of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In terms of general magnon wave function ψ (r) they are: H4 = Hex4 + Hd4 Hex4 = µ2 Bℓ2 2 ´ � − |ψ|2 |∇ψ|2 + 1 2 � ∇ � |ψ|2��2� dV, Hd4 = µ2 B 2 ˜ � |ψ|2 |ψ′|2 ∂z∂′ z − 1 4 |ψ|2 (ψ∂− + ψ∗∂+) � ψ′∂′ − + ψ′∗∂′ + �� dV dV ′ |r−r′| (108) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 Fourth order Hamiltonian in terms of magnon amplitudes χq (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Employing Fourier transformation to the wave vector representation (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' we find the following expressions for Hex4 and Hd4: Hex4 = µ2 Bℓ2 2A2 ´ � q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q4 � −χq1χ∗ q2 � dxχq3dxχ∗ q4 + q3q4χq3χ∗ q4 � + 1 2dx(χq1χ∗ q2)dx(χq3χ∗ q4) + 1 2(q1 − q2)(q3 − q4)χq1χ∗ q2χq3χ∗ q4 � ei(q1−q2+q3−q4)rdV = µ2 Bℓ2 4A ´ � q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q4 � dxχq1χ∗ q2dxχq3χ∗ q4 + χq1dxχ∗ q2χq3dxχ∗ q4 −(q2 1 + q2 2)χq1χ∗ q2χq3χ∗ q4 � δq1−q2+q3−q4dx = µ2 Bℓ2 4A ´ � q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q4 � dxχq1χ∗ q2dxχq3χ∗ q4 + χq1dxχ∗ q2χq3dxχ∗ q4 − 1 2(q2 1 + q2 2 + q2 3 + q2 4)χq1χ∗ q2χq3χ∗ q4 � δq1−q2+q3−q4dx (109) Hd4 = 2πµ2 B A3 ˜ � q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q � q2 zχq1χ∗ q2χ′ q3χ′∗ q4ei[(q1−q2)r+(q3−q4)r′+q(r−r′)]− 1 4χq1χ∗ q2 � χq3 (dx + qy) + χ∗ −q3 (dx − qy) � � χ′ q4 (dx′ − qy) + χ′∗ −q4 (dx′ + qy) � ×ei[(q1−q2)r+q3r+q4r′+q(r−r′)]� Gq(x − x′)dV dV ′ (110) After integration over y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' z and y′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' z′ the 4th-order dipolar Hamiltonian trans- forms into the sum over momenta and integral over transverse coordinates: Hd4 = 2πµ2 B A ˜ � q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='q � q2 zχq1χ∗ q2χ′ q3χ′∗ q4δq1−q2+qδq3−q4−q − 1 4χq1χ∗ q2 � χq3 (dx + qy) + χ∗ −q3 (dx − qy) � � χ′ q4 (dx′ − qy) + χ′∗ −q4 (dx′ + qy) � ×δq1−q2+q3+qδq4−q} Gq(x − x′)dxdx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (111) In these calculation we used the symmetry with respect to permutations of running momenta participating in the sum and the relation between Fourier component of 1/ |r − r′| and one-dimensional Green function Gq (x − x′) (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (29)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 Fourth order Hamiltonian in terms of the magnon amplitudes ηqνn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Employing the Bogoliubov transformation (38), we represent the 4-th order Hamiltonian in terms of the homogeneous fourth order polynomials of the form (the subscripts qi, ni in the coefficients I4 are omitted for brevity): H4 = � qinkρl(i,k,l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4) I(ρ1ρ2ρ3ρ4) 4 � � 4 � j=1 η(ρj) qjnj � � δq1+q2+q3+q4, (112) where η(+) qn = ηqn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' η(−) qn = η∗ qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It is obvious that the matrix I4 can be made invariant under permutation of four its composite indices γj = (ρjqjnj) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' j = 1, 2, 3, 4 since the product in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (112) is invariant under such permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, it is more reasonable to denote the matrix elements of the matrix I4 as (I4)γ1γ2γ3γ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The table of coefficients (I4)γ1γ2γ3γ4 is given in the Appendix [Hamiltonian of the 4-th order].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 Interaction of condensate magnons in thick films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Here we show the results of calculations of the interaction between condensate of magnons that have momenta either Q = Qˆz or −Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' When the conden- sate exists, the chemical potential µ is equal to the minimal magnon energy ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore the wave functions of the condensates ψ±Q do not depend on time (we remind that the time dependence of the wave function is given by exp � − i(∆−µ)t ℏ � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Further for brevity we denote the wave functions of the two condensates as ψ± and present them in terms of the densities of condensates n± and their time-independent phases φ± as ψ± = √n±eiφ±f (x) , (113) where f (x) = √ 2 cos πx d is the transverse wave function corresponding to the ground state of a magnon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The total wave function is ψ (r) = ψ+eiQr + ψ−e−iQr = �√n+ei(Qz+φ+) + √n−ei(−Qz+φ−)� f (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (114) Introducing notation n = n+ + n− for the total density of condensate and Φ (z) = 2Qz + φ+ − φ− for the phase difference of the two condensates, we find the square of modulus of the wave function: |ψ (r)|2 = � n + 2√n+n− cos Φ (z) � f 2 (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (115) The square of gradient of the wave function is |∇ψ|2 = Q2 � n − 2√n+n− cos Φ (z) � f 2 (x) + � n + 2√n+n− cos Φ (z) � � df dx �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (116) 29 The fourth order exchange Hamiltonian contains two terms − |ψ|2 |∇ψ|2 and 1 2 � ∇ |ψ|2�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Assuming that densities of condensates n± and their phases φ± vary in plane on the distances much larger than period of density oscillation L = 2π/Q, the density of interaction energy of condensates is equal to the exact value of interaction energy averaged over period of oscillation L inte- grated over the transverse coordinate x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For thick films the terms in ∇ψ and ∇ |ψ|2containing derivatives df dx can be neglected in comparison with the terms containing derivatives over z or equivalently the value Q since Qd ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Per- forming simple operations of averaging and integration for exchange interaction we find: Hex4 V = −3µ2 Bℓ2 16 Q2 � n2 − 6n+n− � (117) Analyzing in similar way the interaction energy generated by dipolar Hamilto- nian of the 4-th order, we should find the average of the integrand in the third equation (108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' To make it, we will use the identity: 1 |r − r′| = 1 π ∞ ¨ −∞ dqydqzeiq∥(r∥−r′ ∥)Gq∥ (x − x′) , (118) where the subscript ∥ at a vector means that it is parallel to the surfaces of the film, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=', they have only y and z−components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' we remind that the 1-dimensional Green function of the Helmholtz equation Gq (x) is defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The proof of the identity (118) is given in the Appendix [1/r-G-identity].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the dipolar Hamiltonian of the 4-th order can be rewritten as follows: Hd4 = µ2 B 2π ˝ dV dV ′d2q∥eiq∥(r∥−r′ ∥) � |ψ|2 |ψ′|2 ∂z∂′ z − 1 8 � |ψ|2 + |ψ′|2� × (ψ∂− + ψ∗∂+) � ψ′∂′ − + ψ′∗∂′ + �� Gq∥ (x − x′) (119) Note that we symmetrized the integrand over the variables r and r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Except of the exponential function eiq∥(r∥−r′ ∥) the integrand does not depend of y and y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore the integration over y′ gives 2πδ (qy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The partial derivatives ∂± = ∂x ± iqy become equal each to other and equal to ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The magnitude of derivatives ∂z, ∂z′ is equal to Q, whereas the magnitude of the derivatives ∂x, ∂x′ is equal to 2π/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For thick films Q ≫ 1/d, therefore, the first term in the square brackets of this equation dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In this approximation we find Hd4 = µ2 B ´ dV ´ d/2 −d/2 dx′ ´ ∞ −∞ dz′ ´ ∞ −∞ dqz � n + 2√n+n− cos Φ (z) � � n + 2√n+n− cos Φ (z′) � [f (x) f (x′)]2 q2 zeiqz(z−z′)G|qz| (x − x′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (120) Since the integrand does not depend on y, the integration over this variable gives the linear size of sample Ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us make change of variables Z = z+z′ 2 , ζ = z − z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The Jacobian of this transformation is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The only term 30 in the product of two square brackets in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (120) that together with expo- nential factor eiqz(z−z′) gives non-zero average is 4n+n− cos Φ (z) cos Φ (z′) = 2n+n− [cos (Φ (z) + Φ (z′)) + cos (Φ (z) − Φ (z′))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' From these two terms only the second gives nonzero average over z: ∞ ˆ −∞ dζeiqzζ2 cos (2Qζ) = 2π [δ (qz − 2Q) + δ (qz + 2Q)] (121) This result allows us to perform also integration over qz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Besides of that the integrand does not depend on Z and integration over this variable gives the linear size Lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' These integrations strongly simplify the expression for Hd4: Hd4 = 4πµ2 BLyLzQn+n− d/2 ¨ −d/2 f 2 (x) f 2 (x′) e−2Q|x−x′|dxdx′ (122) The calculation of the double integral in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (122) is elementary and gives: ˜ d/2 −d/2 f 2 (x) f 2 (x′) e−2Q|x−x′|dxdx′ = − −3d5Q5−5π2d3Q3+π4(−2dQ−e−2dQ+1) 2Q2(d2Q2+π2)2 , (123) In the limit of thick film Qd ≫ 1 the leading term is equal to 3d/2Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This dependence of the integral in (123) on parameters as d/Q could be predicted without detailed calculation since the exponent e−Q|x−x′| cut in the square of integration a band of the width ∼ 1/Q along the diagonal, whereas the average value of f 2 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' However, strong fluctuations of f 2 from 0 to 1 with period 1/8 of the diagonal requires explicit calculation to get exact numerical coefficient at the leading term: Hd4 V = 6πµ2 Bn+n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (124) Thus, we have found the density of interaction energy between condensates of different minima (the inter-minima interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It can be written as U4int = Bn+n− (125) with B = 6πµ2 B > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It is repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Note that the terms of the same form in the exchange interaction energy (117) has coefficient B which differs from dipolar value by a factor ∼ Q2ℓ2 ∼ ℓ/d ≪ 1 that can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Another term that enters Hex4/V but is absent in Hd4/V is interaction of the condensate magnons within one minimum U4inn = A 2 � n2 + + n2 − � (126) with A = − 3 8µ2 BQ2ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the interaction within one minimum is attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The magnitude |A| is much smaller than B: |A| /B = π1/2 27/2 ℓ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For YIG film 5μm thick at room temperature |A| /B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 Quasi-equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the experiment by Demokritov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [12] the low energy magnons in the YIG film were generated by a microstrip resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' A photon of frequency ωres emit- ted by the resonator decays into two magnons with practically opposite momenta and frequency ωp = ωres/2 (in classical electrodynamics this process is called parametric resonance or parametric pumping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The resonator frequency is cho- sen to be less than 4∆/ℏ, where ∆ ≈ 2µBH is the minimal energy of magnons (gap in the spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then the decays of pumped magnons are forbidden, whereas their collisions with other low energy magnons remain possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' These collisions establish the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The relaxation time τr is just the time be- tween collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' An important role is played by the processes of the Cherenkov radiation of a low-energy magnon by a thermal magnon and inverse process of the absorption of the low-energy magnon by a thermal magnon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' These processes determine the lifetime of low-energy magnons τl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In YIG at room temperature τr ≪ τl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It means that during the relaxation the number of magnons is con- served and they go to equilibrium with the finite chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The role of pumping is to restore the stationary number of magnons in exchange of absorbed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We will call such a stationary state quasi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us consider the balance of magnons following Bun’kov and Volovik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [21] The occupation number of a low-energy magnon with energy ε in the quasi- equilibrium state is n (ε) = T ε−µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The occupation number of the magnon with the same energy in equilibrium without pumping is n0 (ε) = T ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The total density npm (T, µ) of pumped magnons is npm (T, µ) = ∞ ˆ 0 [n (ε) − n0 (ε)] ¯g (ε) dε, (127) where ¯g (ε) is the magnon density of state per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It can be rewritten as npm (T, µ) = ∞ ˆ 0 Tµ ε (ε − µ) ¯g (ε) dε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (128) The density of magnons pumped per unit time is determined by the pumped power W per unit volume as 2W ℏωres .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In a stationary state it must be equal to the density of pumped magnons that disappear per unit time npm τl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the established density of pumped magnons is npm = 2W ℏωres τl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (129) Replacing npm by the integral in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (128), we obtain equation relating the chemical potential µ to the pumped power W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This equation implies that µ grows monotonically with W growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At a critical value of the pumped 32 power W (c) = ℏωres 2τl ∞ ˆ ∆ T∆ ε (ε − ∆) ¯g (ε) dε, (130) chemical potential reaches its maximum possible value µmax = ∆ and the density of pumped magnons reaches its critical value n(c) pm = ∞ ˆ ∆ T∆ ε (ε − ∆) ¯g (ε) dε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (131) Chemical potential cannot grow more since at µ > ∆, the occupation number of magnons with energy between ∆ and µ would be negative that is nonsense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, at W > W (c) the chemical potential remains unchanged µ = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The excessive magnons go to the state with minimal energy ∆ and form the BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The condensate density is nc = 2 � W − W (c)� ℏωres τl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (132) All these calculations assumed that the integrals are converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' There are two possible sources of divergence: large energies ε → ∞ and ε close to ∆ for W ≥ W (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For large ε the exchange interaction dominates, the magnon energy is quadratic function of momentum and ¯g (ε) ∝ √ε,whereas the denominator of integrand in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (128) asymptotically approaches ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the integral converges at ε → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This result physically means that the pumped magnons after relaxation remain in the range of low energy ∼ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Paradoxically their energy escapes into the range ι ∼ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Indeed, the pumped energy is Epm = ∞ ˆ 0 Tµ ε (ε − µ)ε¯g (ε) dε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (133) This integral diverges at ε → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It happens because we applied low-energy Rayleigh-Jeans approximation n (ε) = T ε−µ, n0 (ε) = T ε for the occupation num- bers of magnons, which at high energy must be replaced by the Planck-Bose- Einstein distribution n (ε) = � exp ε−µ T − 1 �−1 , n0 (ε) = � exp ε T − 1 �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the integral (133) is cut-off at ε ∼ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Neglecting µ in denominator of in- tegrand, we find the rough estimate of the pumped energy per unit volume µT 3/2/ � (µBM)3/2 ℓ3� that corresponds to the change of the magnons temper- ature by δT ≈ ∆/kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For YIG film in external magnetic field H = 600Oe and at room temperature, the resulting increase of temperature is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='04K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The convergence at the points of minimum energy ϵ = ∆ follows from the fact that, in the continuous limit, they are isolated points in 3-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, the density of states near each minimum goes to zero as √ ϵ − ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 Spontaneous violation of the reflection symmetry in the quasi- equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the state of quasi-equilibrium its energy (more accurately its Helmholtz free energy) must be minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At fixed temperature and volume, the free energy has minimum when the occupation numbers obey the Bose-Einstein law and excessive magnons occupy the state with minimal energy ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In ferromagnetic films there are two such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, the ground state of the ideal magnon gas is highly degenerate: the condensate energy Eid = V nc∆ depends only on the total number of magnons in condensate Nc = V nc = N+ + N− and does not depend on how these magnons are distributed between two minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This Nc + 1−fold degeneration is lifted by magnons interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [20] As it was derived in the subsection,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 the 4-th order interaction density of energy is U4 = A 2 � n2 + + n2 − � + Bn+n−, (134) with A < 0 and B > 0 for thick films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The interaction energy U4 has minimum equal to U4 = − A 2 n2 either at n+ = n, n− = 0 or at n+ = 0, n− = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In both cases the symmetry with respect to reflection in the plane z = 0 combined with the time reversal is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Unfortunately such a most asymmetric state contradicts to the experiment and to a more sophisticated theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us start with the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In 2012 in the work by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Novik-Boltyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [22] the Münster experimental team led by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Demokritov discovered a stripe interference structure of the magnetization Mz in the YIG sample (see the interference picture in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=') It can be interpreted as the measurement of |ψ|2 = ��√n+eiQz+φ+ + √n−e−iQz+φ−�� = n + 2√n+n− cos (2Qz + φ+ − φ−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This equation clearly shows that the interference picture can be observed only if both n+ and n− are not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In order to explain this result, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Saslow and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Pokrovsky[23] proposed to consider the additional term in the 4-th order interaction Hamiltonian of purely dipolar origin of the form C 2 �� ψ∗ +ψ2 +ψ− + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' � + (+ ↔ −) � (135) where the abbreviation c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' stays for complex conjugate, C is a real constants whose magnitude in terms of parameters is of the same order as |A|, however the numerical constant in C is by a factor 1/2π3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='016 smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This term is contained in the earlier neglected terms of the 4-th order dipolar interac- tion containing derivatives over x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The real processes associated with this term would be decay of one condensate magnon in three and inverse process of merg- ing three condensate magnon in one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' All such processes are forbidden by the energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' However, they determine additional (anomalous) 4-order interaction energy: H4an V = Cn√n+n− cos (φ+ + φ−) (136) 34 Figure 11: Measurement of the BLS intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Dashed circles indicate the posi- tions of two defects causing an appearance of two vortices of positive circulation in different components of the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The vortices show themselves as forks in the interference pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='Reprinted by permission from Macmillan Publishers Ltd: Scientific Reports [22], Copyright 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Note that this energy depends on a different combination of phases φ+ + φ− than the Goldstone phase φ+ − φ− whose variation does not change energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The minimum energy is reached at φ+ + φ− = π or 0 depending on the sign of the coefficient C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' On the line C = 0 the transition from 0− to π−phase or vice versa proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In both these phases the minimum anomalous interaction energy is negative: min �H4an V � = − |C| n√n+n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (137) Thus, the total 4−th order interaction energy acquires the form: U4 ≡ H4 V = A 2 � n2 + + n2 − � + Bn+n− − |C| n√n+n− (138) Its minimization at a fixed n gives: n √n+n− = 2 (B − A) |C| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (139) Let us denote R = B−A |C| + �� B−A |C| �2 − 1 and Θ = |C| 2(B−A)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The value R is very big, whereas the value Θ ≈ 1 4R2 is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The two solutions of this equation are either n+ = (1 − Θ) n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' n− = Θn, (140) 35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 y(μm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='0 0 1 2 3 4 5 6 7 8 z(μm)or n+ and n− interchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In each solution one of two condensate densities is much larger than another, but the smaller one turns into zero only if C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The total interaction energy in this phase is U4 = n2 � A 2 + |C| 2R (1 − Θ) − |C| � Θ (1 − Θ) � ≈ n2 � A 2 − C2 4B � < 0 (141) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='6 Instability of homogeneous asymmetric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We have found that the homogeneous phase with the violated reflection sym- metry has negative interaction energy proportional to n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It means that the interaction energy decreases when the volume occupied by the condensate de- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In the weakly non-ideal attractive Bose-gas of N particles with the coupling constant g < 0 and mass m of particle this tendency leads to the me- chanical instability of the gas and its collapse at a critical value of number of particles Nc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At this value the isothermal compressibility κT = − 1 V � ∂V ∂P � T is zero and at N > Nc becomes negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Due to quantum uncertainty, the kinetic energy per particle can be written as K/N = ℏ2 2mV 2/3 , whereas the interaction energy is U = gN 2 2V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the total energy is E (N, V ) = N ℏ2 2mV 2/3 + gN 2 2V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (142) The pressure is P = −∂E ∂V = ℏ2N 3mV 5/3 + gN 2 2V 2 (143) and the compressibility is κT = 5ℏ2N 9mV 5/3 + gN 2 V 2 (144) Equation κT = 0 determines the critical number of particles Nc = − 5ℏ2V 1/3 9mg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At N > Nc, the compressibility is negative and the gas becomes mechanically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It starts to contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Since this process proceeds simultaneously in the total volume occupied by the gas, the process will stop when the volume wil be divided into N/Nc cells each containing Nc particles and isolated each from other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The volume of such a cell is v = V Nc/N, therefore the critical number in a cell is different than the critical number in the entire volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It should be found from equation Nc = 5ℏ2 9m|g| � V Nc N �1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It is convenient to express the coupling constant g in terms of the Born scattering length as as g = ℏ2 m as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then Nc = 53/2 27 1 n1/6|as|1/2 , where n = N/V is the average density of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For a weakly interacting Bose gas n1/3 |as| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore Nc ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The collapse was observed in cooled gases of alkali atoms 7Li [24] and 85Rb [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At finite temperature the pressure from excitations must be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It changes the critical values for starting the collapse, but the collapse persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Gases of 36 cooled attracting alkali atoms after the collapse flew out the magnetic or laser trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Our calculations relate to the Bose gas of quasiparticles that cannot avoid the system in which they exist like excitons in semiconductors or spin waves in magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Theoretical predictions for starting parameters of collapse in weakly attract- ing Bose gas at finite temperature were made by Mueller and Baym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [26] Dynamic approach to the same problem was developed by Pitaevskii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [27] For the magnon condensation in a ferromagnetic film, the problem is effec- tively two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It is because the minimum of energy corresponds to the transverse standing wave, period of which fits between surfaces of the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The effective masses are strongly anisotropic (see subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The curve of constant kinetic energy is the ellipsis ℏ2k2 y 2my + ℏ2k2 z 2mz = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore we ex- pect that the collapsed magnon condensate will be limited by an ellipsis with semi-axes Ry, Rz whose ratio is Ry/Rz = � mz/my.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then the kinetic energy of collapsed condensate can be estimated as K = N � ℏ2 2myR2y + ℏ2 2mzR2z � = Nℏ2 √mymzRyRz , (145) whereas the condensate potential energy is U = gN 2 2πRyRzd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (146) The pressure at zero temperature is P = N V 2 � πℏ2d √mymz + g 2N � , (147) where V = πRyRzd is the volume of the condensate cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The compressibility of the magnon gas in the film differs from the pressure only by numerical factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, the pressure and compressibility simultaneously become zero when N reaches a critical value Nc = − 2πℏ2d √mymzg = 2πd |as| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (148) The film will be divided into N/Nc almost isolated cells each containing Nc magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let the cell be a rectangle with the sides Ry, Rz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' If A is the area of the sample, then the area of a cell is RyRz = ANc N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' From this equation and require- ment Ry/Rz = � mz/my we find Ry = � mz my �1/4 � ANc N ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Rz = � my mz �1/4 � ANc N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' According to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (148), this result can be rewritten as Ry = �mz my � � 2π n |as|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Rz = �my mz �1/4 � 2π n |as|, (149) where n = N/(Ad) is the average density of magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The collapse destroys the homogeneous coherent condensate transforming it into a set of isolated islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 37 For YIG with n = 1018cm−3 we find Nc ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='14 × 105, Ry ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='16 × 10−6cm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Rz ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='58 × 10−7cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' There is no experimental evidence of the cell structure in YIG films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3 New experiments and our new theoretical ideas about slow inter-minima relaxation and laser effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Two recent articles by the Münster University experimental team led by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Demokritov [28, 29] revealed several important facts about the Bose-Einstein condensation of magnons (BECM) under permanent pumping first discovered in 2006 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Existing theories of this phenomenon predict an attractive interaction between magnons [30, 31, 23] and a strong spontaneous violation of the reflection symmetry [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' However these theories implicitly assumed that all relaxation processes were fast compared to the lifetime of magnons, whereas one of them, the relaxation between two energy minima, is slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We predict the properties of the stationary state of the magnon gas with condensate, that is far from equilibrium with respect to variables responsible for inter-minima coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The momentum-flip relaxation time is no less than 1 hour, which exceeds even the time of the experiment without considering the lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It means that the equilibrium between condensates in different minima is never reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' As a result, the condensates’ stationary state is far from equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In this regard, it is analogous to the laser stationary state, and, like the laser, the magnon condensate state can produce coherent magnon radiation [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The very slow inter-minima relaxation implies that the appearance of the stationary condensate in a ferromagnetic film is a dynamic phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Since the inter-minima equilibrium is not established, the pumping, which is symmetric with respect to the two minima, creates equal numbers of magnons in the two condensates n+ = n− = nc/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, the inter-minima repul- sion energy Bn+n− = Bn2 c/4 strongly exceeds the magnitude of in-minimum attraction |A| 2 (n2 + + n2 −) = |A|n2 c/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This consideration explains why experi- menters observe repulsion of magnons in the stationary state with the conden- sate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' With a somewhat more sophisticated point of view, the mirror symmetry of the pumping does not necessarily lead to the same symmetry of the conden- sate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In principle, dynamic violation of mirror symmetry is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' But in this case, there is no reason why it should be strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This issue requires further theoretical investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It is difficult to avoid a slight asymmetry of the device in real-world experi- ments, which favors a slightly asymmetric stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Such a device asym- metry could explain the asymmetry observed in the experiment II by Borisenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' If the asymmetry is relatively small, then in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (139) the term Bn+n− is dominant and positive, but it completely conceals the possibility of dynamic spontaneous violation of the reflection symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Because the inter-minima equilibrium is not established, the consistent the- ory of the stationary state with condensate necessitates solving the Boltzmann 38 kinetic equation for magnons and the Gross-Pitaevskii equations for the two condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It can be accomplished using either a variational technique based on the idea of maximal entropy production or by solving a problem with proper initial conditions that asymptotically approaches a stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This kinetic approach may help to bridge another gap between the existing theories [21, 20] and experiment [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The theory proofs that the pumped magnons are accumulated in the low-energy region assuming the temperature of accumulated magnons to be the same as initial temperature of the system (room temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The temperature of low energy magnons is approximately three times higher, according to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' If the temperature is a slow-varying function of energy (momentum) that saturates to the system’s room temperature at some intermediate energy between µBH and the room temperature, the controversy may be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We are thankful to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Nattermann, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Saslow, Fuxiang Li, Chen Sun to- gether with whom were obtained many results mentioned in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Our gratefulness is due to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Demokritov and participants of his experimental team V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Demidov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Borisenko, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Divinskii, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Novik-Boltyk for many useful dis- cussions of the experimental results and cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We thank J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Ketterson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Lim for explanation of their experiment and discussion of its results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We are indebted to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Hillebrands, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Serga and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Bozhko for discussion of their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Many theoretical problems were discussed with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Slavin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' L’vov and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Volovik, who also informed us on a vast literature on the subjecr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Our thanks to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' We remember thankfully the discussion with deceased L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Pitaevskii on the instability of attractive Bose condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Appendix 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Motion of minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The dependence of frequency on wave vector is determined by equation (82) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For the reader’s convenience we reproduce it: ω2 = µ2 BH2 � 1 + k2� � 1 + k2 + χ − χk2 z k2 � (150) Here k2 = k2 ∥ + k2 x, where kx is a positive quantized transverse component of wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Generally to find minimum of frequency for a given mode with fixed quantum numbers and direction of propagation, it is necessary to take in account the dependence of quantized kx on k∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This dependence can be neglected in thick films with d ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Indeed according to the main text, quantized values of kx are equal to kx,ν,n = 2πn d +µν,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Here µν,n = 2 d arctan fν,n � k∥ � , where fν,n � k∥ � is a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' According to this definition, µν,n varies in the limits � − π d , π d � when k∥ changes at least by 1/ √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Therefore, the derivative dkx dk∥ ≲ 1 √ d ≪ 1 and the values k∥ and kx can be considered as independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In this approximation 39 the value of parallel wave vector k∥0 at which frequency has minimum can be found from equation: ∂ω2 ∂ � k2 ∥ � = 2k2 + 2 + χ sin2 θ − χk2 x cos2 θ k4 = 0 (151) At small kx i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' at n ≪ d/2π, the value k2 satisfying eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (151) is also small and equal to k2 0 ≈ k2 ∥0 ≈ � χ 2 + χ sin2 θkx cos θ (152) It is however much larger than k2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The value of frequency in minimum is ωmin ≈ � 1 + χ sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' The equation for k2 0 valid in the range of larger kx comparable with 1 can be found by the following scaling transformation: k2 0 = 2 + χ sin2 θ 2 w (ξ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' ξ = 4χk2 x cos2 θ � 2 + χ sin2 θ �3 , (153) where function w (ξ) obeys cubic equation: w3 + w2 = ξ (154) At small ξ, this equation gives the result (152).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This equation shows that at small kx, the wave vector corresponding to minimal frequency k∥0 grows with kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' To study the motion of minimum in a broader interval of kx it is useful to look at the derivative dk2 ∥0 d(k2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' According to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (151), it can be expressed as follows: dk2 ∥0 d(k2x) = − ∂2ω2 ∂ � k2 ∥ � ∂(k2x) ∂2ω2 � ∂ � k2 ∥ ��2 (155) From this equation it follows that maximal value of k∥0 can be found from equation: ∂2ω2 ∂ � k2 ∥ � ∂ (k2x) = 2 − χcos2 θ k4 + 2χk2 x cos2 θ k6 = 0 (156) It is cubic equation for k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It must be solved together with equation of frequency minimum (151).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Eliminating k2 x from these two equations, we arrive at a closed equation for k2: 6k6 + 2 � 2 + χ sin2 θ � k4 − χ cos2 θk2 = 0 (157) Dividing this equation by k2 ̸= 0, we obtain a quadratic equation for k2, whose solution reads: k2 m = �� 2 + χ sin2 θ �2 + 6χ cos2 θ − � 2 + χ sin2 θ � 6 (158) 40 The value of k2 x corresponding to maximal value of k∥0 can be found by elimi- nating k6 from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (151,157).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It reads: � k2 x � m = 1 3χ cos2 θ �� 2 + χ sin2 θ � k4 m + χ cos2 θk2 m � (159) The maximal value of k2 ∥0 is equal to � k2 ∥0 � max = k2 m − � k2 x � m = 2 3k2 m − � 2 + χ sin2 θ � k4 m 3χ cos2 θ At further increase of kx, the position of minimum k∥0 decreases and finally becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At this point, k2 = k2 x and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (151) turns into quadratic equation for k2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Its solution reads: � k2 x � f = �� 2 + χ sin2 θ �2 + 8χ cos2 θ − � 2 + χ sin2 θ � 4 At this value of kx, minimum merges with a local maximum at k∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At larger values of kx, the only minimum of frequency is at k∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Appendix 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Hamiltonian of the 4-th order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' According to the subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2, the 4-th order Hamiltonian is: H4 = 4 � i,k,l=1 � qinkρl I(ρ1ρ2ρ3ρ4) 4q1n1,q2n2,q3n3,q4n4 � � 4 � j=1 η(ρj) qjnj � � δq1+q2+q3+q4, (160) where η(+) qn = ηqn, η(−) qn = η∗ −qn and upper indices ρl (l = 1, 2, 3, 4) take values +, − independently each from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In terms of complex indices γi = (ρiqini) the Hamiltonian H4 can be rewritten as H4 = � γi Iγ1γ2γ3γ4ηγ1ηγ2ηγ3ηγ4δq1+q2+q3+q4 (161) Since the product of four ηj is symmetric at any permutation P of four j, it is possible to replace the initial coefficients Iγ1γ2γ3γ4 by the symmetrized coefficients Is γ1γ2γ3γ4 = 1 24 � P IγP 1γP 2γP 3γP 4, (162) where Pj means the number appearing on j−th place at permutation P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For example, for the permutation 1, 2, 3, 4 → 4, 3, 2, 1 one finds P1 = 4, P2 = 3, P3 = 2, P4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Let us now analyze what are constraints for symmetrized coefficients fol- lowing from the fact that the energy is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' To make notations more compact 41 further we omit the subscript 4 and round brackets in upper part of initial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (161) turns into H4 = � qknlρm Isρ1ρ2ρ3ρ4 q1n1q2n2q3n3,q4n4 4 � j=1 ηρj qjnj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (163) Since η− qjnj = � η+ −qjnj �∗ , the energy is real if the following relations are satis- fied: Is−ρ1−ρ2−ρ3−ρ4 q1n1q2n2q3n3,q4n4 = � Isρ1ρ2ρ3ρ4 −q1n1−q2n2−q3n3,−q4n4 �∗ (164) However, the initial non-symmetrized coefficients Iρ1ρ2ρ3ρ4 q1n1q2n2q3n3,q4n4 calcu- lated according to the rules formulated in the subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' do not obey these relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Nevertheless, not all of them are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In this Ap- pendix we derive the integral presentation for independent coefficients and find relations that allow to find the rest of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' At fixed values qi, ni;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' i = 1, 2, 3, 4, there are 24 = 16 different combinations of ρj = ± that defines coefficients Iρ1ρ2ρ3ρ4 q1n1q2n2q3n3,q4n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Each of them contains contributions from exchange Ie and dipolar Id interactions, in total 32 coeffi- cients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' In each of them ρj take the same value + or − more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' It allows to make the partial symmetrization over repeating indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For a further compactification of notations we denote the pair j ≡ qjnj and j = −qjnj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' j = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Then the resulting relationships for exchange coefficients are: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e1234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I++++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e4321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='�∗ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I+++− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e4321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='�∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='(167) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I−−+− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e1234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I+−++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e4321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='�∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I++−+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e4321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='�∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I−+++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e2143 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='�∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='(169) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='I−−++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e1234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= I++−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e3412 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= I+−−+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e3214 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='= I−++− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='e1432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='(170) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='Altogether there are 10 equations for 16 exchange coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Thus, only 6 of them are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This 6 coefficients can be chosen as: I++++ e1234 = µ2 Bℓ2 4A ´ d/2 −d/2 [(dxu∗ 1) v∗ 2 (dxu∗ 3) v∗ 4 + u∗ 1 (dxv∗ 2) u∗ 3 (dxv∗ 4) − 1 2 ��4 j=1 q2 j � u∗ 1v∗ 2u∗ 3v∗ 4 � dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (171) I+++− e1234 = − µ2 Bℓ2 4A ´ d/2 −d/2 [(dxu∗ 1) v∗ 2 (dxu∗ 3) u4 + u∗ 1 (dxv∗ 2) u∗ 3 (dxu4) − 1 2 ��4 j=1 q2 j � u∗ 1v∗ 2u∗ 3u4 � dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (172) 42 I−+++ e1234 = − µ2 Bℓ2 4A ´ d/2 −d/2 [(dxv1) v∗ 2 (dxu∗ 3) v∗ 4 + v1 (dxv∗ 2) u∗ 3 (dxv∗ 4) − 1 2 ��4 j=1 q2 j � v1v∗ 2u∗ 3v∗ 4 � dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (173) I++−− e1234 = µ2 Bℓ2 4A ´ d/2 −d/2 [(dxu∗ 1) v∗ 2 (dxv3) u4 + u∗ 1 (dxv∗ 2) v3 (dxu4) − 1 2 ��4 j=1 q2 j � u∗ 1v∗ 2v3u4 � dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (174) I+−+− e1234 = µ2 Bℓ2 4A ´ d/2 −d/2 [(dxu∗ 1) u2 (dxu∗ 3) u4 + u∗ 1 (dxu2) u∗ 3 (dxu4) − 1 2 ��4 j=1 q2 j � u∗ 1u2u∗ 3u4 � dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (175) I−+−+ e1234 = µ2 Bℓ2 4A ´ d/2 −d/2 [(dxv1) v∗ 2 (dxv3) v∗ 4 + v1 (dxv∗ 2) v3 (dxv∗ 4) − 1 2 ��4 j=1 q2 j � v1v∗ 2v3v∗ 4 � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' (176) There only 8 relationships for the dipolar part of 4-th order Hamiltonian: I−−−− d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = � I++++ d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 �∗ (177) I−+−− d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = � I−+++ d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 �∗ (178) I+−−− d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = � I+−++ d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 �∗ (179) I−−+− d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = � I++−+ d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 �∗ (180) I−−−+ d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = � I+++− d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 �∗ (181) I−−++ d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = � I++−− d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 �∗ (182) I+−−+ d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = � I+−+− d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 �∗ (183) I−++− d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = � I−+−+ d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 �∗ (184) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' only 8 of them are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' This 8 coefficients can be chosen as: I++++ d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = πµ2 B 2A ˜ dxdx′× � (q1z + q2z)2 (u∗ 1v∗ 2 + v∗ 1u∗ 2) (u′∗ 3 v′∗ 4 + v′∗ 3 u′∗ 4 ) G|q1+q2| (x − x′) −u∗ 1v∗ 2u∗ 3u′∗ 4 (dx + q4y)2 G|q4| (x − x′) +u∗ 1v∗ 2u∗ 3v′∗ 4 � d2 x − q2 4y � G|q4| (x − x′) +u∗ 1v∗ 2v∗ 3u′∗ 4 � d2 x − q2 4y � Gq4 (x − x′) −u∗ 1v∗ 2v∗ 3v′∗ 4 (dx − q4y)2 Gq4 (x − x′) � (185) 43 I−+++ d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = − πµ2 B 2A ˜ dxdx′× � (q1z + q2z)2 (v1v∗ 2 + v∗ 1v2) (u′∗ 3 v′∗ 4 + v′∗ 3 u′∗ 4 ) G|q1+q2| (x − x′) −v1v∗ 2u∗ 3u′∗ 4 (dx + q4y)2 G|q4| (x − x′) +v1v∗ 2u∗ 3v′∗ 4 � d2 x − q2 4y � G|q4| (x − x′) +v1v∗ 2v∗ 3u′∗ 4 � d2 x − q2 4y � Gq4 (x − x′) −v1v∗ 2v∗ 3v′∗ 4 (dx − q4y)2 Gq4 (x − x′) � (186) I+−++ d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = − πµ2 B 2A ˜ dxdx′× � (q1z + q2z)2 (u∗ 1u2 + v1u∗ 2) (u′∗ 3 v′∗ 4 + v′∗ 3 u′∗ 4 ) G|q1+q2| (x − x′) −u∗ 1u2u∗ 3u′∗ 4 (dx + q4y)2 G|q4| (x − x′) +u∗ 1u2u∗ 3v′∗ 4 � d2 x − q2 4y � G|q4| (x − x′) +u∗ 1u2v∗ 3u′∗ 4 � d2 x − q2 4y � Gq4 (x − x′) −u∗ 1u2v∗ 3v′∗ 4 (dx − q4y)2 Gq4 (x − x′) � (187) I++−+ d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = − πµ2 B 2A ˜ dxdx′× � (q1z + q2z)2 (u∗ 1v∗ 2 + v∗ 1u∗ 2) � v′ 3v′∗ 4 + v′∗ 3 v′ 4 � G|q1+q2| (x − x′) −u∗ 1v∗ 2v3u′∗ 4 (dx + q4y)2 G|q4| (x − x′) +u∗ 1v∗ 2v3v′∗ 4 � d2 x − q2 4y � G|q4| (x − x′) +u∗ 1v∗ 2u3u′∗ 4 � d2 x − q2 4y � Gq4 (x − x′) −u∗ 1v∗ 2u3v′∗ 4 (dx − q4y)2 Gq4 (x − x′) � (188) I+++− d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = − πµ2 B 2A ˜ dxdx′× � (q1z + q2z)2 (u∗ 1v∗ 2 + v∗ 1u∗ 2) � u′∗ 3 u′ 4 + u′ 3u′∗ 4 � G|q1+q2| (x − x′) −u∗ 1v∗ 2u∗ 3v′ 4 (dx + q4y)2 G|q4| (x − x′) +u∗ 1v∗ 2u∗ 3u′ 4 � d2 x − q2 4y � G|q4| (x − x′) +u∗ 1v∗ 2v∗ 3v′ 4 � d2 x − q2 4y � Gq4 (x − x′) −u∗ 1v∗ 2v∗ 3u′ 4 (dx − q4y)2 Gq4 (x − x′) � (189) I++−− d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = πµ2 B 2A ˜ dxdx′× � (q1z + q2z)2 (u∗ 1v∗ 2 + v∗ 1u∗ 2) � v′ 3u′ 4 + u′ 3v′ 4 � G|q1+q2| (x − x′) −u∗ 1v∗ 2v3v′ 4 (dx + q4y)2 G|q4| (x − x′) +u∗ 1v∗ 2v3u′ 4 � d2 x − q2 4y � G|q4| (x − x′) +u∗ 1v∗ 2u3v′ 4 � d2 x − q2 4y � Gq4 (x − x′) −u∗ 1v∗ 2u3u′ 4 (dx − q4y)2 Gq4 (x − x′) � (190) 44 I+−+− d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = πµ2 B 2A ˜ dxdx′× � (q1z + q2z)2 (u∗ 1u2 + u1u∗ 2) � u′∗ 3 u′ 4 + u′ 3u′∗ 4 � G|q1+q2| (x − x′) −u∗ 1u2u∗ 3v′ 4 (dx + q4y)2 G|q4| (x − x′) +u∗ 1u2u∗ 3u′ 4 � d2 x − q2 4y � G|q4| (x − x′) +u∗ 1u2v∗ 3v′ 4 � d2 x − q2 4y � Gq4 (x − x′) −u∗ 1u2v∗ 3u′ 4 (dx − q4y)2 Gq4 (x − x′) � (191) I−+−+ d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='4 = πµ2 B 2A ˜ dxdx′× � (q1z + q2z)2 (v1v∗ 2 + v∗ 1v2) � v′ 3v′∗ 4 + v′∗ 3 v′ 4 � G|q1+q2| (x − x′) −v1v∗ 2v3u′∗ 4 (dx + q4y)2 G|q4| (x − x′) +v1v∗ 2v3v′∗ 4 � d2 x − q2 4y � G|q4| (x − x′) +v1v∗ 2u3u′∗ 4 � d2 x − q2 4y � Gq4 (x − x′) −v1v∗ 2u3v′∗ 4 (dx − q4y)2 Gq4 (x − x′) � (192) All the integrals participating in Id can be calculated explicitly since the in- tegrand is the product of sines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' cosines and exponential function of |x − x′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' However, the large number of different combinations of sines and cosines and the necessity to use different exponents depending on the sign of x − x′ makes real calculation sufficiently tiresome to charge a computer with this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' For the coefficients Ie the calculations are much simpler since they include only sines and cosines and integrals over one variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' However, 6 independent coeffi- cients Ie contain about 30 different integrals, so that charging computer with this task is again justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Appendix 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 1/r-G-identity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' From the Fourier transfromation of 1 |r−r′| we have 1 |r − r′| = 1 (2π)3 ∞ ˚ −∞ dqeiqr 4π q2 = 1 (2π)3 ∞ ˚ −∞ dqeiq∥(r∥−r′ ∥)+iqx(x−x′) 4π q2 ∥ + q2x = 1 (2π)3 ∞ ¨ −∞ dqydqzeiq∥(r∥−r′ ∥) ∞ ˆ −∞ dqxeiqx(x−x′) 4π q2 ∥ + q2x 45 Since ´ ∞ −∞ dqxeiqx(x−x′) 4π q2 ∥+q2x = 4π2 q∥ e−q∥|x−x′| = 8π2Gq∥ Then we get the 1/r- G-identity 1 |r − r′| = 1 π ∞ ¨ −∞ dqydqzeiq∥(r∥−r′ ∥)Gq∥ (x − x′) (193) References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Landau and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Lifshitz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Zs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Sowiet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' 8, 153, 1935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Landau and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Lifshitz, Electrodynamics of Continuous Media, Elsevier, 2nd Edition, 1984, Ch.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Nattermann and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Pokrovsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' B 98, 014436 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Ketterson, Study of micron scale dispersion of spin waves in Yttrium Iron Garnet film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Absrract of presentation at March APS Meeting 2018, Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' [20] Chen Sun, Thomas Nattermann and Valery L Pokrovsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfbvyL/content/2301.01391v1.pdf'} +page_content=' D: Appl.' metadata={'source': 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0000000000000000000000000000000000000000..73178fc8fc4cb83b802305b88159c34ddd3a1c78 --- /dev/null +++ b/ktE3T4oBgHgl3EQfhgqt/content/tmp_files/2301.04572v1.pdf.txt @@ -0,0 +1,5011 @@ +Pre-Equilibrium Evolution of Conserved Charges with ICCING +Initial Conditions +P. Carzon,1, ∗ M. Martinez,2 J. Noronha-Hostler,1 +P. Plaschke,3, † S. Schlichting,3 and M. Sievert4 +1Illinois Center for Advanced Studies of the Universe & Department of Physics, +University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA +2Department of Physics, North Carolina State University, Raleigh, NC 27695, USA +3Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany +4Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA +(Dated: January 12, 2023) +Abstract +Heavy-ion collisions can be well described through relativistic viscous hydrodynamics, but ques- +tions still remain when hydrodynamics is applicable because the initial state may begin very far- +from-equilibrium. Thus, a pre-equilibrium evolution phase is used to bridge the gap between the +initial state and hydrodynamics. KøMPøST is one such pre-equilibrium model that propagates +the energy-momentum tensor by decomposing it into the background and fluctuations around that +background, whose evolution is captured by Green’s functions. We extend this formalism to in- +clude conserved charges and calculate the corresponding non-equilibrium Green’s functions in the +relaxation time approximation. The ICCING algorithm initializes conserved charges in the initial +state by sampling g → q¯q splitting probabilities and is, thus, perfectly positioned to implement +Green’s functions for charge propagation. We show that this method alters the initial state charge +geometries and is applicable in central to mid-central collisions. +∗Email: pcarzon2@illinois.edu +†Email: pplaschke@physik.uni-bielefeld.de +1 +arXiv:2301.04572v1 [nucl-th] 11 Jan 2023 + +Contents +I. Introduction +3 +II. Green’s Functions from Kinetic Theory +7 +A. Boltzmann Equation in Relaxation Time Approximation +7 +B. Background Evolution +9 +C. Perturbations around Bjorken flow +12 +1. Evolution equations for perturbations in the transverse plane +13 +2. Solving the equations of motion in the transverse plane +15 +D. Non-equilibrium Green’s Functions +16 +1. Non-equilibrium Green’s Functions of the Energy-Momentum Tensor +17 +2. Non-equilibrium Green’s Functions of the Current of Conserved Charges +17 +3. Numerical Results for the non-equilibrium Green’s Functions +18 +E. Green’s Functions in Coordinate Space +20 +III. Applications +21 +A. Using Green’s Functions in ICCING +22 +IV. Impact on Eccentricities +28 +V. Conclusion & Outlook +35 +Acknowledgements +36 +A. Background evolution +38 +1. Evolution Equations for the spherical harmonic moments +38 +2. Initial Conditions +39 +3. Relation of Intensive and Extensive Quantities +40 +4. Background Evolution in Conformal Systems +41 +B. Perturbations Around Bjorken Flow +46 +1. Linearised equations of motion and Landau matching +46 +2. Evolution Equations for the Perturbed Moments +47 +3. Initial Energy Perturbations +52 +2 + +4. Initial Charge Perturbations +53 +5. Numerics +53 +C. Non-Equilibrium Green’s Functions of Energy-Momentum Tensor and +Current of Conserved Charges +55 +1. Green’s Functions of the Energy-Momentum Tensor +55 +2. Green’s Functions of the Current of Conserved Charges +55 +3. Numerical Results for the Non-Equilibrium Green’s Functions of the +Energy-Momentum Tensor +56 +4. Green’s Functions of the Energy-Momentum Tensor in Coordinate Space +56 +5. Green’s Functions of the Current of Conserved Charges in Coordinate Space +58 +D. Identities For Spherical Harmonics and Associated Legendre Polynomials 58 +E. Eccentricities, Cumulants, and Anisotropic Flow +62 +1. Standard Initial State Eccentricities +62 +2. Cumulants +63 +References +63 +I. +INTRODUCTION +Ultra-relativistic heavy-ion collisions provide an opportunity to study the extreme limits +of deconfined quarks and gluons. In the very early stages of the collisions, the energy is +predominantly composed of saturated gluons emerging from the low-x wave functions of the +colliding nuclei [1]. This initial state is followed by pre-equilibrium dynamics, leading to the +formation of a quark-gluon plasma (QGP) characterized by deconfined quarks and gluons +acting as a nearly perfect fluid [2, 3]. The measured distributions of final-state hadrons +resulting from the freeze-out of this fluid thus encode a complex superposition of the features +of the initial-state geometry, pre-equilibrium dynamics, and hydrodynamic evolution. Thus, +simulations of all stages of heavy-ion collisions are crucial to reconstruct the early stages of +the collision and interpret experimental data (see [4] and citations within). +To simulate heavy-ion collisions, one starts with an initial state characterization of the +energy-momentum tensor T µν and currents Jµ of conserved charges which is far from thermo- +3 + +dynamic equilibrium. The initial state at proper time τ0 and any pre-equilibrium dynamics +which follow determine the initial conditions at proper time τhydro > τ0 of the hydrodynamic +equations of motion. This hydrodynamic evolution continues until the system has frozen out +into baryons and mesons. Apples-to-apples comparisons to experiments are possible after +a further simulation of the hadronic gas phase, where the system is described in terms of +hadrons and their interactions [5–7]. +Because the QGP behaves as a nearly perfect liquid [4], the geometric structure from the +initial-state T µν and Jµ leaves an observable imprint on the final state hadron distributions +[8–15]. This both makes models of the initial conditions particularly important in the pre- +diction of experimental observables and also allows us to constrain these models by direct +confrontation with data. Specifically, observables which are less sensitive to the hydrody- +namic phase, such as the cumulant ratio vn {4} /vn {2} in central collisions, can provide a +direct window into initial state effects [16–18]. +Until recently, initial-state models have primarily focused on descriptions of the energy +density ϵ = T 00. Recent progress has systematically included more initial state variables +including initial flow T 0i and initial shear T ij [19–26]. Initial conditions of conserved charge +densities ρ = J0 [27–31] have also been developed, though primarily concerned with baryon +density ρB due to its role in the search for the QCD critical point. Recently, an open-source +Monte Carlo event generator, known as ICCING (Initial Conserved Charges in Nuclear +Geometry), was developed which is capable of initializing all three conserved charge densities +[32, 33]: baryon density, strangeness density, and electric charge density (BSQ). ICCING +is a model-agnostic algorithm which constructs the initial conditions for the BSQ charge +densities for a given energy density, by treating the energy density as being composed of +gluons and stochastically sampling their probability to split into quark-antiquark (q¯q) pairs. +The g → q¯q splitting probabilities can be specified according to any desired microscopic +model, among many other parts of the code. The ICCING algorithm is constructed in a +modular fashion to account for a variety of different physical inputs relevant throughout +the code. The sampling process is done by seeding a random point from the input energy +distribution and selecting a fraction of energy from a circle centered on that point. The radius +of this ‘gluon’, the probability distribution from which the fraction of energy is sampled, +and the minimum amount of energy allowed for a gluon are external inputs set by the user +for this step of the process. This extends to the rest of the algorithm which is explained in +4 + +full in Refs. [32, 33]. +To constrain the parameters used in ICCING initial conditions from experimental data, +a necessary next step would be to evolve these initial conditions using a 2+1D viscous +hydrodynamics code that simultaneously solves all of the hydrodynamic equations of motion, +including all of the conserved currents as well as the energy-momentum tensor. This task +is technically challenging, not only because of the challenges associated with evolving the +new conserved currents themselves, but also because of the need for a fully four-dimensional +equation of state. These issues have started to be addressed and are expected to appear soon +[34]. There is a further challenge in the connection of the initial state to the hydrodynamic +evolution since the former is far from equilibrium. Recent work [25] has shown that moving +directly from initial conditions to hydro produces a large fraction of fluid cells that violate +nonlinear causality constraints [35], about 30%, while there is uncertainty about the causal +status of the remaining cells. This study found that including a pre-equilibrium evolution +stage reduces the number of acausal cells but does not fully eliminate them. Generally, +a proper description of the pre-equilibrium dynamics is not only theorectically desirable, +but can also affect flow observables in small and large collision systems [36–38]. +Based +on a microscopic description in QCD kinetic theory, the authors of [24, 39] developed the +non-equilibrium linear response formalism KøMPøST which allows to propagate the energy- +momentum tensor T µν from early times up to the point where a fluid dynamical description +becomes applicable. The KøMPøST code was used in [25] as the pre-equilibrium stage. +Coupling the ICCING code to a pre-equilibrium stage would further extend its usefulness. +So far the pre-equilibrium description in KøMPøST itself does not contain what is needed to +evolve the conserved charge densities from ICCING, but the methods that are used, namely +non-equilibrium Green’s functions, could be applied to our case. +The main idea of KøMPøST is to extract the energy-momentum tensor T µν(x) at time +τ = τhydro from an initial state model. For this the system is evolved by using effective kinetic +theory from an initial time τ0 to τhydro. +Fluctuations δT µν(x) of the energy-momentum +tensor around the background energy-momentum tensor T µν +BG(x) are considered within this +framework. In practise the perturbations are assumed to be small and therefore can be +linearised. This gives rise to linear response theory, where the complete energy-momentum +tensor T µν(x) can be obtained by a sum of T µν +BG(x) and a term involving non-equilibrium +Green’s functions, which capture the evolution of the perturbations. This provides a powerful +5 + +method to compute T µν(x) as numerical simulations only need to be done once to obtain +the background evolution and the Green’s functions. In the past few years different groups +have begun to incorporate KøMPøST within their fluid dynamics simulations, making direct +connections to experimental data [40–42]. +While KøMPøST is based on QCD kinetic theory, more recently studies have been made +in which the same Green’s functions are computed in simpler models, such as the Boltzmann +equation in Relaxation Time Approximation (RTA) [43, 44]. Assuming the relaxation time +approximation drastically simplifies the theoretical description and allows an efficient way +to compute the non-equilibrium Green’s functions. In the relaxation time approximation +the general formalism of KøMPøST can also be expanded in a much simpler way to include +conserved charges and compute Green’s functions for the corresponding current. +These +Green’s functions for charge and energy propagation can be included in ICCING by some +careful reworking and provide a meaningful pre-equilibrium evolution for the conserved +charge densities. +To test the effect of these new pre-equilibrium charge evolution equations, we will look +at the event averaged 2-particle eccentricities (See App. E) which describe the geometry of +the initial state and have been shown to be good predictors of the final state flow harmonics +[10] except in peripheral collisions where non-linear corrects become significant [17, 18, 45]. +Because of this linear mapping, it is possible to cancel out many of the medium effects +by taking the ratio of 4-particle to 2-particle cumulants, which also are a measure of the +fluctuations of a certain type of initial state geometry. These are well understood for the +energy density and have only started to be studied for BSQ charge densities [32]. +In this paper we couple the Green’s functions coming from relaxation time approximation +to the ICCING algorithm by treating energy and charge differences after gluon splittings as +small perturbation around the background. For that we first introduce the basics about our +Green’s functions calculation in Sec. II. Sec. III is dedicated to the applications of these +response functions in the ICCING algorithm, while we present our results in detail in Sec. +IV. Conclusions are found in Sec. V. +Besides this we provide additional calculations in the appendix about the background +evolution (App. A), the perturbations around this background (App. B) and about the +Green’s functions (App. C). In App. D we mention technical aspects regarding spherical +harmonics and in App. E the definition is given for the initial state eccenctricities and +6 + +cumulants. +II. +GREEN’S FUNCTIONS FROM KINETIC THEORY +In order to introduce the non-equilibrium Green’s functions we follow the same idea as +[39] by dividing the space-time dynamic into a background evolution and perturbations +around this background. On the technical side we follow [43], where the authors solved the +equations of motions in term of moments of the distribution functions. This formalism will +be extended to include conserved charges. +Although we are not primarily interested in the background evolution in this study, +we need to address it briefly since its dynamics enters the time evolution of the energy and +number density. Therefore Sec. II A and Sec. II B are dedicated to introduce the background +evolution. Further results on our study regarding the background evolution can be found +in App. A. Afterwards we will consider the dynamics of small space-time perturbations in +Sec. II C (see App. B for further details). The evolution of these perturbations will be +captured in terms of non-equilibrium Green’s functions, which are introduced in Sec. II D. +For completeness, the Green’s functions that are not relevant for our study are presented in +App. C. +A. +Boltzmann Equation in Relaxation Time Approximation +Starting point of our analysis is the Boltzmann equation in relaxation time approximation +(RTA) +pµ∂µf = C[f] = −pµuµ(x) +τR +[f − feq(pµβµ(x), µ(x))] , +(1a) +pµ∂µfa = C[fa] = −pµuµ(x) +τR +[fa − fa,eq(pµβµ(x), µa(x))] , +(1b) +where x = (x0, x, x3) describes a four-dimensional vector in Minkowski space, β(x) = +uµ(x)/T(x) with uµ(x) being the local-rest frame velocity obtained by Landau matching, +T(x) being the effective temperature and a standing for the quarks up, down and strange. +7 + +By f and fa we denote the singlet and valence distribution functions +f = νgfg + νq +� +a +� +fqa + f qa +� +, +(2a) +fa = νq +� +fqa − f qa +� +, +(2b) +with g standing for gluon, q standing for quark, a = u, d, s, such that Nf = 3, and +νg = 16, νq = 6 the spin-color degeneracy factor. The corresponding equilibrium distri- +bution functions are the Bose–Einstein distribution function for gluons and the Fermi-Dirac +distribution function for quarks and antiquarks. Each quark flavor has its own chemical po- +tential µa(x) which governs the evolution of the valence charge distribution Eq. (1b), while +the flavor singlet distribution (1a) evolves according to the effective chemical potential µ(x) +for all flavors. We also emphasize that we assume the same relaxation time τR for every +species; while this is a fairly restrictive assumption, we use it as a first step to explore the +geometrical impact of charge diffusion. As we consider the evolution in a conformal system, +τR is proportional to the inverse temperature such that [46] +τRT(τ) = 5 ˜ηT +e + P = const , +(3) +where ˜η is the shear viscosity, e the energy density, P the pressure and T the effective +temperature of the system. +The velocity uµ(x) is the local rest-frame velocity which is +determined via the Landau-matching conditions +T µν(x)uν(x) = e(x)uµ(x) , +(4a) +which ensures energy-momentum conservation [47]. In addition to this, we also have the +matching conditions for the conserved charges na(x), +N µ +a (x)uµ(x) = na(x) , +(5a) +such that the local thermodynamic variables T(x) and µa(x) can be determined from +e(x) = eeq(T(x), µ(x)) , +(6a) +na(x) = na,eq(T(x), µ(x)) . +(6b) +We emphasize that the above-mentioned quantities T µν(x) respectively N µ +a (x) are defined +to have contributions from all species of particles such that +T µν = T µν +g ++ +� +a +� +T µν +a ++ T +µν +a +� +, +(7a) +N µ +a = N µ +qa − N +µ +qa . +(7b) +8 + +As we are interested in longitudinally boost-invariant expanding systems it is convenient to +work in Milne coordinates +τ = +� +(x0)2 − (x3)2 +, +η = arctanh +�x3 +x0 +� +, +(8) +such that gµν = diag(+1, −1, −1, −τ 2) and +� +−g(x) = τ. Furthermore we consider the +quarks and gluons to be massless, such that their momentum can be parameterized as +pµ = (pT cosh(y), p, pT sinh(y)) +(9) +with y = arctanh(p3/p0) being the momentum space rapidity and pT ≡ |p|. In the Milne +coordinates the Boltzmann equation takes the following form +� +pτ∂τ + pi∂i + pη∂η +� +f(x, p) = −pµuµ(x) +τR +[f(x, p) − feq(pµβµ(x), µ(x))] , +(10a) +� +pτ∂τ + pi∂i + pη∂η +� +fa(x, p) = −pµuµ(x) +τR +[fa(x, p) − fa,eq(pµβµ(x), µa(x))] , +(10b) +where +pτ = pT cosh(y − η) +, +pη = 1 +τ pT sinh(y − η) , +(11) +and i = x, y. It turns out that, when analyzing the dynamics of a boost invariant medium, +it is more convenient to work with the (dimensionless) longitudinal momentum variable +pη = −τpT sinh(y − η) . +(12) +With respect to this coordinate we arrive at the following form of the Boltzmann equation +� +pτ∂τ + pi∂i − pη +τ 2∂η +� +f(x, p) = −pµuµ(x) +τR +[f(x, p) − feq(pµβµ(x), µ(x))] , +(13a) +� +pτ∂τ + pi∂i − pη +τ 2∂η +� +fa(x, p) = −pµuµ(x) +τR +[fa(x, p) − fa,eq(pµβµ(x), µa(x))] , +(13b) +where pτ = +� +p2 +T + (pη/τ)2 represents the massless on-shell condition. +B. +Background Evolution +In order to investigate the dynamics of the system in the pre-equilibrium phases, we make +the assumption that the system can be divided into a background and perturbations around +9 + +this background. In the pre-equilibrium stage the plasma experiences a rapid longitudinal +expansion. However, in the transverse plane the plasma is initially at rest and the expansion +only builds up on timescales that are comparable to the systems size. Therefore, we can ne- +glect the transverse expansion at early times and consider the idealized situation of Bjorken +flow. Accordingly, the background is assumed to be longitudinally boost-invariant, parity +invariant under spatial reflections along the longitudinal axis as well as azimuthally symmet- +ric and translationally invariant in the transverse plane. The aforementioned symmetries +constrain the distribution functions of the background to the following form +f(x, p) = fBG(τ, pT, |pη|) , +(14a) +fa(x, p) = fa,BG(τ, pT, |pη|) . +(14b) +For such a system the energy-momentum tensor is diagonal in Milne coordinates with entries +T µν +BG = diag +� +e, PT, PT, PL/τ 2� +, +(15) +where e is the energy density, PT the transverse and PL the longitudinal pressure. As the +energy-momentum tensor is diagonal in Milne coordinates, the Landau-matching Eq. (4) is +solved trivially with +uµ = (uτ, u, uη) = (1, 0, 0, 0) , +(16a) +e = eeq . +(16b) +Regarding the conserved charges and their respective currents one finds +N µ +a = (N τ +a , Na, N η +a ) = (na, 0, 0, 0) , +(17) +where na ≡ nqa − nqa. +Finally, the Boltzmann equation takes the familiar form +τ∂τfBG(τ, pT, |pη|) = − τ +τR +� +fBG(τ, pT, |pη|) − feq +� pτ +T(τ), µ(τ) +�� +, +(18a) +τ∂τfa,BG(τ, pT, |pη|) = − τ +τR +� +fa,BG(τ, pT, |pη|) − fa,eq +� pτ +T(τ), µa(τ) +�� +. +(18b) +Our strategy to solve these equations follows [43, 48] and consists of expanding the distri- +bution functions in terms of spherical harmonics Y m +l (φ, θ) according to +E m +l (τ) = τ 1/3 +� +dpη +(2π) +� +d2p +(2π)2pτY m +l (φp, θp)fBG(τ, pT, |pη|) , +(19a) +N m +al (τ) = +� +dpη +(2π) +� +d2p +(2π)2Y m +l (φp, θp)fa,BG(τ, pT, |pη|) , +(19b) +10 + +and solve the equations of motions for these moments. Since the evolution of the background +is not the main focus of this work, we simply note that a detailed analysis for vanishing charge +densities can be found in [43]. While in this work, we will only consider energy-momentum +and charge perturbations on top of a charge neutral background, we provide additional +discussion on the background evolution in the presence of non-vanishing density in App. A. +As we are interested in the evolution around vanishing background charge density, we +should mention that we can extract the background energy density at any given time as +a function of the initial energy density according to the following method1. By following +[43, 49, 50], we can compute the energy density at late times +� +τ 4/3e(τ) +� +∞ as a function of +the initial energy density as +� +τ 4/3e(τ) +� +∞ = C∞ +� +4π˜η/s +T(τ0)τ 1/4 +0 +�4/9 +(eτ)0 , +(20) +where the constant C∞ ≈ 0.9 [43, 49] quantifies how efficiently the initial energy is converted +into thermal energy, ˜η/s is the shear-viscosity to entropy density ratio, which is constant for +a conformal system with vanishing net charge density. By (eτ)0 we denote the initial energy +density per unit area and rapidity +(eτ)0 ≡ +dE0 +dη d2x = dE0,g +dη d2x + +� +a +� dE0,qa +dη d2x + dE0,qa +dη d2x +� +(21) +which becomes constant in the limit τ → 0, in kinetic theory. +Inverting the Landau matching condition Eq. (4) for vanishing chemical potential gives +T(τ) = +� 30 +π2νeff +e(τ) +�1/4 +(22) +with νeff = νg + 7 +82Nfνq the overall effective degeneracy factor of all partons, such that +� +τ 4/3e(τ) +� +∞ can be expressed as +� +τ 4/3e(τ) +� +∞ = C∞(4π˜η/s)4/9 +�π2νeff +30 +�1/9 +(eτ)8/9 +0 +. +(23) +In the next step we introduce an attractor curve E( ˜w), which depends on the dimensionless +time-variable [49] +˜w = T(τ)τ +4π˜η/s . +(24) +1 For the sake of readability we drop the subscript BG for the energy density for a moment. +11 + +This attractor curve smoothly interpolates between free-streaming at early times and viscous +hydrodynamics at late times +E( ˜w ≪ 1) = C−1 +∞ ˜w4/9 , +(25a) +E( ˜w ≫ 1) = 1 − +2 +3π ˜w4/9 . +(25b) +The attractor curve connects the asymptotic value +� +τ 4/3e(τ) +� +∞ to its counterpart at any +given time +� +τ 4/3e(τ) +� +according to +E( ˜w) = +� +τ 4/3e(τ) +� +(τ 4/3e(τ))∞ +, +(26) +and has been calculated in [43, 49] for the Boltzmann equaton in RTA and in [49, 50] for +Yang-Mills and QCD kinetic theory. Based on this attractor curve, we can therefore relate +the initial energy density to the energy density at a later time via +eBG(e(τ0)) = C∞(4π˜η/s)4/9 +�π2νeff +30 +�1/9(eτ)8/9 +0 +τ 4/3 E( ˜w) , +(27) +assuming that the system can locally be described by conformal Bjorken flow up to this time +scale. +C. +Perturbations around Bjorken flow +So far we addressed the evolution of a homogeneous, boost invariant background. Now +we will consider linearized perturbations around this background, caused by small space- +time dependent variations of the initial energy or charge densities. We will linearize the +kinetic equations, such that we can derive an evolution equation for the perturbations of the +distribution functions δf and δfa +� +pτ∂τ + pi∂i − pη +τ 2∂η +� +δf(x, p) += −pτ +τR +δf(x, p) + pµδuµ(x) +τR +� +(feq − f(x, p)) + +pτ +T(τ)f +(1,0) +eq +� +− pτ +τR +δT(x) +T(τ) +�T(τ) +τR +∂τR +∂T (feq − f(x, p)) + +pτ +T(τ)f +(1,0) +eq +� ++ pτ +τR +� +a +δµa(x) +� +f +(0,1) +qa,eq + f +(0,1) +qa,eq +� +(28a) +12 + +and +� +pτ∂τ + pi∂i − pη +τ 2∂η +� +δfa(x, p) += −pτ +τR +δfa(x, p) + pµδuµ(x) +τR +� +(fa,eq − fa(x, p)) + +pτ +T(τ)f +(1,0) +a,eq +� +− pτ +τR +δT(x) +T(τ) +�T(τ) +τR +∂τR +∂T (fa,eq − fa(x, p)) + +pτ +T(τ)f +(1,0) +a,eq +� ++ pτ +τR +δµa(x)f +(0,1) +a,eq , +(28b) +where +δf(x, p) = νgδfg(x, p) + νq +� +a +� +δfqa(x, p) + δf qa(x, p) +� +, +(29a) +δfa(x, p) = νq +� +δfqa(x, p) − δf qa(x, p) +� +. +(29b) +Here we used a shorter notation for the derivatives, namely +f +(n,m)(x, y) ≡ ∂n +∂xn +∂m +∂ymf(x, y) . +(30) +As +feq = feq +� pτ +T(τ), µ(τ) +� +, +(31) +fa,eq = fa,eq +� pτ +T(τ), µ(τ) +� +, +(32) +the derivatives are with respect to pτ/T(τ) for the (1, 0)-derivative and with respect to µ(τ) +for the (0, 1)-derivative. +The perturbations of the rest-frame velocity, δuµ(x), the temperature, δT(x), and the +chemical potential, δµa(x), are determined by the linearised Landau-matching conditions. +Details can be found in App. B 1. +1. +Evolution equations for perturbations in the transverse plane +From now on we will concentrate on perturbations in the transverse plane, i.e. only for the +transverse coordinates x. To solve the equations of motion we will expand the perturbations +in a Fourier basis such that +δfi(τ, x, p, |pη|) = +� +d2k +(2π)2δfi,k(τ, p, |pη|)eik·x +(33) +13 + +for i +∈ +{g, qa, qa} and where δfi,k(τ, p, |pη|) +≡ +δfi(τ, k, p, |pη|). +The definition of +δfk(τ, p, |pη|) and δfa,k(τ, p, |pη|) is analogous to the cases before. Decomposing the ve- +locity perturbation in the transverse plane into components parallel, δu∥ +k(τ), and transverse, +δu⊥ +k (τ), to the wave vector in the transverse plane k we therefore find +δT(τ, x) = +� +d2k +(2π)2δTk(τ)eik·x , +(34a) +δµa(τ, x) = +� +d2k +(2π)2δµa,k(τ)eik·x , +(34b) +δui(τ, x) = +� +d2k +(2π)2 +� +δu∥ +k(τ)δji + δu⊥ +k (τ)ϵji� kj +|k|eik·x , +(34c) +δuτ(τ, x) = 0 +, +δuη(τ, x) = 0 . +(34d) +Note that δuη(τ, x) = 0 vanishes identically due to the assumption that boost invariance +along the beam axis is not broken for the perturbations. This assumption could be relaxed +in future work, but is important here for coupling to a 2+1D geometry as in ICCING. +Denoting +k · p +|k|pτ = δij kipj +|k|pτ = cos(φpk) sin(θp) , +(35a) +k × p +|k|pτ = ϵij kipj +|k|pτ = sin(φpk) sin(θp) , +(35b) +where φpk ≡ φp−φk is the angle between p and k in the transverse plane and sin(θp) = pT/pτ +and inserting the Fourier integrals above into Eq. (28) allows us to find an evolution equation +for δfk and δfa,k, such that we have +τ∂τδfk(τ, p, |pη|) += − +� +iτ|k| k · p +|k|pτ + τ +τR +� +δfk(τ, p, |pη|) +− τ +τR +� +δu∥ +k(τ) k · p +|k|pτ + δu⊥ +k (τ)k × p +|k|pτ +�� +(feq − f(x, p)) + +pτ +T(τ)f +(1,0) +eq +� +− τ +τR +δTk(τ) +T(τ) +�T(τ) +τR +∂τR +∂T (feq − f(x, p)) + +pτ +T(τ)f +(1,0) +eq +� ++ τ +τR +� +a +δµa,k(x) +� +f +(0,1) +qa,eq + f +(0,1) +qa,eq +� +(36a) +14 + +and +τ∂τδfa,k(τ, p, |pη|) += − +� +iτ|k| k · p +|k|pτ + τ +τR +� +δfa,k(τ, p, |pη|) +− τ +τR +� +δu∥ +k(τ) k · p +|k|pτ + δu⊥ +k (τ)k × p +|k|pτ +�� +(fa,eq − fa(x, p)) + +pτ +T(τ)f +(1,0) +a,eq +� +− τ +τR +δTk(τ) +T(τ) +�T(τ) +τR +∂τR +∂T (fa,eq − fa(x, p)) + +pτ +T(τ)f +(1,0) +a,eq +� ++ τ +τR +δµa,k(x)f +(0,1) +a,eq . +(36b) +In Eqs. (36) we sorted the terms by the several perturbations. The first term on the right +hand side corresponds to free-streaming, while the second term describes the relaxation +of the perturbations. In the following lines one sees that the perturbations of the velocity, +temperature and chemical potential cause a change of the equilibrium distribution, while the +velocity and temperature perturbations also affect the relaxation of the out-of-equilibrium +background. +2. +Solving the equations of motion in the transverse plane +In order to solve Eq. (36) we follow the same strategy as for the background. Therefore, +we define the perturbed moments according to +δE m +l,k(τ) = τ 1/3 +� +dpη +(2π) +� +d2p +(2π)2pτY m +l (φpk, θp)δfk(τ, p, |pη|) , +(37a) +δN m +al,k(τ) = +� +dpη +(2π) +� +d2p +(2π)2Y m +l (φpk, θp)δfa,k(τ, p, |pη|) . +(37b) +As the derivation of the equations of motion for the moments are not of primary interest +here, we will shift the explicit calculation into App. B. +Similar to the background, we are able to obtain the components of δT µν +k +and δN µ +a,k as +combinations of low order moments. A full list can be found in App. B 2. Furthermore +we can also relate the perturbations of the intensive quantities to the perturbation of the +extensive quantities for na = 0 according to Eq. (A15) by +δTk +T += δek +4e , +(38a) +δµa,k = 6 +νq +δna.k +T 2 +. +(38b) +15 + +Therefore we can replace δTk and δµa,k with δek and δna,k, which is useful since we can +express these quantities by low order moments again +τ 4/3δek(τ) = +√ +4πδE 0 +0,k(τ) , +(39a) +τ 4/3(e + PT)δu∥ +k(τ) = − +� +2π +3 +� +δE +1 +1,k(τ) − δE −1 +1,k(τ) +� +, +(39b) +τ 4/3(e + PT)δu⊥ +k (τ) = i +� +2π +3 +� +δE +1 +1,k(τ) + δE −1 +1,k(τ) +� +, +(39c) +τδna,k = +√ +4πδN 0 +a0,k , +(39d) +This results in a closed set of equations since all appearing perturbations can be written as +linear combinations of moments. +The equations of motions will be solved numerically. For this we truncate the evolution +at lmax=512. In order to find a reasonable value we compared our results to the analytical +free-streaming equations. This comparison shows that convergence is reached much faster +for δE m +l,k than for δN m +al,k. More details can be found in Fig. 14 in App. B 5. +We note that, in addition to [43] we also need to invert the matching conditions Eq. (6). +This we do numerically at each time step. More details on this can be found in App. A 3. +D. +Non-equilibrium Green’s Functions +In principle, we can obtain all information about the evolution of the system from the +moments. However, we find it more convenient to consider Green’s functions of the energy- +momentum tensor and the charge current. Therefore, we will consider linear response func- +tions ˜Gµν +αβ for the energy-momentum tensor respectively and ( ˜Fab)µ +α for the charge current. +Here ˜G and ˜F describe the Green’s functions in momentum space. As we will show below, +the Green’s functions can be related to macroscopic quantities (Eq. (44), Eq. (49)), which +are however related to low order moments according to Eqs. (39). This is another powerful +property of our formalism as it is easy to quantify the systems response to perturbations +once one have solved the equations of motions. +For our study, only ˜Gττ +ττ and ( ˜Fab)τ +τ are relevant, such that we only present the result for +these two here. The other Green’s functions can be found in App. C. +16 + +1. +Non-equilibrium Green’s Functions of the Energy-Momentum Tensor +We will follow the construction of the response functions according to [24, 39] and express +δT µν +k (τ) as +δT µν +k (τ) +e(τ) += 1 +2 +˜Gµν +αβ(k, τ, τ0)δT αβ +k (τ0) +e(τ0) +. +(40) +In the following we will omit the explicit dependence on τ0 for better readability since we +are mainly interested in the limit τ0/τR → 0, where the kinetic framework describes the +equilibration process from directly after the collision until the onset of the hydrodynamic +regime. Besides this, we also introduce the propagation phase κ by +κ = |k|(τ − τ0) . +(41) +When expressing the evolution equations for the moments in terms of κ, this change of vari- +able introduces additional terms in the time derivative [43], which were taken into account. +Similarly to [39], we will decompose the response functions into a basis of Lorentz scalars +(s), vectors (v) and tensors (t). For ˜Gττ +ττ this means +˜Gττ +ττ(k, τ) = ˜Gs +s(κ, x) . +(42) +Since the normalization of the linearized perturbation is arbitrary, we adopt the convention +δe(τ0) +e(τ0) = 1 +(43) +such that we can express the decomposed response function in terms of δT µν +k (τ) (see [39]) +according to +˜Gs +s(κ, x) = δT ττ +k (x) +e(x) += δeκ(x) +e(x) . +(44) +2. +Non-equilibrium Green’s Functions of the Current of Conserved Charges +For the Green’s functions corresponding to the conserved charges, we follow the same +strategy. Before we compute them, we first recall that in Bjorken flow +τn(τ) = const . +(45) +17 + +In particular we have τn(τ) = τ0n(τ0), i.e. we need a slightly different definition for the +charge Green’s functions +τδN µ +a,k(τ) = ( ˜Fab)µ +α(k, τ, τ0) τ0δN α +b,k(τ0) . +(46) +Note that in general different flavours can couple to each other via the response function. +However, for vanishing densities, there is no coupling between the response for different +quark flavors and all flavors will have the same response functions, such that the response +matrix is proportional to the identity in flavor space, i.e. ( ˜Fab)µ +α = ˜F µ +α δab. +We will decompose the charge Green’s functions also in a scalar-vector-tensor basis such +that we have +˜F τ +τ (k, τ) = ˜F s +s (κ, x) . +(47) +It is possible to express the response functions in terms of δN µ +a,k. Adapting the normalisation +τ0δN τ +k(τ0) = 1 +(48) +we find +˜F s +s (κ, x) = τδN τ +κ(x) = τδnκ(x) . +(49) +Note that we also drop the index a on the components δN i +k as they will be the same for all +species for vanishing background number charge densities. +3. +Numerical Results for the non-equilibrium Green’s Functions +We present the results for the response functions ˜Gs +s and ˜F s +s in Fig. 1, where we plotted +the different response functions in dependence of the propagation phase κ and time ˜w (Eq. +(24)). The different panels correspond to the different response functions and are labelled +by the components of the energy-momentum tensor and the charge current that they affect, +e.g. ˜Gs +s is labelled by “energy response“ as this function describes the response of δT ττ +k (τ), +which corresponds to the energy. Each curve in each panel corresponds to the response +function at a different time as it is indicated by the colour code. Besides this we also plotted +the free-streaming behaviour for each response function, which corresponds to the black line +at ˜w = 0. +18 + +-1.5 +-1 +-0.5 + 0 + 0.5 + 1 + 1.5 + 0 + 5 + 10 + 15 + 20 +Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] +Energy response: G~ +s +s(w~,k∆τ) +Wave number: κ=k∆τ +free streaming +0 +1 +2 +3 +4 +5 +Figure 1: Left: Evolution of the energy-momentum Green’s function ˜Gs +s in response to initial +energy perturbations. Right: Evolution of the charge Green’s function ˜F s +s in response to initial +charge perturbations. The different curves in each panel correspond to different times ˜w. +Due to the fact that we consider the perturbations for the charge density and chemical +potential around vanishing background densities, the evolution of the response function ˜Gs +s +does not change in comparison to the results in [43]. This can already be seen at the level of +the equation of motion in Eq. (B24a) for vanishing densities. For early times ( ˜w ≪ 1) one +observes the free-streaming behaviour, characterized by wave-like modes with both peaks +of excess density and troughs of depleted density (the diffusion wake). Towards later times +larger κ-modes become damped, which can be explained by viscous effects of the medium. +At the onset of the hydrodynamic regime ( ˜w ∼ 1) only long wave-length modes survive, +which indicates that the free-streaming initial conditions are getting washed out during +the evolution of the system. The shift of the peak for later times towards larger values of +the propagation phase can be understood by noting that at early times, shortly after the +collision, the system is highly anisotropic and expands in the transverse plane with a phase- +velocity close to the speed of light. As the system evolves in time, it will become more and +more isotropic and the phase-velocity will approach the speed of sound resulting in the shift +of the peak. +For the charge density response ˜F s +s we see that for early times ( ˜w ≪ 1) the behaviour is +similar to the one of ˜Gs +s. However for later times the damping of the modes sets in earlier +than for ˜Gs +s, such that for κ ≳ 10 there are already no visible deviations from zero any +more. Due to the damping of these functions we see that at ˜w ∼ 1, when the hydrodynamic +19 + +freestreaming +5 +1 +Evolution time: W=TTefr(t)[4nTef/(e+P)] +4 +0.5 +3 +0 +2 +0.5 +1 +-1 +0 +0 +5 +10 +15 +20 +Wavenumber:K=k△tFigure 2: Left: Evolution of the energy Green’s function ˜Gs +s in response to initial energy perturba- +tions in coordinate space. Right: Evolution of the charge Green’s function ˜F s +s in response to initial +charge perturbations in coordinate space. The different curves in each panel correspond to different +times ˜w. +regime sets in, again only long wave-length modes will survive. Moreover, the absence of +oscillations in the spectrum signals the transition from propagating to diffusive behavior of +the charge perturbations, as will become evident in coordinate space. +We note that these results are obtained for perturbations around zero density. Therefore +further studies are necessary to clarify the impact of perturbations around non-vanishing +densities. +In particular, considering non-vanishing densities will remove the degeneracy +between the flavours leading to interesting phenomena like cross-diffusion [51–53]. +E. +Green’s Functions in Coordinate Space +So far we computed the Green’s functions in Fourier space, which provides useful insight +into the underlying physics and dynamics of such far-from-equilibrium systems. However, +the Green’s functions in position space will provide additional and useful information to +understand the system’s evolution. Similar to the decomposition in Fourier space, we can +decompose the Green’s functions in coordinate space as well into a basis of scalars, vectors, +and tensors, such that for the two Green’s functions we find +Gττ +ττ(r, τ) = Gs +s(|r|, τ) , +(50a) +F τ +τ (r, τ) = F s +s (|r|, τ) . +(50b) +20 + +2 +freestreaming +5 +1.5 +Evolution time: w=T(t)t / (4πn/s) +4 +1 +0.5 +3 +0 +2 +0.5 +1 +-1 +-1.5 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Distance:Ax/△t3.5 +free streaming +5 +3 +2.5 +4 +2 +3 +1.5 +2 +0.5 +0 +1 +-0.5 +-1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Distance:Ax/△tfor the energy-momentum tensor and charge current, respectively. The relation to their +counterparts in Fourier space is given by the following Fourier-Hankel transforms +Gs +s(|r|, τ) = 1 +2π +� +d|k| |k|J0(|k||r|) ˜Gs +s(|k|, τ) , +(51a) +F s +s (|r|, τ) = 1 +2π +� +d|k| |k|J0(|k||r|) ˜F s +s (|k|, τ) , +(51b) +where Jν are the Bessel functions of the first kind. +The results for the Green’s functions in coordinate space are presented in Fig. 2. The +corresponding Green’s function is plotted as a function of ∆x/∆τ, i.e. the propagation +distance in units of the elapsed time, while the color coding indicates the evolution time ˜w. +In the evolution of Gs +s the propagation of sound waves is clearly visible. In the free streaming +evolution and also still at early times the waves propagate with (almost) the speed of light. +Towards later times, the peak shifts to smaller values of ∆x/∆τ approaching the speed of +sound cs = +� +1/3 and exhibits a negative contribution at small ∆x/∆τ which corresponds +to the diffusion wake. +Since at early times the net charge density is carried by free-streaming particles, the charge +response F s +s has the same free-streaming behavior. One observes, that this behavior of the +charge Green’s function F s +s persists up to ˜w ∼ 0.5. +Subsequently, the Green’s function +transitions to a different behavior, where one can clearly see the diffusion of charges that +results in a pronounced peak centered around ∆x/∆τ = 0, and no longer the free propagation +of charges. +III. +APPLICATIONS +Now that we have obtained both the energy-momentum and charge dependent Green’s +functions, we can couple them to ICCING initial conditions. +The upgraded version of +ICCING 2.0 will be available on GitHub2 upon publication. We will study the impacts of +combining linearized pre-equilibrium Green’s functions with initial geometries produced by +ICCING. As we will show, the assumption of linear response leads to nontrivial effects on +the resulting energy and charge perturbations. +2 https://github.com/pcarzon/ICCING +21 + +A. +Using Green’s Functions in ICCING +The starting point of the construction of an initial state profile of the energy-momentum +tensor and the conserved charges, is the generation of an initial energy density profile based +on the initial-state model TRENTO.3 Then, this energy density at τ0 is used to compute +the background energy density at any proper times τ > τ0. To describe the fluctuations of +conserved charges about this background, we use ICCING [32, 33]. The ICCING algorithm is +a Monte Carlo event generator run subsequent to TRENTO which simulates the fluctuations +due to g → q¯q pair production. In particular, ICCING generates 2 + 1D distributions of +the fluctuating BSQ charge densities: baryon number, strangeness, and electric charge. By +incorporating the Green’s functions into the energy and charge redistribution algorithm +in ICCING, we can study the impact of perturbations in both energy and charge on the +evolution. +A single gluon splitting produces two types of perturbations relative to the background. +The first is a negative energy perturbation (“hole”) due to removing a gluon from the back- +ground. The second is a positive energy perturbation, displaced relative to the gluon, which +deposits the energy density corresponding to the quark/anti-quark pair. All three (quark, +antiquark, and gluon) can be treated as perturbation around the background. The propa- +gation of the perturbations generated by ICCING are treated via the Green’s function Gs +s +until some time τhydro when hydrodynamics becomes applicable. +In addition to the perturbations in the energy densities, the charge densities of quarks +created by the gluon splitting process can also be evolved by the same formalism. Since +TRENTO does not provide any charge information, we can treat the quark charges as +perturbations around a vanishing background charge density, which is exactly how the charge +Green’s functions are constructed. First we evolve the background according to the energy +attractor E( ˜w) using Eq. (27), then we can use the Green’s functions Gs +s and F s +s in order +to describe the propagation of energy and respectively charge perturbations that occur +whenever a splitting happens. +3 In practice, the output of TRENTO is a “reduced thickness function” which is taken to be proportional to +the entropy density. The coefficient of proportionality is fixed by comparison to experimental multiplicities +in central collisions, and the properly-normlaized entropy density is converted to an energy density using +the lattice QCD based equation of state from Refs. [54, 55]. +22 + +We can compactly express this as +e(τhydro, x) = eBG(eTrento(τ0), x) +� +1 + +� +⊙ +d2x0 +(∆τ)2(∆τ)2Gs +s +�|x − x0| +∆τ +, ˜w +� +1 +eTrento(τ0, x0) +× (δeq(τ0, x0) + δeq(τ0, x0) − δeg(τ0, x0)) +� +, +(52a) +ni(τhydro, x) = +� +⊙ +d2x0 +(∆τ)2(∆τ)2F s +s +�|x − x0| +∆τ +, ˜w +�τ0 +τ [δnq(τ0, x0) − δnq(τ0, x0)] . +(52b) +where we integrate the Green’s function evolution over all x0 in the past causal light cone +|x0 − x| < ∆τ. +Furthermore x is the point of interest and ∆τ ≡ τhydro − τ0. We have implemented the +pre-equilibrium evolution given in Eq. (52) in a new C++ class GreensFunctions.h which +interacts with the Event class Event.h in nontrivial ways and can be found at the GitHub +link above upon publication. +The energy and BSQ charge densities of a single, peripheral ICCING PbPb event at +√sNN = 5.02 TeV evolved for 1 fm/c using the Green’s functions are shown in Fig. 3 for +our default parameter set (see Table 2 in [56], with the exception of τ0 which here equals +0.1 fm). +The behaviour of the baryon and electric charge distributions are similar to default IC- +CING [56] and follow the bulk geometry while the strangeness distribution is more rarefied +due to the larger mass threshold required to produce strange quarks. A significant difference +is seen in the size of the charge fluctuations, whose radius now depends on the evolution +time. In order to understand the effects of applying the Green’s functions, we can look at +individual quark splittings, the background evolution, and radius dependence separately. +To start, it is important to have a grasp on the full effect the evolution has on a single +quark/anti-quark pair. The energy density and charge density for a quark splitting evolved +for 0.2 fm/c and 1 fm/c are shown in Fig. 4. The top panel of Fig. 4 occurs soon after +the quark splits and the bottom panel is at the end of the evolution. For small evolution +times, we clearly see three different types of perturbations in the top row of Fig. 4: (mostly) +positive-energy perturbations corresponding to the deposition of the q¯q pair, and a (mostly) +negative-energy perturbation corresponding to the subtraction of the parent gluon from the +background. The quarks also come with associated positive / negative charge densities being +deposited, whereas the gluon subtraction has no impact on the charge densities. +The dominant effect seen in Fig. 4 is that the energy and charge perturbations grow in size +23 + +Figure 3: Density distributions for an ICCING event with Green’s function evolution of energy and +charge perturbations from g → q¯q splittings after 1fm/c of evolution. +over time and have a wave-like structure leading to non-trivial interference. Note that the +central positions of the quarks do not change due to the evolution prescribed by the Green’s +functions; rather, they are determined from the g → q¯q splitting function used by ICCING. +The Green’s functions do not interact with any part of the quark sampling algorithm in +ICCING and only determine how the energy and charge densities of the perturbations are +distributed. +Next, we can look at how the Green’s functions distribute the energy and charge for +the quarks. To illustrate the spatial profiles of energy and charge density produced by the +Green’s functions, Fig. 5 shows the results of a single g → q¯q splitting in a low-temperature +24 + +-12 +-5 +0 +5 +12-12 +-5 +0 +5 +12 +-12 +-12 +Energy Density (GeV I fm3) +Baryon Density (fm-3) +-5 +5 +(wy) +0 +5 +0 +4 +8 +13 +17 +21 +-0.040-0.027 +-0.014 +0 +0.015 +0.031 +0.045 +12 +12 +-12 +-12 +Strangeness Density (fm-3) +Charge Density (fm-3) +-5 +-5 +(wy) +0 +5 +5 +-0.070 +-0.047 +-0.024 +0 +0.022 +0.044 +0.066 +-0.069 +-0.047 +-0.023 +0 +0.018 +0.036 +0.053 +12 +12 +5 +Y +-5 +-12 +-5 +0 +12-12 +5 +0 +12 +x (fm) +x (fm)Figure 4: Density distributions for a single strange quark splitting compared for evolution times of +τhydro = 0.2fm, on the left, and τhydro = 1fm, on the right. +region (top) and a high-temperature region (bottom). The Green’s functions have different +behavior depending on the local value of the initial energy density e(τ0). This dependence +arises because the natural unit of time ˜w depends on the effective temperature T (see +Eq. (24)). As a result, splittings which occur in hot spots transition more quickly from +propagating behavior for small ˜w to diffusive behavior for large ˜w. This is clearly seen in the +charge density plots of Fig. 5. The top panel, where the splitting occurs at low temperatures, +retains significant spatial structure of the charge distribution associated with the propagating +modes of the Green’s function. But if the splitting occurs at higher temperatures (bottom +panel), the charge density is distributed according to the diffusive modes from Fig. 1. This +results in charge distributions which wash out the spatial structure of the Green’s functions +as seen in Fig. 5. +There is also a connection here to the Knudsen number (Kn) since Kn = τR/τ ∝ (τT)−1 +so ˜ω ∝ Kn−1 [43] such that at late times one expects a smaller Kn number. This provides +physical intuition for the dependence of the charge fluctuation on the location of splitting, +25 + +Energy Density (GeV/fm3 +Charge Density (fm-3) +(wy) +y +-2 +-0.06 +-0.04 +-0.02 +0 +0.02 +0.04 +-0.10 +-0.05 +0 +0.05 +0.10 +-3 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +2 +3 +x (fm) +x (fm)Energy Density (GeVifm3 +Charge Density (fm-3) +1 +0 +(wy) +y +2 +-0.015-0.010-0.005 +0 +0.0050.010 +-0.02 +-0.01 +0 +0.01 +0.02 +-3 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +1 +2 +3 +x (fm) +x (fm)Figure 5: Density distributions for a single strange quark splitting from different areas of the event: +a cold region on the top, and a hot region on the bottom. Here the separation of the two quarks is +artificially increased to better illustrate the behaviour of the Green’s functions. +hotter spots in the medium will have larger Kn and thus produce more Gaussian charge +densities while cold spots will have smaller Kn and produce charge densities in a shock wave +form. The implications of this difference in behavior based on the location of splitting may +become more important when analyzing events across systems of different energy. +While this result is interesting and an effect of the physics included in the Green’s func- +tions, it could be worrying since one of the core assumptions of the ICCING algorithm is +that all charge must be correlated with some energy. The problem with a linearized treat- +ment of the Green’s functions is that large negative corrections to the energy density could +overturn the background energy density, resulting in grid points with net negative energy. +This problem would be nonphysical and require some sort of remedy. Moreover, even if the +net energy density is not driven negative by a large perturbation, one may still be unable to +match a fluid cell with very low energy density but very high charge density to a reasonable +equation of state. +These potential problems arising from large perturbations could be solved by going back to +26 + +Energy Density (GeVifm3) +Charge Density (fm-3 +0 +(wy) +-0.015-0.010-0.005 +0 +0.005 +0.010 +-0.02 +-0.01 +0 +0.01 +0.02 +-2 +0 +0 +4 +2 +x (fm) +x (fm)Energy Density (GeV/fm3) +Charge Density (fm- +(fm) +-0.75 -0.50 -0.25 +0.25 +0.500.75 +-0.10 +-0.05 +0 +0.05 +0.10 +0 +2 +0 +2 +4 +x (fm) +x (fm)Figure 6: Comparison of εn{2} across energy and BSQ distributions for different Green’s function +evolution times. +the linearized approximation made in Sec. II C, which is broken when the local redistributed +energy or charge density is greater than or close to the background. There are several ways in +which to solve this problem, the first of which would be by artificially damping the magnitude +of the perturbations relative to the background in order ensure that the linearization remains +valid. This would introduce a new problem though, since the artificial damping may not +affect a q¯q pair equally. If the quark is deposited in the periphery of the event, but the +antiquark is deposited closer to the center, then the positive and negative charge densities +will be damped in different amounts, leading to a violation of charge conservation. To correct +this, one could suppress the quark and anti-quark in the same way mirroring the effect. +Another possible solution – the one we pursue here – is to veto any quark splittings that +would create energy density perturbations that are large with respect to the background. +This is a very simple solution but adds a new complication because it effectively eliminates +quark production at the edge of the event, and reducing the "cold quark" Green’s functions +contribution. Another unintended effect of this solution would be that a quark/anti-quark +pair produced near the periphery with a smaller radius, for example from evolving for only +0.5 fm/c, would survive the veto, but a pair evolved for longer time would be rejected. +Evidently, this problem could be cured by also including the transverse expansion of the +background energy density in the pre-equilibrium stage, but that is beyond the scope of +this paper and is left for future work. Despite its shortcomings, the solution of vetoing +quark splittings if the energy perturbation is not small compared to the background has +been chosen here both for its simplicity and its flexibility. +27 + +1.0 +- Trento +: Background* +- ICCING (Default) +0.8 +: - ICCING (Green's Functions) +0.6 +(z)23 +0.4 +0.2 +PbPb [5TeV1, *△t = 1.0 fm +0.0 +20 +40 +60 +80 +100 +0 +Centrality (%)1.0 +- Trento +: Background* +- ICCING (Default) +0.8 +- ICCING (Green's Functions) +0.6 +E3(2) +0.4 +0.2 +PbPb [5TeVl, *△t = 1.0 fm +0.0 +20 +40 +60 +80 +100 +0 +Centrality (%)Because our procedure should be only a small effect on the total energy density, we do +not expect large changes to the energy density eccentricities, which are defined in App. E. +Thus, before looking at the eccentricities of the charge densities, we should look at the effect +that the different processes have on the energy density with the hope that any affect is +minimal. Since the energy density eccentricities are a good predictor of the final state and +these initial conditions have been used extensively in comparisons to experimental data, the +hope is for a minimal effect. In Fig. 6, the energy ellipticity and triangularity is plotted +for the original trento event, the locally evolved background used for the Green’s function +evolution, the trento event after default ICCING, and the full ICCING coupled to Green’s +functions simulation. When comparing the evolved background with the trento profile, we +observe small changes at the percent level which can be attributed to the phenomenon of +inhomogenous longitudinal cooling [36, 38]. In essence, thermalization proceeds more quickly +in more highly energetic regions, leading to a slightly faster decrease of the energy density +of the hotter regions of the QGP as compared to the colder regions of the QGP. However, +as we see in Fig. 6, in practice this effect is rather small. Similarly, we see that for default +ICCING, there is a slight modification in peripheral events and the most central events that +should not have a significant effect on the agreement with experimental data. Adding both +the modifications from the ICCING sampling and the Green’s functions evolution, has no +significant effect on the energy geometry beyond the background evolution. This indicates +that any small changes in energy density distribution generated by ICCING are quickly +washed out and won’t make it to the final state. Additionally, previous comparisons to +experimental data for all charge particles should still be valid. +IV. +IMPACT ON ECCENTRICITIES +Now let us look at the contributions, that different parts of the Green’s function evolution +have on the event averaged eccentricities. Because we will be dealing with time dependent +quantities, we will define the time evolution for applying the Green’s function: +∆τ ≡ τhydro − τ0 +(53) +where τ0 is our initial time when we begin the Green’s function evolution and τhydro is +where we stop the evolution and switch to hydrodynamics. We will start with studying the +28 + +consequences of our perturbative cutoff effect on the eccentricities, then we will compare +the Green’s function expansion to a trivial Gaussian smearing to determine any non-trivial +effects. After these two effects are studied we will explore the time dependence of the Green’s +function on various eccentricities, which are the main results for this work. +Eccentricities of charge are defined the same as for energy, see App. E, except that the +center of mass is taken to be that of the energy density and the observable is calculated for +the positive and negative charge densities separately, since otherwise the observable would +be zero. When there is no charge density from quark/anti-quark splittings, the eccentricity +is defined as zero. While adequate, the definition of the eccentricities for energy are not the +best possible estimators for the the charge density and more development can be made in +this direction [56]. +First, we study the effect of the suppression of gluon splitting to ensure positive energy +densities. In order to avoid problematic regions with negative energy density, we restrict +quarks from splitting if their redistributed energy densities approach a certain threshold +compared to the background. The selection criteria is examined for each point in the quark +densities and is determined by +Eq/Ebg < P, +(54a) +where P is the perturbative cutoff. In Fig. 7, several values were selected for an evolution +time of ∆τ = 1fm/c to illustrate the effect this cutoff has on the charge densities. Both +ε2 {2} and ε3 {2} are shown. The effect of applying the perturbative cutoff vs no cutoff +at all is clearly the dominate effect. This signifies that there are many quark/anti-quark +pairs above 40% Centrality that produce negative energy and thus the mismatch between +the locally evolved background and non-local quark perturbations is quite significant. For +P = 0.9 and evolution of 1fm/c, the percentage of events that produce no quarks at all in +the 65 − 75% Centrality class is 98.9%. The peak of the charge eccentricities thus signifies +that number effects dominate the charge geometry. With such an extreme response to this +perturbation parameter it is reasonable to assume the model, as it is currently formulated, +breaks down at this point and should not be used beyond an evolution time of 1fm/c. +However, we do not find a significant difference between the most generous value of a 0.9 +cutoff vs the 0.5 mark with only small change when P goes to 0.1. In the remaining results +we will explore only the 0.9 cutoff since we do not anticipate a strong dependence on the +29 + +Figure 7: Comparison of εn{2} for different perturbation cutoff values with a Green’s function +evolution of ∆τ = 1.0fm/c. +The solid line is with a Green’s function but no cutoff, default +ICCING is not shown. +cutoff for other observables as well. +Next, we will try to disentangle the effect of the expanding radius from the structure +introduced to the quark densities based on the background energy. In our Green’s function +approach, the overall size of the quarks expand over time and that may be the dominate +(albeit trivial) effect of applying the Green’s function. Thus, to determine any non-trivial +consequences of the Green’s function, we apply a simple Gaussian smearing to the quarks, +as illustrated in Fig. 8, and compare the Gaussian smearing, defined as: +Gs +s(r, t) = F s +s (r, t) = exp(−r2/R(t)2) +πR2(t) +, +(55) +to the Green’s function method. In Fig. 9, we see that there is a negligible difference between +the Green’s function and a simple Gaussian smearing, implying that the dominant effect, +when looking at event averaged geometry, is the size of the density perturbations and not +the structure introduced by the Green’s Functions. +In Fig. 10, we show ε2{2} adding back in the perturbation cutoff and evolving for +∆τ = 1.0fm/c. +The solid curves are from default ICCING without any pre-evolution, +the dashed curves add in evolution but only allow the radius of the quark and gluon density +perturbations to change while holding the density profile fixed, and the dotted curves add +in the full Green’s functions. Several things are happening in Fig. 10 that need to be dis- +entangled. First, the Gaussian smearing with the peturbative cut-off (compared to default +ICCING with no time evolution) has the general effect of shifting the peak in ε2 {2} to +lower centralities and also leading to a larger ε2 {2} in central collisions. The shift from the +30 + +1.0 +-Energy +No Correction +Baryon (+) +Eq/EBg <0.9 +Strange (+) +... EqEBg < 0.5 +0.8 +-Charge (+) +- Eq/EBg < 0.1 +0.6 +(z) +0.4 +0.2 +PbPb [5TeVl, △t = 1.0 fm +0.0 +0 +20 +40 +60 +80 +100 +Centrality (%)1.0 +-Energy + No Correction +Baryon (+) + E/EBg <0.9 +Strange (+) +0.8 +-Charge (+) +- Eq/EBg < 0.1 +0.6 +E3(2] +0.4 +0.2 +PbPb [5TeV1, △t = 1.0 fm +0.0 +20 +40 +60 +0 +80 +100 +Centrality (%)Figure 8: Illustrative density profiles of the Gaussian smearing option at ∆τ = 1.0fm/c which +separates structure introduced by the Green’s functions from the radial dependence. +Figure 9: Comparison of ε2{2} between default ICCING, Gaussian Smearing, and Green’s Functions +with an evolution of ∆τ = 1.0fm/c. The perturbation cutoff is not used here. +Gaussian smearing compared to default ICCING occurs both because as the quarks grow in +size the positive and negative densities will cancel out more and wash out the geometry in +regions with low densities (i.e. peripheral collisions) and because quark/anti-quark splitting +is suppressed due to the perturbation parameter. +Now that we know the effect of just a trivial Gaussian smearing, the next question is what +31 + +Energy Density (GeV/fm3 +Charge Density (fm-3 +2 F +1 +(wy) +0 +-0.03 -0.02 +-0.01 +0 +0.01 +0.02 +-0.03 -0.02 -0.01 +0 +0.01 +0.02 +0.03 +-2 E +-3 +-2 +-1 +0 +1 +2 +3 +4 -3 +-2 +-1 +0 +1 +2 +3 +4 +x (fm) +x (fm)1.0 +-Energy +ICCING (Default) +-Baryon (+) + ICCING (Gaussian Smearing) +-Strange (+) +.... ICCING (Green's Functions) +0.8 +Charge (+) +0.6 +(z) +0.4 +0.2 +PbPb [5TeVl, △t = 1.0 fm +0.0 +20 +40 +60 +80 +100 +0 +Centrality (%)Figure 10: Including perturbation cutoff of 0.9 for comparison between Default ICCING, Gaussian +smearing, and Green’s functions for ∆τ = 1.0fm/c. +effect does the non-trivial Green’s function have? In Fig. 10 we can see that the Green’s +function shifts the peak of the eccentricities even further to lower centralities for all BSQ +densities. To understand this effect, let us break down the fundamental differences between +a trivial Gaussian smearing and the Green’s functions. There are two differences between +the Gaussian smearing and Green’s functions density perturbations one of which is that the +density profile of the Gaussian smearing is smooth and mostly uniform with sharp edges +and only negative values of energy coming from the gluon hole, as shown in Fig. 8. The +Green’s functions, on the other hand, have a density profile that has a wave structure with +the largest energy density values coming from the ring at the edge of the quarks and gluons, +as previously shown in Fig. 4. The Green’s function density profiles also contain negative +energy at the center of the quarks and a large amount around the edge of the gluon. Applying +the perturbative cutoff removes any net negative energy from the final output in these two +methods, which strongly affects Green’s function method because of the concentration of +the energy density around the edge of the quarks. While the Gaussian smearing also breaks +perturbative assumptions the effect is much smaller than for the Green’s function. The +32 + +1.0 +-Energy +ICCING (Default) +-Baryon (+) + ICCING (Gaussian Smearing) +-Strange (+) +.... ICCING (Green's Functions) +0.8 +Charge (+) +0.6 +(z) +0.4 +0.2 +PbPb [5TeV] +△T = 1.0 fm, Eq/EBg < 0.9 +0.0 +20 +40 +60 +80 +100 +0 +Centrality (%)structure of the Green’s function density perturbations is relatively ’microscopic’ and so +when compared against the Gaussian smearing, without the perturbation cutoff in Fig. 9, +there is no difference. Since the perturbation cutoff is defined here as ’microscopic’, then a +difference is seen in Fig. 10 when including the more complicated structure of the Green’s +functions. +The sensitivity to ’microscopic’ differences in the density perturbations may +disappear with different choices of the perturbation cutoff method. However, the unique +structure of the Green’s function density perturbations will still be important when coupling +to hydrodynamics since there would be a non-trivial change to gradients. +Finally putting all the pieces together, Fig. 11 shows the Green’s functions evolution with +the perturbative cutoff for different evolution times, ∆τ = 0.5 fm and ∆τ = 1 fm, for both +elliptical (left) and triangular eccentricities (right). For an evolution time of ∆τ = 0.5fm/c, +we consistently see a shift in the peak of all BSQ charge eccentricities toward the left, +reflecting an increase in the dominance of number effects on the geometry as supported by +the rarity of quark producing events. However, for ∆τ = 0.5fm/c most central collisions +do not appear to be strongly affected by the expansion. For an evolution of 1.0fm/c, there +is a much greater suppression from the perturbation correction and this significantly affects +all centrality classes, such that the most central collisions see enhanced eccentricities but +peripheral collisions are suppressed. The shift in the peak towards smaller centrality classes +(combined with a suppressed eccentricity in peripheral collisions) indicates that the model +starts to break down the further in time the evolution is pushed. Thus, there appears to +be a small window in which we can apply the Green’s function expansion and still obtain +a reasonable number of quark/anti-quark pairs (after applying the perturbative cutoffs). +Generally, we find very similar qualitative behaviors in both ε2 and ε3. However, ε3 is much +more sensitive to BSQ densities and has the most significant difference between the energy +eccentricities vs the BSQ eccentricities. Therefore, high-order harmonics will likely provide +the best observable when comparison to experimental data. +One of the most important quantities for direct comparisons of initial state models to +experimental data is εn{4}/εn{2} because medium effects cancel in the most central collisions +(especially for n = 3 [56]). The eccentricity ratios, εn{4}/εn{2}, are shown in Fig. 12 for +n = 2 (left) and n = 3 (right). The ratio εn{4}/εn{2} measure the fluctuations of geometry +with values close to 1 indicating few fluctuations wheres small values indicate a large amount +of fluctuations. Comparing elliptical and triangular flow, we find quite different results. For +33 + +Figure 11: Comparison of εn{2} across energy and BSQ distributions for different Green’s function +evolution times using the perturbation cutoff. +elliptical flow, for more centrality to mid-central collisions the fluctuations appear to be +nearly identical to the energy density fluctuations (although ultra central collisions have +some small differences). However, for peripheral collisions where the perturbative cutoff +plays a strong role, then we see there is always a centrality wherein large deviations are seen +compared to the energy density distribution. Electric charge and baryon density fluctuations, +for default ICCING, are nearly identical to the energy density fluctuations. However, the +longer you have a Green’s function evolution, then you see deviations at lower and lower +centralities (i.e. for ∆τ = 1 fm, the deviation occurs at ∼ 40% centrality). Naturally for +strangeness this effect is larger because one is dealing with a smaller number of quark/anti- +quark pairs. +The effect for triangular flow is quite different. Generally, we find that the application +of ICCING leads to an overall decrease in the triangular flow fluctuations, regardless of the +BSQ charge. Additionally, the Green’s function evolution appears to enhance that effect +further for ∆τ. In contrast, for elliptical flow we did not see this effect and the fluctuations +were the same (at least within some centrality classes) before and after applying ICCING. +That being said, we do find that the effect of certain centralities being strongly affected by +the perturbative cutoff showing up in triangular flow as well. These are features that could +eventually be looked for in experimental data, if measurements of vn{4}/vn{2} are made +with identified particles. +34 + +1.0 +-Energy +- Default +-Baryon (+) + At = 0.5 fm +Strange (+) +... At = 1.0 fm +0.8 +Charge (+) +0.6 +(z) +0.4 +0.2 +PbPb [5TeV], Eq/EBg < 0.9 +0.0 +20 +40 +60 +80 +100 +0 +Centrality (%)1.0 +-Energy +- Default +Baryon (+) +- At = 0.5 fm +Strange (+) +...- At = 1.0 fm +0.8 +Charge (+) +0.6 +E3(2) +0.4 +0.2 +PbPb [5TeV], Eq/EBg < 0.9 +0.0 +20 +40 +60 +0 +80 +100 +Centrality (%)Figure 12: Comparison of εn{4}/εn{2} across energy and BSQ distributions for different Green’s +function evolution times using the perturbation cutoff. +V. +CONCLUSION & OUTLOOK +We extended the method to compute non-equilibrium Green’s functions of the energy- +momentum tensor developed in [43] by adding conserved charges and computing the corre- +sponding Green’s functions for the charge current for perturbations around vanishing back- +ground charge densities. Using the ICCING model that initializes conserved charges through +g → q¯q splittings, we successfully coupled these Green’s function to ICCING allowing for a +pre-equilibrium phase with conserved charges. The inclusion of this pre-equilibrium evolu- +tion is a non-trivial addition to the ICCING algorithm and the successful implementation +demonstrates the flexibility of the algorithm. +In order to quantify the system’s response to initial perturbations in terms of Green’s +functions, it is necessary to consider the background evolution (see App. A 4) which is used +in the energy evolution of the system. We find that the systems dynamics can be quantified +in terms of the moments δE m +l,k, δN m +al,k (Eq. (37)). Furthermore the Green’s functions can be +obtained directly from these moments, which makes this method a powerful tool to obtain +the response functions. When comparing the energy and charge Green’s function we see +distinct differences in their behaviors. In the energy case we find the propagation of sound +waves, where in free-streaming and even at early times they propagate with almost the +speed of light, while at later times this shifts towards the speed of sound. In contrast, for +the evolution of conserved charges we find a transition from free-streaming propagation in +the beginning to a diffusive behavior at late time as the system continues to thermalize. +To understand the effect the Green’s functions have on initial state charge geometries, we +35 + +1.0 +PbPb @ 5TeV, EglEBg < 0.9 +0.8 +E2(4)/e2(2] +0.6 +-Energy +Baryon (+ +0.4 +Strange (+ + Charge (+) +0.2 + Default +△t = 0.5 fm +△t = 1.0 fm +0.0 +0 +20 +40 +60 +80 +100 +Centrality (%)1.0 +PbPb @ 5TeV, EqlEBg < 0.9 +0.8 +E3(4)/E3(2) +0.6 +Energy +Baryon (+) +0.4 +Strange (+) +Charge (+) +0.2 +Default +At = 0.5 fm +△t = 1.0 fm +0.0 +0 +20 +40 +60 +80 +100 +Centrality (%)compare between the default version of ICCING and ICCING with the Green’s functions, +supplemented by an approximation which simplifies the spatial structure of the Green’s +functions to simple Gaussian smearing. We see that for εn{2} there is no difference when +including the complicated structure of the Green’s functions to the Gaussian smearing, +although that structure becomes important for observables sensitive to ’microscopic’ differ- +ences. A mismatch between the evolution of the background, described as a local process, +and the charge perturbations, described as a non-local process, leads to the possibility of +sites with negative energy. This issue is fixed by suppressing quark/anti-quark production +that would violate some perturbative condition. This pertubative corrective measure signif- +icantly suppresses quark/anti-quark production in peripheral events but less so in central to +mid-central. The primary difference then between default ICCING and ICCING with the +Green’s functions arises from a combination of smearing effects that occur during an ex- +pansion in time and the suppression of non-perturbative quark-anti-quark pairs. This leads +to large eccentricities in central collisions but nearly vanishing eccentricities in peripheral +collisions. +This work constitutes the first step toward including charge evolution in KøMPøST and +illustrates the effect pre-equilibrium evolution has on conserved charge densities. An imple- +mentation of this method in KøMPøST would solve the mismatch between the background +and perturbation evolutions which are local and non-local, respectively. Exploring the effect +of the pre-equilibrium evolution of conserved charges on the hydrodynamic evolution of the +system and on final state observables would be beyond the scope of this paper but is an +interesting open question that will be explored in a future work. It would also be interesting +to compute these Green’s functions for charges in QCD kinetic theory and compare them to +the approximation introduced in this paper. Another possible direction is extending these +Green’s functions around a non-vanishing background which would be useful when looking +at systems that contain baryon stopping. +Acknowledgements +P.P. and S.S acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, Ger- +man Research Foundation) through the CRC-TR 211 ’Strong-interaction matter under ex- +treme conditions’– project number 315477589 – TRR 211. J.N.H. and P.C. acknowledges +36 + +the support from the US-DOE Nuclear Science Grant No. DE-SC0020633 and the support +from the Illinois Campus Cluster, a computing resource that is operated by the Illinois Cam- +pus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing +Applications (NCSA), and which is supported by funds from the University of Illinois at +Urbana-Champaign. M.S. is supported by a start-up grant from New Mexico State Uni- +versity. The authors also acknowledge computing time provided by the Paderborn Center +for Parallel Computing (PC2) and the National Energy Research Scientific Computing Cen- +ter, a DOE Office of Science User Facility supported by the Office of Science of the U.S. +Department of Energy under Contract No. DE-AC02-05CH11231. +37 + +Appendix A: Background evolution +1. +Evolution Equations for the spherical harmonic moments +In order to solve Eq. (18), we will adopt the ideas of [48], where, instead of finding +solutions for the distribution functions, one studies the moments of the distribution function. +For the distribution functions fBG and fa,BG we will consider the following moments +E m +l (τ) = τ 1/3 +� +dpη +(2π) +� +d2p +(2π)2pτY m +l (φp, θp)fBG(τ, pT, |pη|) , +(A1a) +N m +al (τ) = +� +dpη +(2π) +� +d2p +(2π)2Y m +l (φp, θp)fa,BG(τ, pT, |pη|) . +(A1b) +In Eq. (A1) the angles are defined by tan φp = p1/p2 and cos θp = pη/(τpτ), while Y m +l +are +the spherical harmonics given by +Y m +l (φ, θ) = ym +l P m +l (cos θ)eimφ +(A2a) +with +ym +l = +� +(2l + 1)(l − m)! +4π(l + m)! +(A2b) +and +P m +l (x) = (−1)m +2ll! +� +1 − x2�m/2 dl+m +dxl+m +� +x2 − 1 +�l . +(A2c) +being the associated Legendre polynomials. +Based on the explicit form of the spherical harmonics one can find the non-vanishing +components of the background energy-momentum tensor by low order moments as well as +the tracelessness condition +e(τ) = +√ +4π +τ 4/3 E 0 +0(τ) , +(A3a) +PT(τ) = +√ +4π +τ 4/3 +� +1 +3E 0 +0(τ) − +� +1 +45E 0 +2(τ) +� +, +(A3b) +PL(τ) = +√ +4π +τ 4/3 +� +1 +3E 0 +0(τ) + +� +4 +45E 0 +2(τ) +� +. +(A3c) +Furthermore, by plugging in the equilibrium distribution function, we find that +E m +l +�� +eq(τ) = τ 4/3 +√ +4πe(τ)δl0δm0 , +(A4) +38 + +representing the rotational symmetry of the equilibrium. In the same way we are able to +reconstruct the components of the charge current via low-order moments. The net particle +number of specie a is given by +na(τ) = +√ +4π +τ +N 0 +a0(τ) . +(A5) +The chemical potential can be extracted by inverting the Landau conditions Eq. (4) and +Eq. (5). +Applying τ∂τ to the definitions of the moments and using identities for Legendre Poly- +nomials (see App. D) leads directly to the equations of motion, which are given as +τ∂τE m +l (τ) = bm +l,−2E m +l−2(τ) + bm +l,0E m +l (τ) + bm +l,+2E m +l+2(τ) − τ +τR +� +E m +l (τ) − E m +l (τ)|eq +� +, +(A6a) +τ∂τN m +al (τ) = Bm +l,−2N m +al−2(τ) + Bm +l,0N m +al (τ) + Bm +l,+2N m +al+2(τ) − τ +τR +� +N m +al (τ) − N m +al (τ)|eq +� +. +(A6b) +The appearing coefficients bm +l and Bm +l +are given by +bm +l,−2 = (2 + l)(l + m − 1)(l + m) +(1 − 4l2) +� +(2l + 1)(l − m − 1)(l − m) +(2l − 3)(l + m − 1)(l + m) , +(A7a) +bm +l,0 = −5 +3 +l(l + 1) − 3m2 +4l(l + 1) − 3 , +(A7b) +bm +l,+2 = (l − 1) +(2l + 3) +� +(l − m + 1)(l − m + 2)(l + m + 1)(l + m + 2) +(2l + 1)(2l + 5) +. +(A7c) +resp. +Bm +l,−2 = (1 + l)(l + m − 1)(l + m) +(1 − 4l2) +� +(2l + 1)(l − m − 1)(l − m) +(2l − 3)(l + m − 1)(l + m) , +(A8a) +Bm +l,0 = −l(l + 1) − 3m2 +4l(l + 1) − 3 , +(A8b) +Bm +l,+2 = +l +(2l + 3) +� +(l − m + 1)(l − m + 2)(l + m + 1)(l + m + 2) +(2l + 1)(2l + 5) +. +(A8c) +2. +Initial Conditions +In order to solve the equations of motion for E m +l +and N m +al +we need to specify the initial +conditions for the moments. For early time dynamics at τ ≪ τR the system cannot maintain +considerable longitudinal momenta. Therefore the initial distribution is naturally of the form +39 + +that the transverse momentum is much larger than the longitudinal one. Taking also into +account previous results [57–63] one sees that the case of a (longitudinal) support in form of a +Dirac delta function corresponds to a non-equilibrium attractor of the kinetic equations, i.e. +that different initial conditions will approach the same curve for later times. We therefore +choose +fBG(τ0, pT, |pη|) = (2π)3δ(pη) +� +1 +νg +dN0,g +dη d2p d2x + 1 +νq +� +a +� +dN0,qa +dη d2p d2x + +dN0,qa +dη d2p d2x +�� +≡ (2π)3δ(pη) +d ˜N0 +dη d2p d2x , +(A9a) +fa,BG(τ0, pT, |pη|) = (2π)3δ(pη) 1 +νq +� +dN0,qa +dη d2p d2x − +dN0,qa +dη d2p d2x +� +≡ (2π)3δ(pη) +d ˜N0,a +dη d2p d2x , +(A9b) +where we choose the normalization such that the initial energy and charge densities are kept +constant +dE0 +dη d2x = dE0,g +dη d2x + +� +a +� dE0,qa +dη d2x + dE0,qa +dη d2x +� += lim +τ0→0 τ0e(τ0) = (eτ)0 = const , +(A10a) +dN0,a +dη d2x = dN0,qa +dη d2x − dN0,qa +dη d2x = lim +τ0→0 τ0na(τ0) = (naτ)0 = const . +(A10b) +On the level of the moments the initial conditions are given by +E m +l (τ0) = τ 1/3 +0 +(eτ)0 ym +l P m +l (0)δm0 , +(A11a) +N m +al (τ0) = (naτ)0 ym +l P m +l (0)δm0 . +(A11b) +3. +Relation of Intensive and Extensive Quantities +In contrast to the cases in [43], we additionally need to invert +e = T 4 +�νgπ2 +30 − 3νq +π2 +� +a +� +Li4 +� +−z−1 +a +� ++ Li4(−za) +�� +, +(A12a) +na = νq +T 3 +π2 +� +Li3 +� +−z−1 +a +� +− Li3(−za) +� += νq +6π2 +� +π2T 2µa + µ3 +a +� +, +(A12b) +with za ≡ exp(µa/T) to determine the temperature T and the chemical potentials µa as a +function of energy density e and number density na. This is done numerically for each time +40 + +step. The relations are given by +δT = −T χuχdχsδe − 3nuχdχsδnu − 3ndχuχsδnd − 3nsχuχdδns +9n2 +uχdχs + 9n2 +dχuχs + 9n2 +sχuχd − 4eχuχdχs +, +(A13a) +δµu = +[9(n2 +dχs + n2 +sχd) + (3nuµu − 4e)χdχs]δnu +9n2 +uχdχs + 9n2 +dχuχs + 9n2 +sχuχd − 4eχuχdχs ++ +−3αundχsδnd − 3αunsχdδns + αuχdχsδe +9n2 +uχdχs + 9n2 +dχuχs + 9n2 +sχuχd − 4eχuχdχs +, +(A13b) +δµd = +[9n2 +sχu + (9n2 +u + (3ndµd − 4e)χu)χs]δnd +9n2 +uχdχs + 9n2 +dχuχs + 9n2 +sχuχd − 4eχuχdχs ++ +−3αdnuχsδnu − 3αdnsχuδns + αdχuχsδe +9n2 +uχdχs + 9n2 +dχuχs + 9n2 +sχuχd − 4eχuχdχs +, +(A13c) +δµs = +[9n2 +dχu + (9n2 +u + (3nsµs − 4e)χu)χd]δns +9n2 +uχdχs + 9n2 +dχuχs + 9n2 +sχuχd − 4eχuχdχs ++ +−3αsnuχdδnu − 3αsndχuδnd + αsχuχdδe +9n2 +uχdχs + 9n2 +dχuχs + 9n2 +sχuχd − 4eχuχdχs +. +(A13d) +Here χa is the susceptibility, which is given by +χa ≡ νq +6 +�3µ2 +a +π2 + T 2 +� +(A14) +For the special case of perturbations around na = 0 the susceptibilities reduce to χa = νq +6 T 2 +which yields then +δT = T δe +4e , +(A15a) +δµa = 6 +νq +δna +T 2 . +(A15b) +4. +Background Evolution in Conformal Systems +In this section we will consider the evolution of a conformal system. In conformal systems +τR is proportional to the inverse temperature such that [46] +τRT(τ) = 5 ˜ηT +e + P = const , +(A16) +where ˜η is the shear viscosity, e the energy density, P the pressure and T the effective +temperature of the system. +At this point we introduce the dimensionless time variable x = τ/τR, as this produces a +natural time scale for the evolution of the system. Due to the change of variable we need to +transform also the appearing derivatives according to +τ∂τ = τ ∂x +∂τ ∂x = τ +� 1 +τR +− τ +τ 2 +R +(∂ττR) +� +∂x = +� +1 − 1 +τR +τ∂ττR +� +x∂x ≡ a(x)x∂x , +(A17) +41 + +where we will call +a(x) ≡ 1 − x∂ττR = 1 − 1 +τR +τ∂ττR +(A18) +the scale factor. +As τR is not constant, the scale factor will take a complicated form. +However, it can be related to the moments again as we find +a(x) = 1 + τ∂τT +T += 1 − χuχdχs˜a(x)e + 3n2 +uχdχs + 3n2 +dχuχs + 3n2 +sχuχd +9n2 +uχdχs + 9n2 +dχuχs + 9n2 +sχuχd − 4eχuχdχs +, +(A19) +where +˜a(x) = −4 +3 + b0 +0,0E 0 +0 + b0 +0,+2 +E 0 +2(x) +E 0 +0(x) . +(A20) +The appearing quantities can be expressed in terms of the low order moments (see Eq. +(A3a), (A5)). We note at this point that for na = 0 the scale factor reduces to +a(x) = 1 + τ∂τT +T += 1 − χuχdχs˜a(x)e +−4eχuχdχs += 2 +3 + 1 +4 +� +b0 +0,0E 0 +0 + b0 +0,+2 +E 0 +2(x) +E 0 +0(x) +� +, +(A21) +which is the form used in [43]. +Using the change of variables the equations of motion can be written in terms of x as +a(x)x∂xE m +l = bm +l,−2E m +l−2 + bm +l,0E m +l + bm +l,+2E m +l+2 − x +� +E m +l − E m +l |eq +� +, +(A22a) +a(x)x∂xN m +al += Bm +l,−2N m +al−2 + Bm +l,0N m +al + Bm +l,+2N m +al+2 − x +� +N m +al − N m +al |eq +� +. +(A22b) +For our analysis we are varying the initial charge number densities in order to see its impact +on the evolution of the background. Nevertheless, we keep the ratios between the three +species the same, namely +nu,BG +nd,BG += 8 +7 , +ns,BG = 0.0 , +(A23) +as these ratios correspond to typical values in heavy-ion collisions. +Our results for a conformal system can be seen in Fig. 13. In conformal systems without +conserved charges it was found that the evolution is controlled by the dimensionless time +variable ˜w = τT(τ)/(4π˜η/s) [49]. We will generalise this to systems with conserved charges +and choose to present the different quantities as functions of the dimensionless time variable +˜w = τT(τ) +4π +(e + P) +˜ηT += 5 +4πx , +(A24) +42 + + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 0.01 + 0.1 + 1 + 10 +Chemical Potential/Temperature: µu/T +Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] +(µu/T)eq=0.02 +(µu/T)eq=0.11 +(µu/T)eq=0.24 +(µu/T)eq=0.37 +(µu/T)eq=0.56 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +Longitudinal Pressure/Energy: PL/e +Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] +(µu/T)eq=0.02 +(µu/T)eq=0.11 +(µu/T)eq=0.24 +(µu/T)eq=0.37 +(µu/T)eq=0.56 + 0.09 + 0.1 + 0.11 + 0.12 + 0.13 + 0.14 + 0.3 + 0.35 + 0.4 + 0.45 + 0.5 + 0.09 + 0.1 + 0.11 + 0.12 + 0.13 + 0.14 + 0.3 + 0.35 + 0.4 + 0.45 + 0.5 + 0 + 0.5 + 1 + 1.5 + 2 + 0.01 + 0.1 + 1 + 10 +Energy attractor: τ4/3 Tττ/(eτ4/3)∞ +Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] +(µu/T)eq=0.02 +(µu/T)eq=0.11 +(µu/T)eq=0.24 +(µu/T)eq=0.37 +(µu/T)eq=0.56 +Navier-Stokes -- 1-2/(3πw~) + 0.15 + 0.16 + 0.17 + 0.18 + 0.19 + 0.2 +0.02 + 0.01 + 0.15 + 0.16 + 0.17 + 0.18 + 0.19 + 0.2 +0.02 + 0.01 +Figure 13: Background evolution for conformal systems. +Top: µu/T ratio for different initial +values for nu, Bottom left: Longitudinal pressure over energy for different values of (µu/T)eq; The +coloured dashed curves in the PL/e plot correspond to the hydrodynamical behaviour at later times, +PL/e = 1 +3 − +4 +9π ˜w for ˜w → ∞, Bottom right: Energy attractor for different values of (µu/T)eq. The +coloured dashed curves in the +� +τ 4/3e +� +/ +� +τ 4/3e +� +∞ plot correspond to the free streaming behaviour +of the energy attractors at early times, +� +τ 4/3e +� +/ +� +τ 4/3e +� +∞ = +1 +C∞ ˜w4/9 for ˜w ≪ 1. More details on +how to fit the dashed curves in the two plots are given in the text. Inset plots are given in order to +show that there are deviations between the curves. On the two axes of the inset plots are the same +quantities plotted as for the larger plot, but the labels are omitted for better readability. +where T(τ) = +� +30 +νeffπ2e(τ) +�1/4 +and νeff = νg + 3 · 7 +4νq. +At the top of Fig. 13 we show the different curves we obtain for the ratio µu/T. We see +that at late times, when hydrodynamics is applicable, the ratio becomes constant according +43 + +to +�µa +T +� +eq = 6 +νq +na +T = 6 +νq +�νeffπ2 +30 +� 3 +4�na +e +3 +4 +� +eq += const . +(A25) +In the bottom left panel we show the ratio of longitudinal pressure and energy, PL/e. We +see that the ratio is essentially zero at early times as the longitudinal pressure needs to build +up first and that at late times a smooth transition to the hydrodynamical behaviour +�PL +e +� +vHydro += 1 +3 − +4 +9π ˜w +(A26) +is provided around time ˜w ∼ 1.5. In the corresponding figure this behaviour is indicated by +the coloured dashed curves. Note that the validity of the hydrodynamic limit Eq. (A26) is +guaranteed for small values of (µu/T)eq, while for larger values of (µu/T)eq this is a priori +not clear and needs further studies. We also see the effect of the chemical potential. As +one can see in the inset plot, the PL/e-ratio increases slower for for increasing (µu/T)-ratio. +Nevertheless the differences are very small, which can be explained by the assumption that +we choose the same relaxation scale for all particles. It is expected to improve the results +if one assumes different time scales for gluons and for quarks like following the approach +of [64]. Regarding the bottom right panel of Fig. 13 we show the results for the energy +attractor +� +τ 4/3e +� +/ +� +τ 4/3e +� +∞ , +(A27) +where +� +τ 4/3e +� +∞ = lim +τ→∞ τ 4/3e(τ) = const +(A28) +describes the asymptotic energy density scaled with τ 4/3. It is convenient to consider τ 4/3e +as this becomes constant at late times as ideal hydrodynamics predicts. The value of the +constant can be obtained by the numerical solution of the equations of motion and depends +on the chemical potential which is considered as the energy evolution couples to the charge +number via the scale factor. In the figure we also show the free-streaming behaviour, which +we can parametrise according to [43] +� +τ 4/3e +� +(τ 4/3e)∞ += C−1 +∞ ˜w +4 +9 +(A29) +44 + +at early times (corresponds to the dashed coloured curves) and the hydrodynamical be- +haviour +� +τ 4/3e +� +(τ 4/3e)∞ += 1 − +2 +3π ˜w +(A30) +at late times (corresponding to the black dashed curve) [43]. We emphasise that in the +case of a conformal system we also observe a smooth transition from the early time free- +streaming regime to the late time viscous hydrodynamical regime, which starts to describe +the evolution around times ˜w ∼ 1.5. +Regarding the free-streaming behaviour we fit the energy attractor at early times for the +curves corresponding to different chemical potentials using +� +τ 4/3e +� +(τ 4/3e)∞ += C−1 +∞ ˜w +4 +9 +(A31) +to extract the values of C∞. The results can be seen in Tab. (I). We emphasise that the role +of C∞ is as follows. At late times the system can be described by viscous hydrodynamics. +However, the approach to this regime depends on the theory, such that different theories +approach viscous hydrodynamics differently. This difference in the approach to the late time +behaviour results in a mismatch of the ratios of initial energy density to the final energy +density (see [43] for a comparison of KøMPøST QCD kinetic theory to results obtained in +conformal relaxation time approximation without conserved charges), where C∞ is used to +express the late time energy density in terms of the initial energy density, such that we find +Eq. (A31) at early times. In Yang-Mills kinetic theory one finds C∞ ≈ 0.9 [24, 39, 49]. +Looking at Tab. (I) we see that in conformal relaxation time approximation with conserved +charges the value of C∞ increases as we increase the initial charge number density, however +it will stay below the value in [24, 39, 49]. +Table I: Values of C∞ obtained by numerical fits. +(µu/T)eq +0.02 +0.11 +0.24 +0.37 +0.56 +C∞ +0.87643 0.87712 0.87927 0.88374 0.89349 +45 + +Appendix B: Perturbations Around Bjorken Flow +1. +Linearised equations of motion and Landau matching +By linearising the kinetic equations around the boost invariant and homogeneous back- +ground one finds an evolution equation for the perturbation of the distribution functions δf +and δfa +� +pτ∂τ + pi∂i − pη +τ 2∂η +� +δf(x, p) = −pτ +τR +δf(x, p) + pµδuµ(x) +τR +� +(feq − f(x, p)) + +pτ +T(τ)f +(1,0) +eq +� +− pτ +τR +δT(x) +T(τ) +�T(τ) +τR +∂τR +∂T (feq − f(x, p)) + +pτ +T(τ)f +(1,0) +eq +� ++ pτ +τR +� +a +δµa(x) +� +f +(0,1) +qa,eq + f +(0,1) +qa,eq +� +(B1a) +and +� +pτ∂τ + pi∂i − pη +τ 2∂η +� +δfa(x, p) = −pτ +τR +δfa(x, p) + pµδuµ(x) +τR +� +(fa,eq − fa(x, p)) + +pτ +T(τ)f +(1,0) +a,eq +� +− pτ +τR +δT(x) +T(τ) +�T(τ) +τR +∂τR +∂T (fa,eq − fa(x, p)) + +pτ +T(τ)f +(1,0) +a,eq +� ++ pτ +τR +δµa(x)f +(0,1) +a,eq , +(B1b) +The perturbations of the rest-frame velocity, δuµ(x), the temperature, δT(x), and the chem- +ical potential, δµa(x), are obtained by the linearised Landau-matching. +As the velocity uµ is normalised to uµuµ = +1, we immediately find that uµδuµ = 0, from +which +δuτ = 0 +(B2) +directly follows. The perturbed energy-momentum tensor and the perturbed charge current +are given by +δT µν = +� +d4p +(2π)4 +2π +� +−g(x) +δ(p2)2θ(p0)pµpνδf(x, p) , +(B3a) +δN µ +a = +� +d4p +(2π)4 +2π +� +−g(x) +δ(p2)2θ(p0)pµδfa(x, p) . +(B3b) +46 + +Using this, the perturbed eigenvalue problem for the energy-momentum tensor reads +(uµ + δuµ)(T µν + δT µν) = (e + δe)(uν + δuν) , +(B4a) +while the one for the charge current is given by +(uµ + δuµ)(N µ +a + δN µ +a ) = na + δna . +(B5) +By using the leading order solutions we can deduce the different components, namely +δe = δT ττ , +δuτ = 0 +, +δui = +δT τi +e + PT +, +δuη = δT τη +e + PL +, +δna = δN τ +a . +(B6) +2. +Evolution Equations for the Perturbed Moments +The perturbation of the distribution functions are expanded in terms of spherical har- +monics according to +δE m +l,k(τ) = τ 1/3 +� +dpη +(2π) +� +d2p +(2π)2pτY m +l (φpk, θp)δfk(τ, p, |pη|) , +(B7a) +δN m +al,k(τ) = +� +dpη +(2π) +� +d2p +(2π)2Y m +l (φpk, θp)δfa,k(τ, p, |pη|) , +(B7b) +Similar to the background one can obtain the components of δT µν +k +as combinations of low +order moments [43] +τ 4/3δT ττ +k += +√ +4πδE 0 +0,k , +(B8a) +δij iki +|k|τ 4/3δT τj +k = −i +� +2π +3 +� +δE +1 +1,k − δE −1 +1,k +� +, +(B8b) +ϵij iki +|k|τ 4/3δT τj +k = − +� +2π +3 +� +δE +1 +1,k + δE −1 +1,k +� +, +(B8c) +τ 4/3(−τ)δT τη +k = +� +4π +3 δE 0 +1,k , +(B8d) +δijτ 4/3δT ij +k = +� +16π +9 δE 0 +0,k − +� +16π +45 δE 0 +2,k , +(B8e) +kikj +k2 τ 4/3δT ij +k = +� +4π +9 δE 0 +0,k − +� +4π +45 δE 0 +2,k + +� +2π +15 +� +δE +2 +2,k + δE −2 +2,k +� +, +(B8f) +ϵlj kikl +k2 τ 4/3δT ij +k = −i +� +2π +15 +� +δE +2 +2,k − δE −2 +2,k +� +, +(B8g) +47 + +δij iki +|k|τ 4/3(−τ)δT ηj +k = −i +� +2π +15 +� +δE +1 +2,k − δE −1 +2,k +� +, +(B8h) +ϵij iki +|k|τ 4/3(−τ)δT ηj +k = − +� +2π +15 +� +δE +1 +2,k + δE −1 +2,k +� +, +(B8i) +τ 4/3τ 2δT ηη +k = +� +16π +45 δE 0 +2,k + +� +4π +9 δE 0 +0,k . +(B8j) +It is also possible to obtain the components of δN µ +a,k, which are given by +τδN τ +a,k = +√ +4πδN 0 +a0,k , +(B9a) +δij iki +|k|τδN j +a,k = −i +� +2π +3 +� +δN +1 +a1,k − δN −1 +a1,k +� +, +(B9b) +ϵij iki +|k|τδN j +a,k = − +� +2π +3 +� +δN +1 +a1,k + δN −1 +a1,k +� +, +(B9c) +τ(−τ)δN η +a,k = +� +4π +3 δN 0 +a1,k . +(B9d) +Note that we decomposed transverse components parallel and perpendicular to the wave +vector k. +In order to shorten the notation in the following we define +(∆E)m +l ≡ (Eeq − E + E +(1,0) +eq )m +l , +(B10a) +(∆Na)m +l ≡ (Na,eq − Na + N +(1,0) +a,eq )m +l . +(B10b) +By direct application of the time derivative to the moments we can find their evolution +equations to be +τ∂τδE m +l,k += bm +l,−2δE m +l−2,k + bm +l,0δE m +l,k + bm +l,+2δE m +l+2,k +− i|k|τ +2 +� +um +l,−δE m+1 +l−1,k + um +l,+δE m+1 +l+1,k + dm +l,−δE m−1 +l−1,k + dm +l,+δE m−1 +l+1,k +� +− τ +τR +� +δE m +l,k + δTk +T (E +(1,0) +eq )m +l − +� +a δµa,k +� +(E +(0,1) +qa,eq + E +(0,1) +qa,eq)m +l +�� +− τ +τR +δTk +T +T(τ) +τR +∂τR +∂T (Eeq − E)m +l +− τ +τR +δu∥ +k +2 +� +um +l,−(∆E)m+1 +l−1 + um +l,+(∆E)m+1 +l+1 + dm +l,−(∆E)m−1 +l−1 + dm +l,+(∆E)m−1 +l+1 +� +− τ +τR +δu⊥ +k +2i +� +um +l,−(∆E)m+1 +l−1 + um +l,+(∆E)m+1 +l+1 − dm +l,−(∆E)m−1 +l−1 − dm +l,+(∆E)m−1 +l+1 +� +, +(B11) +48 + +and +τ∂τδN m +al,k += Bm +l,−2δN m +al−2,k + Bm +l,0δN m +al,k + Bm +l,+2δN m +al+2,k +− i|k|τ +2 +� +um +l,−δN m+1 +al−1,k + um +l,+δN m+1 +al+1,k + dm +l,−δN m−1 +al−1,k + dm +l,+δN m−1 +al+1,k +� +− τ +τR +� +δN m +al,k + δTk +T (N +(1,0) +a,eq )m +l − δµa,k(N +(0,1) +a,eq )m +l +� +− τ +τR +δTk +T +T(τ) +τR +∂τR +∂T (Na,eq − Na)m +l +− τ +τR +δu∥ +k +2 +� +um +l,−(∆Na)m+1 +l−1 + um +l,+(∆Na)m+1 +l+1 + dm +l,−(∆Na)m−1 +l−1 + dm +l,+(∆Na)m−1 +l+1 +� +− τ +τR +δu⊥ +k +2i +� +um +l,−(∆Na)m+1 +l−1 + um +l,+(∆Na)m+1 +l+1 − dm +l,−(∆Na)m−1 +l−1 − dm +l,+(∆Na)m−1 +l+1 +� +. +(B12) +where we used App. D in order to express the angle relations in terms of moments. The +coefficients um +l,± and dm +l,± are given by +um +l,− = + +� +(l − m)(l − m − 1) +4l2 − 1 +, +um +l,+ = − +� +(l + m + 1)(l + m + 2) +3 + 4l(l + 2) +, +(B13a) +dm +l,− = − +� +(l + m)(l + m − 1) +4l2 − 1 +, +dm +l,+ = + +� +(l − m + 1)(l − m + 2) +3 + 4l(l + 2) +. +(B13b) +while the bm +l +and Bm +l +are the same as in the background evolution equations (see Eq. +(A7),(A8)). +The derivatives of the moments are defined to be +(E +(1,0) +eq )m +l (τ) = τ 1/3 +� +dpη +(2π) +� +d2p +(2π)2pτY m +l (φpk, θp) pτ +T(τ)f +(1,0) +eq +� pτ +T(τ), µ(τ) +� +, +(B14a) +(E +(0,1) +eq )m +l (τ) = τ 1/3 +� +dpη +(2π) +� +d2p +(2π)2pτY m +l (φpk, θp)f +(0,1) +eq +� pτ +T(τ), µ(τ) +� +, +(B14b) +(N +(1,0) +a,eq )m +l (τ) = +� +dpη +(2π) +� +d2p +(2π)2Y m +l (φpk, θp) pτ +T(τ)f +(1,0) +a,eq +� pτ +T(τ), µ(τ) +� +, +(B14c) +(N +(0,1) +a,eq )m +l (τ) = +� +dpη +(2π) +� +d2p +(2π)2Y m +l (φpk, θp)f +(0,1) +a,eq +� pτ +T(τ), µ(τ) +� +. +(B14d) +A straightforward computation of the derivatives shows that we can relate them to +(E +(1,0) +eq )m +l (τ) = −4(Eeq)m +l (τ) , +(B15a) +(E +(0,1) +eq )m +l (τ) = 3τ 1/3 � +a +(Na,eq)m +l (τ) , +(B15b) +49 + +(N +(1,0) +a,eq )m +l (τ) = −3(Na,eq)m +l (τ) , +(B15c) +(N +(0,1) +a,eq )m +l (τ) = +τ +√ +4πχaδl0δm0 . +(B15d) +However, as we are interested in perturbations around vanishing background density, the +equations will simplify since +(Na,eq)m +l (τ) = 0 +(B16) +for µ(x) = 0. Nevertheless, the susceptibilities +χa = νq +6 +�3µ2 +a +π2 + T 2 +� +(B17) +are non-zero for zero density but reduce to +χa(µa = 0) = νq +6 T 2 . +(B18) +We can also relate the perturbations of the intensive quantities to the perturbation of the +extensive quantities for na = 0 according to Eq. (A15) +δTk +T += δek +4e , +(B19a) +δµa,k = 6 +νq +δna.k +T 2 +. +(B19b) +Therefore we can replace δTk and δµa,k with δek and δna,k, which is useful since we can +express these quantities by low order moments again +τ 4/3δek(τ) = +√ +4πδE 0 +0,k(τ) , +(B20a) +τ 4/3(e + PT)δu∥ +k(τ) = − +� +2π +3 +� +δE +1 +1,k(τ) − δE −1 +1,k(τ) +� +, +(B20b) +τ 4/3(e + PT)δu⊥ +k (τ) = i +� +2π +3 +� +δE +1 +1,k(τ) + δE −1 +1,k(τ) +� +, +(B20c) +τδna,k = +√ +4πδN 0 +a0,k , +(B20d) +This results in a closed set of equations since all appearing perturbations can be written as +linear combinations of moments. +At the level of the equations of motion we see that the dependence on the direction of the +transverse wave-vector k has disappeared and the equations only depend on |k|. This is due +to the decomposition of the distribution functions into spherical harmonics and represents +the azimuthal rotation symmetry of the background in the transverse plane. +50 + +The equations of motion above (Eq. (B11) and Eq. (B12)) are considered at a fixed value +of the wave-number |k|. However, it is more convenient to rewrite the equation of motion +in a mode where we consider it for a fixed value of the propagation phase +κ = |k|(τ − τ0) . +(B21) +By making this change of variable from |k| to κ, we also need to rewrite the time derivative +according to +τ∂τ +�� +k = τ∂τ +�� +|k|(τ−τ0) + +τ +τ − τ0 +|k|(τ − τ0)∂|k|(τ−τ0) +�� +τ . +(B22) +This means, that we find an additional term resulting from the change of variables. +Furthermore, for conformal systems it is convenient to work again with the dimension- +less variable x = τ/τR. Following the same procedure as for the background, we need to +transform the derivative by making use of the scale factor (see App. A 4). By introducing +s(τ) = (τ − τ0)/τ with +a(x)x∂xs(x) = 1 − s(x) +(B23) +we the finally find the equations we are using to compute the Green’s functions +� +s(x)a(x)x∂x + κ∂κ +� +δE m +l,k += s(x) +� +bm +l,−2δE m +l−2,k + bm +l,0δE m +l,k + bm +l,+2δE m +l+2,k +� +− iκ +2 +� +um +l,−δE m+1 +l−1,k + um +l,+δE m+1 +l+1,k + dm +l,−δE m−1 +l−1,k + dm +l,+δE m−1 +l+1,k +� +− xs(x) +� +δE m +l,k + δTk +T (E +(1,0) +eq )m +l +� +− xs(x)δTk +T +T(τ) +τR +∂τR +∂T (Eeq − E)m +l +− xs(x)δu∥ +k +2 +� +um +l,−(∆E)m+1 +l−1 + um +l,+(∆E)m+1 +l+1 + dm +l,−(∆E)m−1 +l−1 + dm +l,+(∆E)m−1 +l+1 +� +− xs(x)δu⊥ +k +2i +� +um +l,−(∆E)m+1 +l−1 + um +l,+(∆E)m+1 +l+1 − dm +l,−(∆E)m−1 +l−1 − dm +l,+(∆E)m−1 +l+1 +� +, +(B24a) +51 + +and +� +s(x)a(x)x∂x + κ∂κ +� +δN m +al,k += s(x) +� +Bm +l,−2δN m +al−2,k + Bm +l,0δN m +al,k + Bm +l,+2δN m +al+2,k +� +− iκ +2 +� +um +l,−δN m+1 +al−1,k + um +l,+δN m+1 +al+1,k + dm +l,−δN m−1 +al−1,k + dm +l,+δN m−1 +al+1,k +� +− xs(x) +� +δN m +al,k − δµa,k(N +(0,1) +a,eq )m +l +� +− xs(x)δTk +T +T(τ) +τR +∂τR +∂T (Na,eq − Na)m +l +− xs(x)δu∥ +k +2 +� +um +l,−(∆Na)m+1 +l−1 + um +l,+(∆Na)m+1 +l+1 + dm +l,−(∆Na)m−1 +l+1 +� +− xs(x)δu⊥ +k +2i +� +um +l,−(∆Na)m+1 +l−1 + um +l,+(∆Na)m−1 +l−1 − dm +l,+(∆Na)m−1 +l+1 +� +, +(B24b) +where +(∆E)m +l ≡ (Eeq − E + E +(1,0) +eq )m +l , +(B25a) +(∆Na)m +l ≡ (Na,eq − Na)m +l . +(B25b) +Note that N (1,0) +a,eq = 0 since it is proportional to (Na,eq)m +l , which is zero for vanishing back- +ground density. +3. +Initial Energy Perturbations +So far we considered the evolution of linearised perturbations on top of a background +modeled as a Bjorken flow. +In order to describe the early time dynamics of heavy-ion +collisions we need suitable initial conditions for the perturbations to solve the equations of +motion. Here we will consider initial energy and charge perturbations. +We follow the idea of [39], which means initial energy perturbations will be associated with +an infinitesimal change of the energy scale of the background distribution. For the initial +distribution function of the perturbations we will find therefore +δfk(τ0, p, |pη|) = − +�|p| +3 ∂|p|f (0) +BG +� +e−ik· p +|p| τ0 . +(B26) +The factor e−ik· p +|p| τ0 takes into account the free-streaming behaviour for times τ < τ0 ≪ τR, +while f (0) +BG is given by Eq. (A9a). We can insert Eq. (B26) into the definition of δE m +l,k to +translate the initial condition to the moments according to +δE m +l,k(τ0) = τ 1/3 +0 +(−i)mJm(|k|τ0)ym +l P m +l (0)(eτ)0 +(B27) +52 + +with (eτ)0 being the asymptotic energy density of the background Eq. (A10a) and Jm(x) +being the Bessel function of the first kind of order m. In agreement with [39] we find for the +energy and velocity perturbations +δek(τ0) +e += +J0(|k|τ0) , +(B28a) +e + PT +e +δu∥ +k(τ0) = −iJ1(|k|τ0) , +(B28b) +e + PT +e +δu⊥ +k (τ0) = 0 . +(B28c) +4. +Initial Charge Perturbations +The natural choice for the moments of conserved charges is an initial perturbation in +terms of the number of quarks respectively the number of anti-quarks. Therefore we choose +initial perturbations of the form +δfa,k(τ0, p, |pη|) = δfqa,k(τ0, p, |pη|) − δf qa,k(τ0, p, |pη|) += +� +1 + 1 +2αa − 1 + 1 +2αa +� +f (0) +a,BGe−ik· p +|p| τ0 += αaf (0) +a,BGe−ik· p +|p| τ0 . +(B29) +In this particular case we choose +αa = δna(τ0) +na(τ0) . +(B30) +Translated to the level of the charge moments the initial conditions are given by +δN m +al,k(τ0) = (−i)mJm(|k|τ0)ym +l P m +l (0)αa(naτ)0 , +(B31) +where (naτ)0 is given by Eq. (A10b). For the perturbation δna we find +δna,k(τ0) +na += αaJ0(|k|τ0) . +(B32) +5. +Numerics +The procedure for finding the Green’s functions numerically is more or less the same as +for the background. However we will consider perturbations around zero densities, i.e. we +53 + +set the initial values for na to zero. At a given lmax we truncate the equations of motions +for the moments. +Regarding lmax our numerical studies have shown that we need take a relatively high value +of l in order to find convergence for the charge moments. To check this, it is convenient to +consider free-streaming. This is due to the fact that we are able to compute analytically +the behaviour of the response functions in free-streaming. Based on this we can use free- +streaming in order to check if the code runs correctly (at least without perturbation-terms +in the equations of motion). +It turns out that we find convergence towards the free-streaming behaviour for the energy +moments a lot faster than for the charge moments in terms of lmax. The results of this studies +can be seen in Fig. 14. This justifies our choice of lmax = 512. +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 +lmax=64 +G~ +s +s(w~,k∆τ) +κ=k∆τ +free streaming +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 +lmax=64 +F~ +as +s(w~,k∆τ) +κ=k∆τ +free streaming +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 +lmax=128 +G~ +s +s(w~,k∆τ) +κ=k∆τ +free streaming +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 +lmax=128 +F~ +as +s(w~,k∆τ) +κ=k∆τ +free streaming +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 +lmax=256 +G~ +s +s(w~,k∆τ) +κ=k∆τ +free streaming +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 +lmax=256 +F~ +as +s(w~,k∆τ) +κ=k∆τ +free streaming +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 +lmax=512 +G~ +s +s(w~,k∆τ) +κ=k∆τ +free streaming +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 +lmax=512 +F~ +as +s(w~,k∆τ) +κ=k∆τ +free streaming +Figure 14: Top row: ˜Gs +s, bottom row: ˜F s +s . From left to right: Corresponding Green’s functions for +lmax = 64,128, 256, 512. The black curve corresponds to analytic free-streaming solutions while the +blue curve corresponds to our data. Clearly, for ˜Gs +s we find convergence to to the free-streaming +very fast (no significant improvement for lmax > 128). For ˜F s +s we see acceptable convergence only +for lmax ≥ 512, which justifies that we choose lmax = 512 for our numerics. +54 + +Appendix C: Non-Equilibrium Green’s Functions of Energy-Momentum Tensor and +Current of Conserved Charges +1. +Green’s Functions of the Energy-Momentum Tensor +We follow the construction of the response functions according to [24, 39] and express +δT µν +k (τ) as +δT µν +k (τ) +e(τ) += 1 +2 +˜Gµν +αβ(k, τ, τ0)δT αβ +k (τ0) +e(τ0) +. +(C1) +We decompose the several response functions into a basis of scalars (s), vectors (v) and +tensors (t). For initial energy perturbations we thus have +˜Gττ +ττ(k, τ) = ˜Gs +s(κ, x) , +(C2a) +˜Gτi +ττ(k, τ) = −i ki +|k| +˜Gv +s(κ, x) , +(C2b) +˜Gij +ττ(k, τ) = δij ˜Gt,δ +s (κ, x) + kikj +|k|2 ˜Gt,k +s (κ, x) . +(C2c) +Since the normalization of the linearized perturbation is arbitrary, we adopt the convention +δe(τ0) +e(τ0) = 1 +(C3) +such that we can express the decomposed response functions in terms of δT µν +k (τ) (see [39]) +according to +˜Gs +s(κ, x) = δT ττ +k (x) +e(x) += δeκ(x) +e(x) , +(C4a) +˜Gv +s(κ, x) = iδij +ki +|k| +δT τj +κ (x) +e(x) +, +(C4b) +˜Gt,δ +s (κ, x) = +� +δij − kikj +|k|2 +�δT ij +κ (x) +e(x) +, +(C4c) +˜Gt,k +s (κ, x) = +� +2kikj +|k|2 − δij +�δT ij +κ (x) +e(x) +. +(C4d) +2. +Green’s Functions of the Current of Conserved Charges +The Green’s functions corresponding to the conserved charges are defined by +τδN µ +k(τ) = ˜F µ +α (k, τ, τ0) τ0δN α +k (τ0) . +(C5) +55 + +Note that we dropped the flavor indices on ˜F µ +α as we consider perturbations around vanishing +background densities. In such a setting the Green’s functions decouple in terms of the flavor. +Following the same argumentation δN µ +k does not depend on the flavor anymore neither. +Like before, we will decompose ˜F µ +α also in a scalar-vector-tensor basis according to +˜F τ +τ (k, τ) = ˜F s +s (κ, x) , +(C6a) +˜F i +τ (k, τ) = −i ki +|k| +˜F v +s (κ, x) . +(C6b) +Adapting the normalisation +τ0δN τ +k(τ0) = 1 +(C7) +we find +˜F s +s (κ, x) = τδN τ +κ(x) , +(C8a) +˜F v +s (κ, x) = i ki +|k|τδN i +κ(x) . +(C8b) +3. +Numerical Results for the Non-Equilibrium Green’s Functions of the Energy- +Momentum Tensor +The results for ˜Gs +s an ˜F s +s are presented in the main text in Sec. II D. In Fig. 15 and +Fig. 16 we will show the results for the other Green’s functions. In addition to the points +discussed in the main part we can very clearly see the isotropy at later times in the figure +for the pressure response ˜Gt,δ +s . After scaling the response function, we see that at early +times the longitudinal pressure is zero while at times when the system can be described +by hydrodynamics ( ˜w ≥ 1), the longitudinal pressure is established and we find the effect +of isotropy as the response function approaches one at zero propagation phase indicating +e = 3P in the hydrodynamic limit. +4. +Green’s Functions of the Energy-Momentum Tensor in Coordinate Space +Similar to the decomposition in Fourier space, we can decompose the Green’s functions +in coordinate space as well into a basis of scalars, vectors and tensors, such that we find +Gττ +ττ(r, τ) = Gs +s(|r|, τ) , +(C9a) +56 + +-0.6 +-0.4 +-0.2 + 0 + 0.2 + 0.4 + 0.6 + 0 + 5 + 10 + 15 + 20 +Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] +Momentum response: G~ +s +v(w~,k∆τ) +Wave number: κ=k∆τ +free streaming +0 +1 +2 +3 +4 +5 +-1.5 +-1 +-0.5 + 0 + 0.5 + 1 + 1.5 + 0 + 5 + 10 + 15 + 20 +Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] +Pressure response: 3G~ +s +t,δ(w~,k∆τ) +Wave number: κ=k∆τ +free streaming +0 +1 +2 +3 +4 +5 +-0.6 +-0.4 +-0.2 + 0 + 0.2 + 0.4 + 0.6 + 0 + 5 + 10 + 15 + 20 +Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] +Shear-stress response: G~ +s +t,k(w~,k∆τ) +Wave number: κ=k∆τ +free streaming +0 +1 +2 +3 +4 +5 +Figure 15: Evolution of the energy-momentum Green’s functions in response to initial energy per- +turbations in the constant κ-mode. The different panels correspond to different response functions; +different curves in each panel corresponds to different times ˜w. +Gτi +ττ(r, τ) = ri +|r|Gv +s(|r|, τ) , +(C9b) +Gij +ττ(r, τ) = δijGt,δ +s (|r|, τ) + rirj +|r|2 Gt,r +s (|r|, τ) . +(C9c) +The relation to their counterparts in Fourier space is given by the following Fourier-Hankel +transforms +Gs +s(|r|, τ) = 1 +2π +� +d|k| |k|J0(|k||r|) ˜Gs +s(|k|, τ) , +(C10a) +Gv +s(|r|, τ) = 1 +2π +� +d|k| |k|J1(|k||r|) ˜Gv +s(|k|, τ) , +(C10b) +Gt,δ +s (|r|, τ) = 1 +2π +� +d|k| |k| +� +J0(|k||r|) ˜Gt,δ +s (|k|, τ) + J1(|k||r|) +|k||r| +˜Gt,k +s (|k|, τ) +� +, +(C10c) +Gt,r +s (|r|, τ) = −1 +2π +� +d|k| |k|J2(|k||r|) ˜Gt,k +s (|k|, τ) . +(C10d) +57 + +Figure 16: Evolution of the charge Green’s functions in response to initial charge perturbations +in the constant κ-mode. The different panels correspond to different response functions; different +curves in each panel corresponds to different times ˜w. +5. +Green’s Functions of the Current of Conserved Charges in Coordinate Space +For the charge Green’s functions the decomposition in coordinate space is given by +F τ +τ (r, τ) = F s +s (|r|, τ) , +(C11a) +F i +τ (r, τ) = ri +|r|F v +s (|r|, τ) . +(C11b) +The relation to their counterparts in Fourier space is given by the Fourier-Hankel transforms +F s +s (|r|, τ) = 1 +2π +� +d|k| |k|J0(|k||r|) ˜F s +s (|k|, τ) , +(C12a) +F v +s (|r|, τ) = 1 +2π +� +d|k| |k|J1(|k||r|) ˜F v +s (|k|, τ) . +(C12b) +Appendix D: Identities For Spherical Harmonics and Associated Legendre Polyno- +mials +While deriving the equations of motion for E m +l +and N m +al +we used several identities for the +associated Legendre polynomials and numerical coefficients. First we will list the appearing +58 + +freestreaming +5 +0.6 +Evolution time: W=TTefr(t)[4nTef/(e+P)] +0.4 +4 +0.2 +3 +0 +2 +-0.2 +-0.4 +1 +-0.6 +0 +0 +5 +10 +15 +20 +Wavenumber:k=k△t-1.5 +-1 +-0.5 + 0 + 0.5 + 1 + 1.5 + 2 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 +Evolution time: w~=T(τ)τ / (4πη/s) +Pressure response: 3∆τ2 Gs +t,δ(∆x,∆τ,w~) +Distance: ∆x/∆τ +free streaming +0 +1 +2 +3 +4 +5 +-1.5 +-1 +-0.5 + 0 + 0.5 + 1 + 1.5 + 2 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 +Evolution time: w~=T(τ)τ / (4πη/s) +Shear-stress response: ∆τ2 Gs +t,r(∆x,∆τ,w~) +Distance: ∆x/∆τ +free streaming +0 +1 +2 +3 +4 +5 +Figure 17: Evolution of the energy Green’s functions in response to initial energy perturbations in +coordinate space. The different panels correspond to different response functions; different curves +in each panel corresponds to different times ˜w. +59 + +2 +free streaming +5 +1.5 +4 +1 +0.5 +3 +0 +2 +0.5 +1 +-1 +-1.5 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Distance:Ax/△tFigure 18: Evolution of the charge Green’s functions in response to initial charge perturbations in +coordinate space. The different panels correspond to different response functions; different curves +in each panel corresponds to different times ˜w. +coefficients +∆m +l,− = + (l + 1)(l + m) +2l + 1 +, +ξm +l,− += l + m +2l + 1 , +∆m +l,+ = − l(l − m + 1) +2l + 1 +, +ξm +l,+ +=l − m + 1 +2l + 1 +, +am +l,−2 = − ξ(2),m +l,−2 − ∆m +l,−ξm +l−1,− +, +bm +l,−2 =am +l,−2 +ym +l +ym +l−2 +, +am +l,0 =1 +3 − ξ(2),m +l,0 +− ∆m +l,−ξm +l−1,+ − ∆m +l,+ξm +l+1,− +, +bm +l,0 +=am +l,0 , +am +l,+2 = − ξ(2),m +l,+2 − ∆m +l,+ξm +l+1,+ +, +bm +l,+2 =am +l,+2 +ym +l +ym +l+2 +, +Am +l,−2 = − ∆m +l,−ξm +l−1,− +, +Bm +l,−2 =Am +l,−2 +ym +l +ym +l−2 +, +Am +l,0 = − ∆m +l,−ξm +l−1,+ − ∆m +l,+ξm +l+1,− +, +Bm +l,0 +=Am +l,0 , +Am +l,+2 = − ∆m +l,+ξm +l+1,+ +, +Bm +l,+2 =Am +l,+2 +ym +l +ym +l+2 +, +um +l,− = + +ym +l +(2l + 1)ym+1 +l−1 +, +dm +l,− += + +ym +l +(2l + 1)ym−1 +l−1 σm +l σ−m+1 +l−1 +, +um +l,+ = − +ym +l +(2l + 1)ym+1 +l+1 +, +dm +l,+ += − +ym +l +(2l + 1)ym−1 +l+1 σm +l σ−m+1 +l+1 +. +(D1a) +60 + +0.8 +freestreaming +5 +0.6 +Evolution time: W=T(t)t / (4Tn/s) +4 +0.4 +3 +0.2 +0 +2 +-0.2 +1 +-0.4 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Distance:Ax/△tand +ξ(2),m +l,−2 = ξm +l,−ξm +l−1,− , +ξ(2),m +l,0 += ξm +l,−ξm +l−1,+ + ξm +l,+ξm +l+1,− , +ξ(2),m +l,+2 = ξm +l,+ξm +l+1,+ , +σm +l = (−1)m(l − m)! +(l + m)! . +(D1b) +These coefficients are now used to formulate the following identities in a compact way. For +the background we use +� +1 − x2� d +dxP m +l (x) = ∆m +l,−P m +l−1 (x) + ∆m +l,+P m +l+1 (x) +(D2) +and +xP m +l (x) = ξm +l,−P m +l−1 (x) + ξm +l,+P m +l+1 (x) +(D3) +together with +x2P m +l (x) = ξ(2),m +l,−2 P m +l−2 (x) + ξ(2),m +l,0 +P m +l (x) + ξ(2),m +l,+2 P m +l+2 (x) . +(D4) +Combining these identities we find +��1 +3 − x2 +� +− x +� +1 − x2� d +dx +� +P m +l (x) = am +l,−2P m +l−2 (x) + am +l,0P m +l (x) + am +l,+2P m +l+2 (x) +(D5) +respectively +−x +� +1 − x2� d +dxP m +l (x) = Am +l,−2P m +l−2 (x) + Am +l,0P m +l (x) + Am +l,+2P m +l+2 (x) . +(D6) +Furthermore we make use of +P −m +l +(x) = σm +l P m +l (x) +(D7) +and +√ +1 − x2P m +l (x) = +1 +2l + 1 +� +P m+l +l−1 (x) − P m+1 +l+1 (x) +� +(D8) +in order to find +sin(θ)e+iφY m +l (φ, θ) = um +l,−Y m+1 +l−1 (φ, θ) + um +l,+Y m+1 +l+1 (φ, θ) , +(D9a) +sin(θ)e−iφY m +l (φ, θ) = dm +l,−Y m−1 +l−1 (φ, θ) + dm +l,+Y m−1 +l+1 (φ, θ) , +(D9b) +which are used to compute the additional terms in the equation of motion for the perturbed +moments. +61 + +Appendix E: Eccentricities, Cumulants, and Anisotropic Flow +1. +Standard Initial State Eccentricities +Quantifying the geometry of the initial state is done using the standard definition of the +complex eccentricity vector En given as +En ≡ εn einψn ≡ − +� +rdrdφ rneinφ f(r, φ) +� +rdrdφ rn f(r, φ) +, +(E1) +where f(r, φ) is an initial state distribution like the entropy or energy density which specifies +the initial state. The magnitude of the eccentricity is εn and ψn is the complex (event-plane) +angle. We can express this quantity in terms of the complex position vector r ≡ x + iy +through rneinφ = rn: +En ≡ − +� +d2r rn f(r) +� +d2r |r|n f(r) , +(E2) +where boldface is used to denote the complex vector. Usually these definitions are specified +as applying only in the center of mass frame. This can be expressed in terms of a general +coordinate system: +En ≡ − +� +d2r (r − rCMS)n f(r) +� +d2r |r − rCMS|n f(r) +(E3) +with the center-of-mass vector +rCMS ≡ +� +d2r r f(r) +� +d2r f(r) = +1 +ftot +� +d2r r f(r). +(E4) +A consequence of this definition is that the directed eccentricity E1 vanishes identically. +This method for describing the initial state is well suited when the quantity f(r) being +described is positive definite like the energy or entropy density. However if f(r) = ρ(r) is +a charge density, particularly when total net charge is zero, it becomes impossible to define +a corresponding frame such that E1 = 0 when the total charge vanishes. Instead, there is +always a nonzero E1 proportional to the dipole moment. Due to this inability to construct +a center-of-charge frame and ensure that E1 vanishes for a conserved charge with qtot = 0, +the usual definitions (E1) or (E2) must be modified. +For a conserved charge density ρX(r) (here we consider baryon number B, strangeness S, +or electric charge Q for X), regions of positive charge with ρX(r) > 0 and negative charge +62 + +with ρX(r) < 0 will be treated separately by decomposing +ρX ≡ ρ(X +) θ(ρX) + ρ(X −) θ(−ρX), +(E5) +where the position argument r is suppressed for brevity. 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C 102, 064910 (2020), +2006.14252. +66 + diff --git a/ktE3T4oBgHgl3EQfhgqt/content/tmp_files/load_file.txt b/ktE3T4oBgHgl3EQfhgqt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d5383d95e38709ef047463328caf05e6ac5c5691 --- /dev/null +++ b/ktE3T4oBgHgl3EQfhgqt/content/tmp_files/load_file.txt @@ -0,0 +1,2113 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf,len=2112 +page_content='Pre-Equilibrium Evolution of Conserved Charges with ICCING Initial Conditions P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Carzon,1, ∗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Martinez,2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha-Hostler,1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Plaschke,3, † S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Schlichting,3 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Sievert4 1Illinois Center for Advanced Studies of the Universe & Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Urbana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' IL 61801,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' USA 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' North Carolina State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Raleigh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' NC 27695,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' USA 3Fakultät für Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Universität Bielefeld,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D-33615 Bielefeld,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Germany 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' New Mexico State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Las Cruces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' NM 88003,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' USA (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 2023) Abstract Heavy-ion collisions can be well described through relativistic viscous hydrodynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' but ques- tions still remain when hydrodynamics is applicable because the initial state may begin very far- from-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Thus, a pre-equilibrium evolution phase is used to bridge the gap between the initial state and hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' KøMPøST is one such pre-equilibrium model that propagates the energy-momentum tensor by decomposing it into the background and fluctuations around that background, whose evolution is captured by Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We extend this formalism to in- clude conserved charges and calculate the corresponding non-equilibrium Green’s functions in the relaxation time approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The ICCING algorithm initializes conserved charges in the initial state by sampling g → q¯q splitting probabilities and is, thus, perfectly positioned to implement Green’s functions for charge propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We show that this method alters the initial state charge geometries and is applicable in central to mid-central collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' ∗Email: pcarzon2@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='edu †Email: pplaschke@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='uni-bielefeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='04572v1 [nucl-th] 11 Jan 2023 Contents I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Introduction 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions from Kinetic Theory 7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Boltzmann Equation in Relaxation Time Approximation 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Background Evolution 9 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Perturbations around Bjorken flow 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Evolution equations for perturbations in the transverse plane 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Solving the equations of motion in the transverse plane 15 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Non-equilibrium Green’s Functions 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Non-equilibrium Green’s Functions of the Energy-Momentum Tensor 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Non-equilibrium Green’s Functions of the Current of Conserved Charges 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Numerical Results for the non-equilibrium Green’s Functions 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions in Coordinate Space 20 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Applications 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Using Green’s Functions in ICCING 22 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Impact on Eccentricities 28 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Conclusion & Outlook 35 Acknowledgements 36 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Background evolution 38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Evolution Equations for the spherical harmonic moments 38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Initial Conditions 39 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Relation of Intensive and Extensive Quantities 40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Background Evolution in Conformal Systems 41 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Perturbations Around Bjorken Flow 46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Linearised equations of motion and Landau matching 46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Evolution Equations for the Perturbed Moments 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Initial Energy Perturbations 52 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Initial Charge Perturbations 53 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Numerics 53 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Non-Equilibrium Green’s Functions of Energy-Momentum Tensor and Current of Conserved Charges 55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions of the Energy-Momentum Tensor 55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions of the Current of Conserved Charges 55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Numerical Results for the Non-Equilibrium Green’s Functions of the Energy-Momentum Tensor 56 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions of the Energy-Momentum Tensor in Coordinate Space 56 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions of the Current of Conserved Charges in Coordinate Space 58 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Identities For Spherical Harmonics and Associated Legendre Polynomials 58 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Eccentricities, Cumulants, and Anisotropic Flow 62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Standard Initial State Eccentricities 62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Cumulants 63 References 63 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' INTRODUCTION Ultra-relativistic heavy-ion collisions provide an opportunity to study the extreme limits of deconfined quarks and gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the very early stages of the collisions, the energy is predominantly composed of saturated gluons emerging from the low-x wave functions of the colliding nuclei [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This initial state is followed by pre-equilibrium dynamics, leading to the formation of a quark-gluon plasma (QGP) characterized by deconfined quarks and gluons acting as a nearly perfect fluid [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The measured distributions of final-state hadrons resulting from the freeze-out of this fluid thus encode a complex superposition of the features of the initial-state geometry, pre-equilibrium dynamics, and hydrodynamic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Thus, simulations of all stages of heavy-ion collisions are crucial to reconstruct the early stages of the collision and interpret experimental data (see [4] and citations within).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To simulate heavy-ion collisions, one starts with an initial state characterization of the energy-momentum tensor T µν and currents Jµ of conserved charges which is far from thermo- 3 dynamic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The initial state at proper time τ0 and any pre-equilibrium dynamics which follow determine the initial conditions at proper time τhydro > τ0 of the hydrodynamic equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This hydrodynamic evolution continues until the system has frozen out into baryons and mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Apples-to-apples comparisons to experiments are possible after a further simulation of the hadronic gas phase, where the system is described in terms of hadrons and their interactions [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Because the QGP behaves as a nearly perfect liquid [4], the geometric structure from the initial-state T µν and Jµ leaves an observable imprint on the final state hadron distributions [8–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This both makes models of the initial conditions particularly important in the pre- diction of experimental observables and also allows us to constrain these models by direct confrontation with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Specifically, observables which are less sensitive to the hydrody- namic phase, such as the cumulant ratio vn {4} /vn {2} in central collisions, can provide a direct window into initial state effects [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Until recently, initial-state models have primarily focused on descriptions of the energy density ϵ = T 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Recent progress has systematically included more initial state variables including initial flow T 0i and initial shear T ij [19–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Initial conditions of conserved charge densities ρ = J0 [27–31] have also been developed, though primarily concerned with baryon density ρB due to its role in the search for the QCD critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Recently, an open-source Monte Carlo event generator, known as ICCING (Initial Conserved Charges in Nuclear Geometry), was developed which is capable of initializing all three conserved charge densities [32, 33]: baryon density, strangeness density, and electric charge density (BSQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' ICCING is a model-agnostic algorithm which constructs the initial conditions for the BSQ charge densities for a given energy density, by treating the energy density as being composed of gluons and stochastically sampling their probability to split into quark-antiquark (q¯q) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The g → q¯q splitting probabilities can be specified according to any desired microscopic model, among many other parts of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The ICCING algorithm is constructed in a modular fashion to account for a variety of different physical inputs relevant throughout the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The sampling process is done by seeding a random point from the input energy distribution and selecting a fraction of energy from a circle centered on that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The radius of this ‘gluon’, the probability distribution from which the fraction of energy is sampled, and the minimum amount of energy allowed for a gluon are external inputs set by the user for this step of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This extends to the rest of the algorithm which is explained in 4 full in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To constrain the parameters used in ICCING initial conditions from experimental data, a necessary next step would be to evolve these initial conditions using a 2+1D viscous hydrodynamics code that simultaneously solves all of the hydrodynamic equations of motion, including all of the conserved currents as well as the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This task is technically challenging, not only because of the challenges associated with evolving the new conserved currents themselves, but also because of the need for a fully four-dimensional equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' These issues have started to be addressed and are expected to appear soon [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' There is a further challenge in the connection of the initial state to the hydrodynamic evolution since the former is far from equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Recent work [25] has shown that moving directly from initial conditions to hydro produces a large fraction of fluid cells that violate nonlinear causality constraints [35], about 30%, while there is uncertainty about the causal status of the remaining cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This study found that including a pre-equilibrium evolution stage reduces the number of acausal cells but does not fully eliminate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Generally, a proper description of the pre-equilibrium dynamics is not only theorectically desirable, but can also affect flow observables in small and large collision systems [36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Based on a microscopic description in QCD kinetic theory, the authors of [24, 39] developed the non-equilibrium linear response formalism KøMPøST which allows to propagate the energy- momentum tensor T µν from early times up to the point where a fluid dynamical description becomes applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The KøMPøST code was used in [25] as the pre-equilibrium stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Coupling the ICCING code to a pre-equilibrium stage would further extend its usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' So far the pre-equilibrium description in KøMPøST itself does not contain what is needed to evolve the conserved charge densities from ICCING, but the methods that are used, namely non-equilibrium Green’s functions, could be applied to our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The main idea of KøMPøST is to extract the energy-momentum tensor T µν(x) at time τ = τhydro from an initial state model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For this the system is evolved by using effective kinetic theory from an initial time τ0 to τhydro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Fluctuations δT µν(x) of the energy-momentum tensor around the background energy-momentum tensor T µν BG(x) are considered within this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In practise the perturbations are assumed to be small and therefore can be linearised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This gives rise to linear response theory, where the complete energy-momentum tensor T µν(x) can be obtained by a sum of T µν BG(x) and a term involving non-equilibrium Green’s functions, which capture the evolution of the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This provides a powerful 5 method to compute T µν(x) as numerical simulations only need to be done once to obtain the background evolution and the Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the past few years different groups have begun to incorporate KøMPøST within their fluid dynamics simulations, making direct connections to experimental data [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' While KøMPøST is based on QCD kinetic theory, more recently studies have been made in which the same Green’s functions are computed in simpler models, such as the Boltzmann equation in Relaxation Time Approximation (RTA) [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Assuming the relaxation time approximation drastically simplifies the theoretical description and allows an efficient way to compute the non-equilibrium Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the relaxation time approximation the general formalism of KøMPøST can also be expanded in a much simpler way to include conserved charges and compute Green’s functions for the corresponding current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' These Green’s functions for charge and energy propagation can be included in ICCING by some careful reworking and provide a meaningful pre-equilibrium evolution for the conserved charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To test the effect of these new pre-equilibrium charge evolution equations, we will look at the event averaged 2-particle eccentricities (See App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' E) which describe the geometry of the initial state and have been shown to be good predictors of the final state flow harmonics [10] except in peripheral collisions where non-linear corrects become significant [17, 18, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Because of this linear mapping, it is possible to cancel out many of the medium effects by taking the ratio of 4-particle to 2-particle cumulants, which also are a measure of the fluctuations of a certain type of initial state geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' These are well understood for the energy density and have only started to be studied for BSQ charge densities [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In this paper we couple the Green’s functions coming from relaxation time approximation to the ICCING algorithm by treating energy and charge differences after gluon splittings as small perturbation around the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For that we first introduce the basics about our Green’s functions calculation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' III is dedicated to the applications of these response functions in the ICCING algorithm, while we present our results in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Conclusions are found in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Besides this we provide additional calculations in the appendix about the background evolution (App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A), the perturbations around this background (App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' B) and about the Green’s functions (App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D we mention technical aspects regarding spherical harmonics and in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' E the definition is given for the initial state eccenctricities and 6 cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' GREEN’S FUNCTIONS FROM KINETIC THEORY In order to introduce the non-equilibrium Green’s functions we follow the same idea as [39] by dividing the space-time dynamic into a background evolution and perturbations around this background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' On the technical side we follow [43], where the authors solved the equations of motions in term of moments of the distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This formalism will be extended to include conserved charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Although we are not primarily interested in the background evolution in this study, we need to address it briefly since its dynamics enters the time evolution of the energy and number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Therefore Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' II A and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' II B are dedicated to introduce the background evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Further results on our study regarding the background evolution can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Afterwards we will consider the dynamics of small space-time perturbations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' II C (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' B for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The evolution of these perturbations will be captured in terms of non-equilibrium Green’s functions, which are introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For completeness, the Green’s functions that are not relevant for our study are presented in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Boltzmann Equation in Relaxation Time Approximation Starting point of our analysis is the Boltzmann equation in relaxation time approximation (RTA) pµ∂µf = C[f] = −pµuµ(x) τR [f − feq(pµβµ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' µ(x))] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (1a) pµ∂µfa = C[fa] = −pµuµ(x) τR [fa − fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq(pµβµ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' µa(x))] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (1b) where x = (x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' x3) describes a four-dimensional vector in Minkowski space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' β(x) = uµ(x)/T(x) with uµ(x) being the local-rest frame velocity obtained by Landau matching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' T(x) being the effective temperature and a standing for the quarks up,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' down and strange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 7 By f and fa we denote the singlet and valence distribution functions f = νgfg + νq � a � fqa + f qa � , (2a) fa = νq � fqa − f qa � , (2b) with g standing for gluon, q standing for quark, a = u, d, s, such that Nf = 3, and νg = 16, νq = 6 the spin-color degeneracy factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The corresponding equilibrium distri- bution functions are the Bose–Einstein distribution function for gluons and the Fermi-Dirac distribution function for quarks and antiquarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Each quark flavor has its own chemical po- tential µa(x) which governs the evolution of the valence charge distribution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (1b), while the flavor singlet distribution (1a) evolves according to the effective chemical potential µ(x) for all flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We also emphasize that we assume the same relaxation time τR for every species;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' while this is a fairly restrictive assumption, we use it as a first step to explore the geometrical impact of charge diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As we consider the evolution in a conformal system, τR is proportional to the inverse temperature such that [46] τRT(τ) = 5 ˜ηT e + P = const , (3) where ˜η is the shear viscosity, e the energy density, P the pressure and T the effective temperature of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The velocity uµ(x) is the local rest-frame velocity which is determined via the Landau-matching conditions T µν(x)uν(x) = e(x)uµ(x) , (4a) which ensures energy-momentum conservation [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In addition to this, we also have the matching conditions for the conserved charges na(x), N µ a (x)uµ(x) = na(x) , (5a) such that the local thermodynamic variables T(x) and µa(x) can be determined from e(x) = eeq(T(x), µ(x)) , (6a) na(x) = na,eq(T(x), µ(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (6b) We emphasize that the above-mentioned quantities T µν(x) respectively N µ a (x) are defined to have contributions from all species of particles such that T µν = T µν g + � a � T µν a + T µν a � , (7a) N µ a = N µ qa − N µ qa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (7b) 8 As we are interested in longitudinally boost-invariant expanding systems it is convenient to work in Milne coordinates τ = � (x0)2 − (x3)2 , η = arctanh �x3 x0 � , (8) such that gµν = diag(+1, −1, −1, −τ 2) and � −g(x) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Furthermore we consider the quarks and gluons to be massless, such that their momentum can be parameterized as pµ = (pT cosh(y), p, pT sinh(y)) (9) with y = arctanh(p3/p0) being the momentum space rapidity and pT ≡ |p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the Milne coordinates the Boltzmann equation takes the following form � pτ∂τ + pi∂i + pη∂η � f(x, p) = −pµuµ(x) τR [f(x, p) − feq(pµβµ(x), µ(x))] , (10a) � pτ∂τ + pi∂i + pη∂η � fa(x, p) = −pµuµ(x) τR [fa(x, p) − fa,eq(pµβµ(x), µa(x))] , (10b) where pτ = pT cosh(y − η) , pη = 1 τ pT sinh(y − η) , (11) and i = x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' It turns out that, when analyzing the dynamics of a boost invariant medium, it is more convenient to work with the (dimensionless) longitudinal momentum variable pη = −τpT sinh(y − η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (12) With respect to this coordinate we arrive at the following form of the Boltzmann equation � pτ∂τ + pi∂i − pη τ 2∂η � f(x, p) = −pµuµ(x) τR [f(x, p) − feq(pµβµ(x), µ(x))] , (13a) � pτ∂τ + pi∂i − pη τ 2∂η � fa(x, p) = −pµuµ(x) τR [fa(x, p) − fa,eq(pµβµ(x), µa(x))] , (13b) where pτ = � p2 T + (pη/τ)2 represents the massless on-shell condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Background Evolution In order to investigate the dynamics of the system in the pre-equilibrium phases, we make the assumption that the system can be divided into a background and perturbations around 9 this background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the pre-equilibrium stage the plasma experiences a rapid longitudinal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, in the transverse plane the plasma is initially at rest and the expansion only builds up on timescales that are comparable to the systems size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Therefore, we can ne- glect the transverse expansion at early times and consider the idealized situation of Bjorken flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Accordingly, the background is assumed to be longitudinally boost-invariant, parity invariant under spatial reflections along the longitudinal axis as well as azimuthally symmet- ric and translationally invariant in the transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The aforementioned symmetries constrain the distribution functions of the background to the following form f(x, p) = fBG(τ, pT, |pη|) , (14a) fa(x, p) = fa,BG(τ, pT, |pη|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (14b) For such a system the energy-momentum tensor is diagonal in Milne coordinates with entries T µν BG = diag � e, PT, PT, PL/τ 2� , (15) where e is the energy density, PT the transverse and PL the longitudinal pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As the energy-momentum tensor is diagonal in Milne coordinates, the Landau-matching Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (4) is solved trivially with uµ = (uτ, u, uη) = (1, 0, 0, 0) , (16a) e = eeq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (16b) Regarding the conserved charges and their respective currents one finds N µ a = (N τ a , Na, N η a ) = (na, 0, 0, 0) , (17) where na ≡ nqa − nqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Finally, the Boltzmann equation takes the familiar form τ∂τfBG(τ, pT, |pη|) = − τ τR � fBG(τ, pT, |pη|) − feq � pτ T(τ), µ(τ) �� , (18a) τ∂τfa,BG(τ, pT, |pη|) = − τ τR � fa,BG(τ, pT, |pη|) − fa,eq � pτ T(τ), µa(τ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (18b) Our strategy to solve these equations follows [43, 48] and consists of expanding the distri- bution functions in terms of spherical harmonics Y m l (φ, θ) according to E m l (τ) = τ 1/3 � dpη (2π) � d2p (2π)2pτY m l (φp, θp)fBG(τ, pT, |pη|) , (19a) N m al (τ) = � dpη (2π) � d2p (2π)2Y m l (φp, θp)fa,BG(τ, pT, |pη|) , (19b) 10 and solve the equations of motions for these moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Since the evolution of the background is not the main focus of this work, we simply note that a detailed analysis for vanishing charge densities can be found in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' While in this work, we will only consider energy-momentum and charge perturbations on top of a charge neutral background, we provide additional discussion on the background evolution in the presence of non-vanishing density in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As we are interested in the evolution around vanishing background charge density, we should mention that we can extract the background energy density at any given time as a function of the initial energy density according to the following method1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' By following [43, 49, 50], we can compute the energy density at late times � τ 4/3e(τ) � ∞ as a function of the initial energy density as � τ 4/3e(τ) � ∞ = C∞ � 4π˜η/s T(τ0)τ 1/4 0 �4/9 (eτ)0 , (20) where the constant C∞ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 [43, 49] quantifies how efficiently the initial energy is converted into thermal energy, ˜η/s is the shear-viscosity to entropy density ratio, which is constant for a conformal system with vanishing net charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' By (eτ)0 we denote the initial energy density per unit area and rapidity (eτ)0 ≡ dE0 dη d2x = dE0,g dη d2x + � a � dE0,qa dη d2x + dE0,qa dη d2x � (21) which becomes constant in the limit τ → 0, in kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Inverting the Landau matching condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (4) for vanishing chemical potential gives T(τ) = � 30 π2νeff e(τ) �1/4 (22) with νeff = νg + 7 82Nfνq the overall effective degeneracy factor of all partons, such that � τ 4/3e(τ) � ∞ can be expressed as � τ 4/3e(τ) � ∞ = C∞(4π˜η/s)4/9 �π2νeff 30 �1/9 (eτ)8/9 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (23) In the next step we introduce an attractor curve E( ˜w), which depends on the dimensionless time-variable [49] ˜w = T(τ)τ 4π˜η/s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (24) 1 For the sake of readability we drop the subscript BG for the energy density for a moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 11 This attractor curve smoothly interpolates between free-streaming at early times and viscous hydrodynamics at late times E( ˜w ≪ 1) = C−1 ∞ ˜w4/9 , (25a) E( ˜w ≫ 1) = 1 − 2 3π ˜w4/9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (25b) The attractor curve connects the asymptotic value � τ 4/3e(τ) � ∞ to its counterpart at any given time � τ 4/3e(τ) � according to E( ˜w) = � τ 4/3e(τ) � (τ 4/3e(τ))∞ , (26) and has been calculated in [43, 49] for the Boltzmann equaton in RTA and in [49, 50] for Yang-Mills and QCD kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Based on this attractor curve, we can therefore relate the initial energy density to the energy density at a later time via eBG(e(τ0)) = C∞(4π˜η/s)4/9 �π2νeff 30 �1/9(eτ)8/9 0 τ 4/3 E( ˜w) , (27) assuming that the system can locally be described by conformal Bjorken flow up to this time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Perturbations around Bjorken flow So far we addressed the evolution of a homogeneous, boost invariant background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Now we will consider linearized perturbations around this background, caused by small space- time dependent variations of the initial energy or charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We will linearize the kinetic equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' such that we can derive an evolution equation for the perturbations of the distribution functions δf and δfa � pτ∂τ + pi∂i − pη τ 2∂η � δf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) = −pτ τR δf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) + pµδuµ(x) τR � (feq − f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq � − pτ τR δT(x) T(τ) �T(τ) τR ∂τR ∂T (feq − f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq � + pτ τR � a δµa(x) � f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) qa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq + f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) qa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � (28a) 12 and � pτ∂τ + pi∂i − pη τ 2∂η � δfa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) = −pτ τR δfa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) + pµδuµ(x) τR � (fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq − fa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � − pτ τR δT(x) T(τ) �T(τ) τR ∂τR ∂T (fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq − fa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � + pτ τR δµa(x)f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (28b) where δf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) = νgδfg(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) + νq � a � δfqa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) + δf qa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (29a) δfa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) = νq � δfqa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) − δf qa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (29b) Here we used a shorter notation for the derivatives, namely f (n,m)(x, y) ≡ ∂n ∂xn ∂m ∂ymf(x, y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (30) As feq = feq � pτ T(τ), µ(τ) � , (31) fa,eq = fa,eq � pτ T(τ), µ(τ) � , (32) the derivatives are with respect to pτ/T(τ) for the (1, 0)-derivative and with respect to µ(τ) for the (0, 1)-derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The perturbations of the rest-frame velocity, δuµ(x), the temperature, δT(x), and the chemical potential, δµa(x), are determined by the linearised Landau-matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Details can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Evolution equations for perturbations in the transverse plane From now on we will concentrate on perturbations in the transverse plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' only for the transverse coordinates x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To solve the equations of motion we will expand the perturbations in a Fourier basis such that δfi(τ, x, p, |pη|) = � d2k (2π)2δfi,k(τ, p, |pη|)eik·x (33) 13 for i ∈ {g, qa, qa} and where δfi,k(τ, p, |pη|) ≡ δfi(τ, k, p, |pη|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The definition of δfk(τ, p, |pη|) and δfa,k(τ, p, |pη|) is analogous to the cases before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Decomposing the ve- locity perturbation in the transverse plane into components parallel, δu∥ k(τ), and transverse, δu⊥ k (τ), to the wave vector in the transverse plane k we therefore find δT(τ, x) = � d2k (2π)2δTk(τ)eik·x , (34a) δµa(τ, x) = � d2k (2π)2δµa,k(τ)eik·x , (34b) δui(τ, x) = � d2k (2π)2 � δu∥ k(τ)δji + δu⊥ k (τ)ϵji� kj |k|eik·x , (34c) δuτ(τ, x) = 0 , δuη(τ, x) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (34d) Note that δuη(τ, x) = 0 vanishes identically due to the assumption that boost invariance along the beam axis is not broken for the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This assumption could be relaxed in future work, but is important here for coupling to a 2+1D geometry as in ICCING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Denoting k · p |k|pτ = δij kipj |k|pτ = cos(φpk) sin(θp) , (35a) k × p |k|pτ = ϵij kipj |k|pτ = sin(φpk) sin(θp) , (35b) where φpk ≡ φp−φk is the angle between p and k in the transverse plane and sin(θp) = pT/pτ and inserting the Fourier integrals above into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (28) allows us to find an evolution equation for δfk and δfa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' such that we have τ∂τδfk(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' |pη|) = − � iτ|k| k · p |k|pτ + τ τR � δfk(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' |pη|) − τ τR � δu∥ k(τ) k · p |k|pτ + δu⊥ k (τ)k × p |k|pτ �� (feq − f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq � − τ τR δTk(τ) T(τ) �T(τ) τR ∂τR ∂T (feq − f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq � + τ τR � a δµa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(x) � f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) qa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq + f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) qa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � (36a) 14 and τ∂τδfa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' |pη|) = − � iτ|k| k · p |k|pτ + τ τR � δfa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' |pη|) − τ τR � δu∥ k(τ) k · p |k|pτ + δu⊥ k (τ)k × p |k|pτ �� (fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq − fa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � − τ τR δTk(τ) T(τ) �T(τ) τR ∂τR ∂T (fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq − fa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � + τ τR δµa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(x)f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (36b) In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (36) we sorted the terms by the several perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The first term on the right hand side corresponds to free-streaming, while the second term describes the relaxation of the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the following lines one sees that the perturbations of the velocity, temperature and chemical potential cause a change of the equilibrium distribution, while the velocity and temperature perturbations also affect the relaxation of the out-of-equilibrium background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Solving the equations of motion in the transverse plane In order to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (36) we follow the same strategy as for the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Therefore, we define the perturbed moments according to δE m l,k(τ) = τ 1/3 � dpη (2π) � d2p (2π)2pτY m l (φpk, θp)δfk(τ, p, |pη|) , (37a) δN m al,k(τ) = � dpη (2π) � d2p (2π)2Y m l (φpk, θp)δfa,k(τ, p, |pη|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (37b) As the derivation of the equations of motion for the moments are not of primary interest here, we will shift the explicit calculation into App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Similar to the background, we are able to obtain the components of δT µν k and δN µ a,k as combinations of low order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A full list can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Furthermore we can also relate the perturbations of the intensive quantities to the perturbation of the extensive quantities for na = 0 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A15) by δTk T = δek 4e , (38a) δµa,k = 6 νq δna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (38b) 15 Therefore we can replace δTk and δµa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k with δek and δna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' which is useful since we can express these quantities by low order moments again τ 4/3δek(τ) = √ 4πδE 0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (39a) τ 4/3(e + PT)δu∥ k(τ) = − � 2π 3 � δE +1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) − δE −1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (39b) τ 4/3(e + PT)δu⊥ k (τ) = i � 2π 3 � δE +1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) + δE −1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (39c) τδna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k = √ 4πδN 0 a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (39d) This results in a closed set of equations since all appearing perturbations can be written as linear combinations of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The equations of motions will be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For this we truncate the evolution at lmax=512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In order to find a reasonable value we compared our results to the analytical free-streaming equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This comparison shows that convergence is reached much faster for δE m l,k than for δN m al,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' More details can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 14 in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' B 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We note that, in addition to [43] we also need to invert the matching conditions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This we do numerically at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' More details on this can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Non-equilibrium Green’s Functions In principle, we can obtain all information about the evolution of the system from the moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, we find it more convenient to consider Green’s functions of the energy- momentum tensor and the charge current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Therefore, we will consider linear response func- tions ˜Gµν αβ for the energy-momentum tensor respectively and ( ˜Fab)µ α for the charge current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Here ˜G and ˜F describe the Green’s functions in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As we will show below, the Green’s functions can be related to macroscopic quantities (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (44), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (49)), which are however related to low order moments according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This is another powerful property of our formalism as it is easy to quantify the systems response to perturbations once one have solved the equations of motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For our study, only ˜Gττ ττ and ( ˜Fab)τ τ are relevant, such that we only present the result for these two here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The other Green’s functions can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Non-equilibrium Green’s Functions of the Energy-Momentum Tensor We will follow the construction of the response functions according to [24, 39] and express δT µν k (τ) as δT µν k (τ) e(τ) = 1 2 ˜Gµν αβ(k, τ, τ0)δT αβ k (τ0) e(τ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (40) In the following we will omit the explicit dependence on τ0 for better readability since we are mainly interested in the limit τ0/τR → 0, where the kinetic framework describes the equilibration process from directly after the collision until the onset of the hydrodynamic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Besides this, we also introduce the propagation phase κ by κ = |k|(τ − τ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (41) When expressing the evolution equations for the moments in terms of κ, this change of vari- able introduces additional terms in the time derivative [43], which were taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Similarly to [39], we will decompose the response functions into a basis of Lorentz scalars (s), vectors (v) and tensors (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For ˜Gττ ττ this means ˜Gττ ττ(k, τ) = ˜Gs s(κ, x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (42) Since the normalization of the linearized perturbation is arbitrary, we adopt the convention δe(τ0) e(τ0) = 1 (43) such that we can express the decomposed response function in terms of δT µν k (τ) (see [39]) according to ˜Gs s(κ, x) = δT ττ k (x) e(x) = δeκ(x) e(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (44) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Non-equilibrium Green’s Functions of the Current of Conserved Charges For the Green’s functions corresponding to the conserved charges, we follow the same strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Before we compute them, we first recall that in Bjorken flow τn(τ) = const .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (45) 17 In particular we have τn(τ) = τ0n(τ0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' we need a slightly different definition for the charge Green’s functions τδN µ a,k(τ) = ( ˜Fab)µ α(k, τ, τ0) τ0δN α b,k(τ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (46) Note that in general different flavours can couple to each other via the response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, for vanishing densities, there is no coupling between the response for different quark flavors and all flavors will have the same response functions, such that the response matrix is proportional to the identity in flavor space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' ( ˜Fab)µ α = ˜F µ α δab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We will decompose the charge Green’s functions also in a scalar-vector-tensor basis such that we have ˜F τ τ (k, τ) = ˜F s s (κ, x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (47) It is possible to express the response functions in terms of δN µ a,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Adapting the normalisation τ0δN τ k(τ0) = 1 (48) we find ˜F s s (κ, x) = τδN τ κ(x) = τδnκ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (49) Note that we also drop the index a on the components δN i k as they will be the same for all species for vanishing background number charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Numerical Results for the non-equilibrium Green’s Functions We present the results for the response functions ˜Gs s and ˜F s s in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 1, where we plotted the different response functions in dependence of the propagation phase κ and time ˜w (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (24)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The different panels correspond to the different response functions and are labelled by the components of the energy-momentum tensor and the charge current that they affect, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' ˜Gs s is labelled by “energy response“ as this function describes the response of δT ττ k (τ), which corresponds to the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Each curve in each panel corresponds to the response function at a different time as it is indicated by the colour code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Besides this we also plotted the free-streaming behaviour for each response function, which corresponds to the black line at ˜w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 5 10 15 20 Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] Energy response: G~ s s(w~,k∆τ) Wave number: κ=k∆τ free streaming 0 1 2 3 4 5 Figure 1: Left: Evolution of the energy-momentum Green’s function ˜Gs s in response to initial energy perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Right: Evolution of the charge Green’s function ˜F s s in response to initial charge perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The different curves in each panel correspond to different times ˜w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Due to the fact that we consider the perturbations for the charge density and chemical potential around vanishing background densities, the evolution of the response function ˜Gs s does not change in comparison to the results in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This can already be seen at the level of the equation of motion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B24a) for vanishing densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For early times ( ˜w ≪ 1) one observes the free-streaming behaviour, characterized by wave-like modes with both peaks of excess density and troughs of depleted density (the diffusion wake).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Towards later times larger κ-modes become damped, which can be explained by viscous effects of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' At the onset of the hydrodynamic regime ( ˜w ∼ 1) only long wave-length modes survive, which indicates that the free-streaming initial conditions are getting washed out during the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The shift of the peak for later times towards larger values of the propagation phase can be understood by noting that at early times, shortly after the collision, the system is highly anisotropic and expands in the transverse plane with a phase- velocity close to the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As the system evolves in time, it will become more and more isotropic and the phase-velocity will approach the speed of sound resulting in the shift of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For the charge density response ˜F s s we see that for early times ( ˜w ≪ 1) the behaviour is similar to the one of ˜Gs s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However for later times the damping of the modes sets in earlier than for ˜Gs s, such that for κ ≳ 10 there are already no visible deviations from zero any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Due to the damping of these functions we see that at ˜w ∼ 1, when the hydrodynamic 19 freestreaming 5 1 Evolution time: W=TTefr(t)[4nTef/(e+P)] 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 3 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1 0 0 5 10 15 20 Wavenumber:K=k△tFigure 2: Left: Evolution of the energy Green’s function ˜Gs s in response to initial energy perturba- tions in coordinate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Right: Evolution of the charge Green’s function ˜F s s in response to initial charge perturbations in coordinate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The different curves in each panel correspond to different times ˜w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' regime sets in, again only long wave-length modes will survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Moreover, the absence of oscillations in the spectrum signals the transition from propagating to diffusive behavior of the charge perturbations, as will become evident in coordinate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We note that these results are obtained for perturbations around zero density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Therefore further studies are necessary to clarify the impact of perturbations around non-vanishing densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In particular, considering non-vanishing densities will remove the degeneracy between the flavours leading to interesting phenomena like cross-diffusion [51–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions in Coordinate Space So far we computed the Green’s functions in Fourier space, which provides useful insight into the underlying physics and dynamics of such far-from-equilibrium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, the Green’s functions in position space will provide additional and useful information to understand the system’s evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Similar to the decomposition in Fourier space, we can decompose the Green’s functions in coordinate space as well into a basis of scalars, vectors, and tensors, such that for the two Green’s functions we find Gττ ττ(r, τ) = Gs s(|r|, τ) , (50a) F τ τ (r, τ) = F s s (|r|, τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (50b) 20 2 freestreaming 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 Evolution time: w=T(t)t / (4πn/s) 4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 3 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 Distance:Ax/△t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 free streaming 5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 4 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 Distance:Ax/△tfor the energy-momentum tensor and charge current, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The relation to their counterparts in Fourier space is given by the following Fourier-Hankel transforms Gs s(|r|, τ) = 1 2π � d|k| |k|J0(|k||r|) ˜Gs s(|k|, τ) , (51a) F s s (|r|, τ) = 1 2π � d|k| |k|J0(|k||r|) ˜F s s (|k|, τ) , (51b) where Jν are the Bessel functions of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The results for the Green’s functions in coordinate space are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The corresponding Green’s function is plotted as a function of ∆x/∆τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' the propagation distance in units of the elapsed time, while the color coding indicates the evolution time ˜w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the evolution of Gs s the propagation of sound waves is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the free streaming evolution and also still at early times the waves propagate with (almost) the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Towards later times, the peak shifts to smaller values of ∆x/∆τ approaching the speed of sound cs = � 1/3 and exhibits a negative contribution at small ∆x/∆τ which corresponds to the diffusion wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Since at early times the net charge density is carried by free-streaming particles, the charge response F s s has the same free-streaming behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' One observes, that this behavior of the charge Green’s function F s s persists up to ˜w ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Subsequently, the Green’s function transitions to a different behavior, where one can clearly see the diffusion of charges that results in a pronounced peak centered around ∆x/∆τ = 0, and no longer the free propagation of charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' APPLICATIONS Now that we have obtained both the energy-momentum and charge dependent Green’s functions, we can couple them to ICCING initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The upgraded version of ICCING 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 will be available on GitHub2 upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We will study the impacts of combining linearized pre-equilibrium Green’s functions with initial geometries produced by ICCING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As we will show, the assumption of linear response leads to nontrivial effects on the resulting energy and charge perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='com/pcarzon/ICCING 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Using Green’s Functions in ICCING The starting point of the construction of an initial state profile of the energy-momentum tensor and the conserved charges, is the generation of an initial energy density profile based on the initial-state model TRENTO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='3 Then, this energy density at τ0 is used to compute the background energy density at any proper times τ > τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To describe the fluctuations of conserved charges about this background, we use ICCING [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The ICCING algorithm is a Monte Carlo event generator run subsequent to TRENTO which simulates the fluctuations due to g → q¯q pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In particular, ICCING generates 2 + 1D distributions of the fluctuating BSQ charge densities: baryon number, strangeness, and electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' By incorporating the Green’s functions into the energy and charge redistribution algorithm in ICCING, we can study the impact of perturbations in both energy and charge on the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A single gluon splitting produces two types of perturbations relative to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The first is a negative energy perturbation (“hole”) due to removing a gluon from the back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The second is a positive energy perturbation, displaced relative to the gluon, which deposits the energy density corresponding to the quark/anti-quark pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' All three (quark, antiquark, and gluon) can be treated as perturbation around the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The propa- gation of the perturbations generated by ICCING are treated via the Green’s function Gs s until some time τhydro when hydrodynamics becomes applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In addition to the perturbations in the energy densities, the charge densities of quarks created by the gluon splitting process can also be evolved by the same formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Since TRENTO does not provide any charge information, we can treat the quark charges as perturbations around a vanishing background charge density, which is exactly how the charge Green’s functions are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' First we evolve the background according to the energy attractor E( ˜w) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (27), then we can use the Green’s functions Gs s and F s s in order to describe the propagation of energy and respectively charge perturbations that occur whenever a splitting happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 3 In practice, the output of TRENTO is a “reduced thickness function” which is taken to be proportional to the entropy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The coefficient of proportionality is fixed by comparison to experimental multiplicities in central collisions, and the properly-normlaized entropy density is converted to an energy density using the lattice QCD based equation of state from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 22 We can compactly express this as e(τhydro, x) = eBG(eTrento(τ0), x) � 1 + � ⊙ d2x0 (∆τ)2(∆τ)2Gs s �|x − x0| ∆τ , ˜w � 1 eTrento(τ0, x0) × (δeq(τ0, x0) + δeq(τ0, x0) − δeg(τ0, x0)) � , (52a) ni(τhydro, x) = � ⊙ d2x0 (∆τ)2(∆τ)2F s s �|x − x0| ∆τ , ˜w �τ0 τ [δnq(τ0, x0) − δnq(τ0, x0)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (52b) where we integrate the Green’s function evolution over all x0 in the past causal light cone |x0 − x| < ∆τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Furthermore x is the point of interest and ∆τ ≡ τhydro − τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We have implemented the pre-equilibrium evolution given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (52) in a new C++ class GreensFunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='h which interacts with the Event class Event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='h in nontrivial ways and can be found at the GitHub link above upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The energy and BSQ charge densities of a single, peripheral ICCING PbPb event at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 TeV evolved for 1 fm/c using the Green’s functions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 3 for our default parameter set (see Table 2 in [56], with the exception of τ0 which here equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1 fm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The behaviour of the baryon and electric charge distributions are similar to default IC- CING [56] and follow the bulk geometry while the strangeness distribution is more rarefied due to the larger mass threshold required to produce strange quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A significant difference is seen in the size of the charge fluctuations, whose radius now depends on the evolution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In order to understand the effects of applying the Green’s functions, we can look at individual quark splittings, the background evolution, and radius dependence separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To start, it is important to have a grasp on the full effect the evolution has on a single quark/anti-quark pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The energy density and charge density for a quark splitting evolved for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 fm/c and 1 fm/c are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 4 occurs soon after the quark splits and the bottom panel is at the end of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For small evolution times, we clearly see three different types of perturbations in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 4: (mostly) positive-energy perturbations corresponding to the deposition of the q¯q pair, and a (mostly) negative-energy perturbation corresponding to the subtraction of the parent gluon from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The quarks also come with associated positive / negative charge densities being deposited, whereas the gluon subtraction has no impact on the charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The dominant effect seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 4 is that the energy and charge perturbations grow in size 23 Figure 3: Density distributions for an ICCING event with Green’s function evolution of energy and charge perturbations from g → q¯q splittings after 1fm/c of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' over time and have a wave-like structure leading to non-trivial interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Note that the central positions of the quarks do not change due to the evolution prescribed by the Green’s functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' rather, they are determined from the g → q¯q splitting function used by ICCING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The Green’s functions do not interact with any part of the quark sampling algorithm in ICCING and only determine how the energy and charge densities of the perturbations are distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Next, we can look at how the Green’s functions distribute the energy and charge for the quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To illustrate the spatial profiles of energy and charge density produced by the Green’s functions, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 5 shows the results of a single g → q¯q splitting in a low-temperature 24 12 5 0 5 12-12 5 0 5 12 12 12 Energy Density (GeV I fm3) Baryon Density (fm-3) 5 5 (wy) 0 5 0 4 8 13 17 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='040-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='014 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='015 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='053 12 12 5 Y 5 12 5 0 12-12 5 0 12 x (fm) x (fm)Figure 4: Density distributions for a single strange quark splitting compared for evolution times of τhydro = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2fm, on the left, and τhydro = 1fm, on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' region (top) and a high-temperature region (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The Green’s functions have different behavior depending on the local value of the initial energy density e(τ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This dependence arises because the natural unit of time ˜w depends on the effective temperature T (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (24)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As a result, splittings which occur in hot spots transition more quickly from propagating behavior for small ˜w to diffusive behavior for large ˜w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This is clearly seen in the charge density plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The top panel, where the splitting occurs at low temperatures, retains significant spatial structure of the charge distribution associated with the propagating modes of the Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' But if the splitting occurs at higher temperatures (bottom panel), the charge density is distributed according to the diffusive modes from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This results in charge distributions which wash out the spatial structure of the Green’s functions as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' There is also a connection here to the Knudsen number (Kn) since Kn = τR/τ ∝ (τT)−1 so ˜ω ∝ Kn−1 [43] such that at late times one expects a smaller Kn number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This provides physical intuition for the dependence of the charge fluctuation on the location of splitting, 25 Energy Density (GeV/fm3 Charge Density (fm-3) (wy) y 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='10 3 2 1 0 1 2 3 3 2 1 2 3 x (fm) x (fm)Energy Density (GeVifm3 Charge Density (fm-3) 1 0 (wy) y 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='015-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='010-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 3 2 1 0 1 2 3 3 2 1 1 2 3 x (fm) x (fm)Figure 5: Density distributions for a single strange quark splitting from different areas of the event: a cold region on the top, and a hot region on the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Here the separation of the two quarks is artificially increased to better illustrate the behaviour of the Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' hotter spots in the medium will have larger Kn and thus produce more Gaussian charge densities while cold spots will have smaller Kn and produce charge densities in a shock wave form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The implications of this difference in behavior based on the location of splitting may become more important when analyzing events across systems of different energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' While this result is interesting and an effect of the physics included in the Green’s func- tions, it could be worrying since one of the core assumptions of the ICCING algorithm is that all charge must be correlated with some energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The problem with a linearized treat- ment of the Green’s functions is that large negative corrections to the energy density could overturn the background energy density, resulting in grid points with net negative energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This problem would be nonphysical and require some sort of remedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Moreover, even if the net energy density is not driven negative by a large perturbation, one may still be unable to match a fluid cell with very low energy density but very high charge density to a reasonable equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' These potential problems arising from large perturbations could be solved by going back to 26 Energy Density (GeVifm3) Charge Density (fm-3 0 (wy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='015-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='010-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='10 0 2 0 2 4 x (fm) x (fm)Figure 6: Comparison of εn{2} across energy and BSQ distributions for different Green’s function evolution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' the linearized approximation made in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' II C, which is broken when the local redistributed energy or charge density is greater than or close to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' There are several ways in which to solve this problem, the first of which would be by artificially damping the magnitude of the perturbations relative to the background in order ensure that the linearization remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This would introduce a new problem though, since the artificial damping may not affect a q¯q pair equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' If the quark is deposited in the periphery of the event, but the antiquark is deposited closer to the center, then the positive and negative charge densities will be damped in different amounts, leading to a violation of charge conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To correct this, one could suppress the quark and anti-quark in the same way mirroring the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Another possible solution – the one we pursue here – is to veto any quark splittings that would create energy density perturbations that are large with respect to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This is a very simple solution but adds a new complication because it effectively eliminates quark production at the edge of the event, and reducing the "cold quark" Green’s functions contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Another unintended effect of this solution would be that a quark/anti-quark pair produced near the periphery with a smaller radius, for example from evolving for only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 fm/c, would survive the veto, but a pair evolved for longer time would be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Evidently, this problem could be cured by also including the transverse expansion of the background energy density in the pre-equilibrium stage, but that is beyond the scope of this paper and is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Despite its shortcomings, the solution of vetoing quark splittings if the energy perturbation is not small compared to the background has been chosen here both for its simplicity and its flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 Trento : Background* ICCING (Default) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content="8 : - ICCING (Green's Functions) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 (z)23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 PbPb [5TeV1, *△t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 20 40 60 80 100 0 Centrality (%)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 Trento : Background* ICCING (Default) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content="8 ICCING (Green's Functions) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 E3(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 PbPb [5TeVl, *△t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 20 40 60 80 100 0 Centrality (%)Because our procedure should be only a small effect on the total energy density, we do not expect large changes to the energy density eccentricities, which are defined in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Thus, before looking at the eccentricities of the charge densities, we should look at the effect that the different processes have on the energy density with the hope that any affect is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Since the energy density eccentricities are a good predictor of the final state and these initial conditions have been used extensively in comparisons to experimental data, the hope is for a minimal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 6, the energy ellipticity and triangularity is plotted for the original trento event, the locally evolved background used for the Green’s function evolution, the trento event after default ICCING, and the full ICCING coupled to Green’s functions simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' When comparing the evolved background with the trento profile, we observe small changes at the percent level which can be attributed to the phenomenon of inhomogenous longitudinal cooling [36, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In essence, thermalization proceeds more quickly in more highly energetic regions, leading to a slightly faster decrease of the energy density of the hotter regions of the QGP as compared to the colder regions of the QGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, as we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 6, in practice this effect is rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Similarly, we see that for default ICCING, there is a slight modification in peripheral events and the most central events that should not have a significant effect on the agreement with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Adding both the modifications from the ICCING sampling and the Green’s functions evolution, has no significant effect on the energy geometry beyond the background evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This indicates that any small changes in energy density distribution generated by ICCING are quickly washed out and won’t make it to the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Additionally, previous comparisons to experimental data for all charge particles should still be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' IMPACT ON ECCENTRICITIES Now let us look at the contributions, that different parts of the Green’s function evolution have on the event averaged eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Because we will be dealing with time dependent quantities, we will define the time evolution for applying the Green’s function: ∆τ ≡ τhydro − τ0 (53) where τ0 is our initial time when we begin the Green’s function evolution and τhydro is where we stop the evolution and switch to hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We will start with studying the 28 consequences of our perturbative cutoff effect on the eccentricities, then we will compare the Green’s function expansion to a trivial Gaussian smearing to determine any non-trivial effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' After these two effects are studied we will explore the time dependence of the Green’s function on various eccentricities, which are the main results for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Eccentricities of charge are defined the same as for energy, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' E, except that the center of mass is taken to be that of the energy density and the observable is calculated for the positive and negative charge densities separately, since otherwise the observable would be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' When there is no charge density from quark/anti-quark splittings, the eccentricity is defined as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' While adequate, the definition of the eccentricities for energy are not the best possible estimators for the the charge density and more development can be made in this direction [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' First, we study the effect of the suppression of gluon splitting to ensure positive energy densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In order to avoid problematic regions with negative energy density, we restrict quarks from splitting if their redistributed energy densities approach a certain threshold compared to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The selection criteria is examined for each point in the quark densities and is determined by Eq/Ebg < P, (54a) where P is the perturbative cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 7, several values were selected for an evolution time of ∆τ = 1fm/c to illustrate the effect this cutoff has on the charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Both ε2 {2} and ε3 {2} are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The effect of applying the perturbative cutoff vs no cutoff at all is clearly the dominate effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This signifies that there are many quark/anti-quark pairs above 40% Centrality that produce negative energy and thus the mismatch between the locally evolved background and non-local quark perturbations is quite significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 and evolution of 1fm/c, the percentage of events that produce no quarks at all in the 65 − 75% Centrality class is 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The peak of the charge eccentricities thus signifies that number effects dominate the charge geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' With such an extreme response to this perturbation parameter it is reasonable to assume the model, as it is currently formulated, breaks down at this point and should not be used beyond an evolution time of 1fm/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, we do not find a significant difference between the most generous value of a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 cutoff vs the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 mark with only small change when P goes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the remaining results we will explore only the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 cutoff since we do not anticipate a strong dependence on the 29 Figure 7: Comparison of εn{2} for different perturbation cutoff values with a Green’s function evolution of ∆τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0fm/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The solid line is with a Green’s function but no cutoff, default ICCING is not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' cutoff for other observables as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Next, we will try to disentangle the effect of the expanding radius from the structure introduced to the quark densities based on the background energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In our Green’s function approach, the overall size of the quarks expand over time and that may be the dominate (albeit trivial) effect of applying the Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Thus, to determine any non-trivial consequences of the Green’s function, we apply a simple Gaussian smearing to the quarks, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 8, and compare the Gaussian smearing, defined as: Gs s(r, t) = F s s (r, t) = exp(−r2/R(t)2) πR2(t) , (55) to the Green’s function method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 9, we see that there is a negligible difference between the Green’s function and a simple Gaussian smearing, implying that the dominant effect, when looking at event averaged geometry, is the size of the density perturbations and not the structure introduced by the Green’s Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 10, we show ε2{2} adding back in the perturbation cutoff and evolving for ∆τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0fm/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The solid curves are from default ICCING without any pre-evolution, the dashed curves add in evolution but only allow the radius of the quark and gluon density perturbations to change while holding the density profile fixed, and the dotted curves add in the full Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Several things are happening in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 10 that need to be dis- entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' First, the Gaussian smearing with the peturbative cut-off (compared to default ICCING with no time evolution) has the general effect of shifting the peak in ε2 {2} to lower centralities and also leading to a larger ε2 {2} in central collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The shift from the 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 Energy No Correction Baryon (+) Eq/EBg <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 Strange (+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' EqEBg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 Charge (+) Eq/EBg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 (z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 PbPb [5TeVl, △t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 0 20 40 60 80 100 Centrality (%)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 Energy No Correction Baryon (+) E/EBg <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 Strange (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 Charge (+) Eq/EBg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 E3(2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 PbPb [5TeV1, △t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 20 40 60 0 80 100 Centrality (%)Figure 8: Illustrative density profiles of the Gaussian smearing option at ∆τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0fm/c which separates structure introduced by the Green’s functions from the radial dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Figure 9: Comparison of ε2{2} between default ICCING, Gaussian Smearing, and Green’s Functions with an evolution of ∆τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0fm/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The perturbation cutoff is not used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Gaussian smearing compared to default ICCING occurs both because as the quarks grow in size the positive and negative densities will cancel out more and wash out the geometry in regions with low densities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' peripheral collisions) and because quark/anti-quark splitting is suppressed due to the perturbation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Now that we know the effect of just a trivial Gaussian smearing, the next question is what 31 Energy Density (GeV/fm3 Charge Density (fm-3 2 F 1 (wy) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='03 2 E 3 2 1 0 1 2 3 4 -3 2 1 0 1 2 3 4 x (fm) x (fm)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 Energy ICCING (Default) Baryon (+) ICCING (Gaussian Smearing) Strange (+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=". ICCING (Green's Functions) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 Charge (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 (z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 PbPb [5TeVl, △t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 20 40 60 80 100 0 Centrality (%)Figure 10: Including perturbation cutoff of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 for comparison between Default ICCING, Gaussian smearing, and Green’s functions for ∆τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0fm/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' effect does the non-trivial Green’s function have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 10 we can see that the Green’s function shifts the peak of the eccentricities even further to lower centralities for all BSQ densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To understand this effect, let us break down the fundamental differences between a trivial Gaussian smearing and the Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' There are two differences between the Gaussian smearing and Green’s functions density perturbations one of which is that the density profile of the Gaussian smearing is smooth and mostly uniform with sharp edges and only negative values of energy coming from the gluon hole, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The Green’s functions, on the other hand, have a density profile that has a wave structure with the largest energy density values coming from the ring at the edge of the quarks and gluons, as previously shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The Green’s function density profiles also contain negative energy at the center of the quarks and a large amount around the edge of the gluon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Applying the perturbative cutoff removes any net negative energy from the final output in these two methods, which strongly affects Green’s function method because of the concentration of the energy density around the edge of the quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' While the Gaussian smearing also breaks perturbative assumptions the effect is much smaller than for the Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 Energy ICCING (Default) Baryon (+) ICCING (Gaussian Smearing) Strange (+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=". ICCING (Green's Functions) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 Charge (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 (z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 PbPb [5TeV] △T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm, Eq/EBg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 20 40 60 80 100 0 Centrality (%)structure of the Green’s function density perturbations is relatively ’microscopic’ and so when compared against the Gaussian smearing, without the perturbation cutoff in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 9, there is no difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Since the perturbation cutoff is defined here as ’microscopic’, then a difference is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 10 when including the more complicated structure of the Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The sensitivity to ’microscopic’ differences in the density perturbations may disappear with different choices of the perturbation cutoff method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, the unique structure of the Green’s function density perturbations will still be important when coupling to hydrodynamics since there would be a non-trivial change to gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Finally putting all the pieces together, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 11 shows the Green’s functions evolution with the perturbative cutoff for different evolution times, ∆τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 fm and ∆τ = 1 fm, for both elliptical (left) and triangular eccentricities (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For an evolution time of ∆τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5fm/c, we consistently see a shift in the peak of all BSQ charge eccentricities toward the left, reflecting an increase in the dominance of number effects on the geometry as supported by the rarity of quark producing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, for ∆τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5fm/c most central collisions do not appear to be strongly affected by the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For an evolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0fm/c, there is a much greater suppression from the perturbation correction and this significantly affects all centrality classes, such that the most central collisions see enhanced eccentricities but peripheral collisions are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The shift in the peak towards smaller centrality classes (combined with a suppressed eccentricity in peripheral collisions) indicates that the model starts to break down the further in time the evolution is pushed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Thus, there appears to be a small window in which we can apply the Green’s function expansion and still obtain a reasonable number of quark/anti-quark pairs (after applying the perturbative cutoffs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Generally, we find very similar qualitative behaviors in both ε2 and ε3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, ε3 is much more sensitive to BSQ densities and has the most significant difference between the energy eccentricities vs the BSQ eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Therefore, high-order harmonics will likely provide the best observable when comparison to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' One of the most important quantities for direct comparisons of initial state models to experimental data is εn{4}/εn{2} because medium effects cancel in the most central collisions (especially for n = 3 [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The eccentricity ratios, εn{4}/εn{2}, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 12 for n = 2 (left) and n = 3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The ratio εn{4}/εn{2} measure the fluctuations of geometry with values close to 1 indicating few fluctuations wheres small values indicate a large amount of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Comparing elliptical and triangular flow, we find quite different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For 33 Figure 11: Comparison of εn{2} across energy and BSQ distributions for different Green’s function evolution times using the perturbation cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' elliptical flow, for more centrality to mid-central collisions the fluctuations appear to be nearly identical to the energy density fluctuations (although ultra central collisions have some small differences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, for peripheral collisions where the perturbative cutoff plays a strong role, then we see there is always a centrality wherein large deviations are seen compared to the energy density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Electric charge and baryon density fluctuations, for default ICCING, are nearly identical to the energy density fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, the longer you have a Green’s function evolution, then you see deviations at lower and lower centralities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' for ∆τ = 1 fm, the deviation occurs at ∼ 40% centrality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Naturally for strangeness this effect is larger because one is dealing with a smaller number of quark/anti- quark pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The effect for triangular flow is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Generally, we find that the application of ICCING leads to an overall decrease in the triangular flow fluctuations, regardless of the BSQ charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Additionally, the Green’s function evolution appears to enhance that effect further for ∆τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In contrast, for elliptical flow we did not see this effect and the fluctuations were the same (at least within some centrality classes) before and after applying ICCING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' That being said, we do find that the effect of certain centralities being strongly affected by the perturbative cutoff showing up in triangular flow as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' These are features that could eventually be looked for in experimental data, if measurements of vn{4}/vn{2} are made with identified particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 Energy Default Baryon (+) At = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 fm Strange (+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' At = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 Charge (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 (z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 PbPb [5TeV], Eq/EBg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 20 40 60 80 100 0 Centrality (%)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 Energy Default Baryon (+) At = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 fm Strange (+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='- At = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 Charge (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 E3(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 PbPb [5TeV], Eq/EBg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 20 40 60 0 80 100 Centrality (%)Figure 12: Comparison of εn{4}/εn{2} across energy and BSQ distributions for different Green’s function evolution times using the perturbation cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' CONCLUSION & OUTLOOK We extended the method to compute non-equilibrium Green’s functions of the energy- momentum tensor developed in [43] by adding conserved charges and computing the corre- sponding Green’s functions for the charge current for perturbations around vanishing back- ground charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Using the ICCING model that initializes conserved charges through g → q¯q splittings, we successfully coupled these Green’s function to ICCING allowing for a pre-equilibrium phase with conserved charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The inclusion of this pre-equilibrium evolu- tion is a non-trivial addition to the ICCING algorithm and the successful implementation demonstrates the flexibility of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In order to quantify the system’s response to initial perturbations in terms of Green’s functions, it is necessary to consider the background evolution (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A 4) which is used in the energy evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We find that the systems dynamics can be quantified in terms of the moments δE m l,k, δN m al,k (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (37)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Furthermore the Green’s functions can be obtained directly from these moments, which makes this method a powerful tool to obtain the response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' When comparing the energy and charge Green’s function we see distinct differences in their behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the energy case we find the propagation of sound waves, where in free-streaming and even at early times they propagate with almost the speed of light, while at later times this shifts towards the speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In contrast, for the evolution of conserved charges we find a transition from free-streaming propagation in the beginning to a diffusive behavior at late time as the system continues to thermalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To understand the effect the Green’s functions have on initial state charge geometries, we 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 PbPb @ 5TeV, EglEBg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 E2(4)/e2(2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 Energy Baryon (+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 Strange (+ Charge (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 Default △t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 fm △t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 0 20 40 60 80 100 Centrality (%)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 PbPb @ 5TeV, EqlEBg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 E3(4)/E3(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 Energy Baryon (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 Strange (+) Charge (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 Default At = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 fm △t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 fm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 0 20 40 60 80 100 Centrality (%)compare between the default version of ICCING and ICCING with the Green’s functions, supplemented by an approximation which simplifies the spatial structure of the Green’s functions to simple Gaussian smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We see that for εn{2} there is no difference when including the complicated structure of the Green’s functions to the Gaussian smearing, although that structure becomes important for observables sensitive to ’microscopic’ differ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A mismatch between the evolution of the background, described as a local process, and the charge perturbations, described as a non-local process, leads to the possibility of sites with negative energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This issue is fixed by suppressing quark/anti-quark production that would violate some perturbative condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This pertubative corrective measure signif- icantly suppresses quark/anti-quark production in peripheral events but less so in central to mid-central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The primary difference then between default ICCING and ICCING with the Green’s functions arises from a combination of smearing effects that occur during an ex- pansion in time and the suppression of non-perturbative quark-anti-quark pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This leads to large eccentricities in central collisions but nearly vanishing eccentricities in peripheral collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This work constitutes the first step toward including charge evolution in KøMPøST and illustrates the effect pre-equilibrium evolution has on conserved charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' An imple- mentation of this method in KøMPøST would solve the mismatch between the background and perturbation evolutions which are local and non-local, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Exploring the effect of the pre-equilibrium evolution of conserved charges on the hydrodynamic evolution of the system and on final state observables would be beyond the scope of this paper but is an interesting open question that will be explored in a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' It would also be interesting to compute these Green’s functions for charges in QCD kinetic theory and compare them to the approximation introduced in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Another possible direction is extending these Green’s functions around a non-vanishing background which would be useful when looking at systems that contain baryon stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Acknowledgements P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='S acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, Ger- man Research Foundation) through the CRC-TR 211 ’Strong-interaction matter under ex- treme conditions’– project number 315477589 – TRR 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' acknowledges 36 the support from the US-DOE Nuclear Science Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' DE-SC0020633 and the support from the Illinois Campus Cluster, a computing resource that is operated by the Illinois Cam- pus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing Applications (NCSA), and which is supported by funds from the University of Illinois at Urbana-Champaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' is supported by a start-up grant from New Mexico State Uni- versity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The authors also acknowledge computing time provided by the Paderborn Center for Parallel Computing (PC2) and the National Energy Research Scientific Computing Cen- ter, a DOE Office of Science User Facility supported by the Office of Science of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Department of Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' DE-AC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 37 Appendix A: Background evolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Evolution Equations for the spherical harmonic moments In order to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (18), we will adopt the ideas of [48], where, instead of finding solutions for the distribution functions, one studies the moments of the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For the distribution functions fBG and fa,BG we will consider the following moments E m l (τ) = τ 1/3 � dpη (2π) � d2p (2π)2pτY m l (φp, θp)fBG(τ, pT, |pη|) , (A1a) N m al (τ) = � dpη (2π) � d2p (2π)2Y m l (φp, θp)fa,BG(τ, pT, |pη|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A1b) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A1) the angles are defined by tan φp = p1/p2 and cos θp = pη/(τpτ), while Y m l are the spherical harmonics given by Y m l (φ, θ) = ym l P m l (cos θ)eimφ (A2a) with ym l = � (2l + 1)(l − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 4π(l + m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A2b) and P m l (x) = (−1)m 2ll!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' � 1 − x2�m/2 dl+m dxl+m � x2 − 1 �l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A2c) being the associated Legendre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Based on the explicit form of the spherical harmonics one can find the non-vanishing components of the background energy-momentum tensor by low order moments as well as the tracelessness condition e(τ) = √ 4π τ 4/3 E 0 0(τ) , (A3a) PT(τ) = √ 4π τ 4/3 � 1 3E 0 0(τ) − � 1 45E 0 2(τ) � , (A3b) PL(τ) = √ 4π τ 4/3 � 1 3E 0 0(τ) + � 4 45E 0 2(τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A3c) Furthermore, by plugging in the equilibrium distribution function, we find that E m l �� eq(τ) = τ 4/3 √ 4πe(τ)δl0δm0 , (A4) 38 representing the rotational symmetry of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the same way we are able to reconstruct the components of the charge current via low-order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The net particle number of specie a is given by na(τ) = √ 4π τ N 0 a0(τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A5) The chemical potential can be extracted by inverting the Landau conditions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Applying τ∂τ to the definitions of the moments and using identities for Legendre Poly- nomials (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D) leads directly to the equations of motion, which are given as τ∂τE m l (τ) = bm l,−2E m l−2(τ) + bm l,0E m l (τ) + bm l,+2E m l+2(τ) − τ τR � E m l (τ) − E m l (τ)|eq � , (A6a) τ∂τN m al (τ) = Bm l,−2N m al−2(τ) + Bm l,0N m al (τ) + Bm l,+2N m al+2(τ) − τ τR � N m al (τ) − N m al (τ)|eq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A6b) The appearing coefficients bm l and Bm l are given by bm l,−2 = (2 + l)(l + m − 1)(l + m) (1 − 4l2) � (2l + 1)(l − m − 1)(l − m) (2l − 3)(l + m − 1)(l + m) , (A7a) bm l,0 = −5 3 l(l + 1) − 3m2 4l(l + 1) − 3 , (A7b) bm l,+2 = (l − 1) (2l + 3) � (l − m + 1)(l − m + 2)(l + m + 1)(l + m + 2) (2l + 1)(2l + 5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A7c) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Bm l,−2 = (1 + l)(l + m − 1)(l + m) (1 − 4l2) � (2l + 1)(l − m − 1)(l − m) (2l − 3)(l + m − 1)(l + m) , (A8a) Bm l,0 = −l(l + 1) − 3m2 4l(l + 1) − 3 , (A8b) Bm l,+2 = l (2l + 3) � (l − m + 1)(l − m + 2)(l + m + 1)(l + m + 2) (2l + 1)(2l + 5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A8c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Initial Conditions In order to solve the equations of motion for E m l and N m al we need to specify the initial conditions for the moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For early time dynamics at τ ≪ τR the system cannot maintain considerable longitudinal momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Therefore the initial distribution is naturally of the form 39 that the transverse momentum is much larger than the longitudinal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Taking also into account previous results [57–63] one sees that the case of a (longitudinal) support in form of a Dirac delta function corresponds to a non-equilibrium attractor of the kinetic equations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' that different initial conditions will approach the same curve for later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We therefore choose fBG(τ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' pT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' |pη|) = (2π)3δ(pη) � 1 νg dN0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='g dη d2p d2x + 1 νq � a � dN0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='qa dη d2p d2x + dN0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='qa dη d2p d2x �� ≡ (2π)3δ(pη) d ˜N0 dη d2p d2x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A9a) fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='BG(τ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' pT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' |pη|) = (2π)3δ(pη) 1 νq � dN0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='qa dη d2p d2x − dN0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='qa dη d2p d2x � ≡ (2π)3δ(pη) d ˜N0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='a dη d2p d2x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A9b) where we choose the normalization such that the initial energy and charge densities are kept constant dE0 dη d2x = dE0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='g dη d2x + � a � dE0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='qa dη d2x + dE0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='qa dη d2x � = lim τ0→0 τ0e(τ0) = (eτ)0 = const ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A10a) dN0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='a dη d2x = dN0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='qa dη d2x − dN0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='qa dη d2x = lim τ0→0 τ0na(τ0) = (naτ)0 = const .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A10b) On the level of the moments the initial conditions are given by E m l (τ0) = τ 1/3 0 (eτ)0 ym l P m l (0)δm0 , (A11a) N m al (τ0) = (naτ)0 ym l P m l (0)δm0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A11b) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Relation of Intensive and Extensive Quantities In contrast to the cases in [43], we additionally need to invert e = T 4 �νgπ2 30 − 3νq π2 � a � Li4 � −z−1 a � + Li4(−za) �� , (A12a) na = νq T 3 π2 � Li3 � −z−1 a � − Li3(−za) � = νq 6π2 � π2T 2µa + µ3 a � , (A12b) with za ≡ exp(µa/T) to determine the temperature T and the chemical potentials µa as a function of energy density e and number density na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This is done numerically for each time 40 step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The relations are given by δT = −T χuχdχsδe − 3nuχdχsδnu − 3ndχuχsδnd − 3nsχuχdδns 9n2 uχdχs + 9n2 dχuχs + 9n2 sχuχd − 4eχuχdχs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A13a) δµu = [9(n2 dχs + n2 sχd) + (3nuµu − 4e)χdχs]δnu 9n2 uχdχs + 9n2 dχuχs + 9n2 sχuχd − 4eχuχdχs + −3αundχsδnd − 3αunsχdδns + αuχdχsδe 9n2 uχdχs + 9n2 dχuχs + 9n2 sχuχd − 4eχuχdχs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A13b) δµd = [9n2 sχu + (9n2 u + (3ndµd − 4e)χu)χs]δnd 9n2 uχdχs + 9n2 dχuχs + 9n2 sχuχd − 4eχuχdχs + −3αdnuχsδnu − 3αdnsχuδns + αdχuχsδe 9n2 uχdχs + 9n2 dχuχs + 9n2 sχuχd − 4eχuχdχs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A13c) δµs = [9n2 dχu + (9n2 u + (3nsµs − 4e)χu)χd]δns 9n2 uχdχs + 9n2 dχuχs + 9n2 sχuχd − 4eχuχdχs + −3αsnuχdδnu − 3αsndχuδnd + αsχuχdδe 9n2 uχdχs + 9n2 dχuχs + 9n2 sχuχd − 4eχuχdχs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A13d) Here χa is the susceptibility, which is given by χa ≡ νq 6 �3µ2 a π2 + T 2 � (A14) For the special case of perturbations around na = 0 the susceptibilities reduce to χa = νq 6 T 2 which yields then δT = T δe 4e , (A15a) δµa = 6 νq δna T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A15b) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Background Evolution in Conformal Systems In this section we will consider the evolution of a conformal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In conformal systems τR is proportional to the inverse temperature such that [46] τRT(τ) = 5 ˜ηT e + P = const , (A16) where ˜η is the shear viscosity, e the energy density, P the pressure and T the effective temperature of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' At this point we introduce the dimensionless time variable x = τ/τR, as this produces a natural time scale for the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Due to the change of variable we need to transform also the appearing derivatives according to τ∂τ = τ ∂x ∂τ ∂x = τ � 1 τR − τ τ 2 R (∂ττR) � ∂x = � 1 − 1 τR τ∂ττR � x∂x ≡ a(x)x∂x , (A17) 41 where we will call a(x) ≡ 1 − x∂ττR = 1 − 1 τR τ∂ττR (A18) the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As τR is not constant, the scale factor will take a complicated form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, it can be related to the moments again as we find a(x) = 1 + τ∂τT T = 1 − χuχdχs˜a(x)e + 3n2 uχdχs + 3n2 dχuχs + 3n2 sχuχd 9n2 uχdχs + 9n2 dχuχs + 9n2 sχuχd − 4eχuχdχs , (A19) where ˜a(x) = −4 3 + b0 0,0E 0 0 + b0 0,+2 E 0 2(x) E 0 0(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A20) The appearing quantities can be expressed in terms of the low order moments (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A3a), (A5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We note at this point that for na = 0 the scale factor reduces to a(x) = 1 + τ∂τT T = 1 − χuχdχs˜a(x)e −4eχuχdχs = 2 3 + 1 4 � b0 0,0E 0 0 + b0 0,+2 E 0 2(x) E 0 0(x) � , (A21) which is the form used in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Using the change of variables the equations of motion can be written in terms of x as a(x)x∂xE m l = bm l,−2E m l−2 + bm l,0E m l + bm l,+2E m l+2 − x � E m l − E m l |eq � , (A22a) a(x)x∂xN m al = Bm l,−2N m al−2 + Bm l,0N m al + Bm l,+2N m al+2 − x � N m al − N m al |eq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A22b) For our analysis we are varying the initial charge number densities in order to see its impact on the evolution of the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Nevertheless, we keep the ratios between the three species the same, namely nu,BG nd,BG = 8 7 , ns,BG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 , (A23) as these ratios correspond to typical values in heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Our results for a conformal system can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In conformal systems without conserved charges it was found that the evolution is controlled by the dimensionless time variable ˜w = τT(τ)/(4π˜η/s) [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We will generalise this to systems with conserved charges and choose to present the different quantities as functions of the dimensionless time variable ˜w = τT(τ) 4π (e + P) ˜ηT = 5 4πx , (A24) 42 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1 1 10 Chemical Potential/Temperature: µu/T Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='11 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='24 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='37 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='56 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 4 Longitudinal Pressure/Energy: PL/e Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='11 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='24 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='37 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='09 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1 1 10 Energy attractor: τ4/3 Tττ/(eτ4/3)∞ Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='11 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='24 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='37 (µu/T)eq=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='56 Navier-Stokes -- 1-2/(3πw~) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01 Figure 13: Background evolution for conformal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Top: µu/T ratio for different initial values for nu, Bottom left: Longitudinal pressure over energy for different values of (µu/T)eq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The coloured dashed curves in the PL/e plot correspond to the hydrodynamical behaviour at later times, PL/e = 1 3 − 4 9π ˜w for ˜w → ∞, Bottom right: Energy attractor for different values of (µu/T)eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The coloured dashed curves in the � τ 4/3e � / � τ 4/3e � ∞ plot correspond to the free streaming behaviour of the energy attractors at early times, � τ 4/3e � / � τ 4/3e � ∞ = 1 C∞ ˜w4/9 for ˜w ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' More details on how to fit the dashed curves in the two plots are given in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Inset plots are given in order to show that there are deviations between the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' On the two axes of the inset plots are the same quantities plotted as for the larger plot, but the labels are omitted for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' where T(τ) = � 30 νeffπ2e(τ) �1/4 and νeff = νg + 3 · 7 4νq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' At the top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 13 we show the different curves we obtain for the ratio µu/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We see that at late times, when hydrodynamics is applicable, the ratio becomes constant according 43 to �µa T � eq = 6 νq na T = 6 νq �νeffπ2 30 � 3 4�na e 3 4 � eq = const .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A25) In the bottom left panel we show the ratio of longitudinal pressure and energy, PL/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We see that the ratio is essentially zero at early times as the longitudinal pressure needs to build up first and that at late times a smooth transition to the hydrodynamical behaviour �PL e � vHydro = 1 3 − 4 9π ˜w (A26) is provided around time ˜w ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the corresponding figure this behaviour is indicated by the coloured dashed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Note that the validity of the hydrodynamic limit Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A26) is guaranteed for small values of (µu/T)eq, while for larger values of (µu/T)eq this is a priori not clear and needs further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We also see the effect of the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As one can see in the inset plot, the PL/e-ratio increases slower for for increasing (µu/T)-ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Nevertheless the differences are very small, which can be explained by the assumption that we choose the same relaxation scale for all particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' It is expected to improve the results if one assumes different time scales for gluons and for quarks like following the approach of [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Regarding the bottom right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 13 we show the results for the energy attractor � τ 4/3e � / � τ 4/3e � ∞ , (A27) where � τ 4/3e � ∞ = lim τ→∞ τ 4/3e(τ) = const (A28) describes the asymptotic energy density scaled with τ 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' It is convenient to consider τ 4/3e as this becomes constant at late times as ideal hydrodynamics predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The value of the constant can be obtained by the numerical solution of the equations of motion and depends on the chemical potential which is considered as the energy evolution couples to the charge number via the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In the figure we also show the free-streaming behaviour, which we can parametrise according to [43] � τ 4/3e � (τ 4/3e)∞ = C−1 ∞ ˜w 4 9 (A29) 44 at early times (corresponds to the dashed coloured curves) and the hydrodynamical be- haviour � τ 4/3e � (τ 4/3e)∞ = 1 − 2 3π ˜w (A30) at late times (corresponding to the black dashed curve) [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We emphasise that in the case of a conformal system we also observe a smooth transition from the early time free- streaming regime to the late time viscous hydrodynamical regime, which starts to describe the evolution around times ˜w ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Regarding the free-streaming behaviour we fit the energy attractor at early times for the curves corresponding to different chemical potentials using � τ 4/3e � (τ 4/3e)∞ = C−1 ∞ ˜w 4 9 (A31) to extract the values of C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The results can be seen in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We emphasise that the role of C∞ is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' At late times the system can be described by viscous hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, the approach to this regime depends on the theory, such that different theories approach viscous hydrodynamics differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This difference in the approach to the late time behaviour results in a mismatch of the ratios of initial energy density to the final energy density (see [43] for a comparison of KøMPøST QCD kinetic theory to results obtained in conformal relaxation time approximation without conserved charges), where C∞ is used to express the late time energy density in terms of the initial energy density, such that we find Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A31) at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In Yang-Mills kinetic theory one finds C∞ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='9 [24, 39, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Looking at Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (I) we see that in conformal relaxation time approximation with conserved charges the value of C∞ increases as we increase the initial charge number density, however it will stay below the value in [24, 39, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Table I: Values of C∞ obtained by numerical fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (µu/T)eq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='56 C∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='87643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='87712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='87927 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='88374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='89349 45 Appendix B: Perturbations Around Bjorken Flow 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Linearised equations of motion and Landau matching By linearising the kinetic equations around the boost invariant and homogeneous back- ground one finds an evolution equation for the perturbation of the distribution functions δf and δfa � pτ∂τ + pi∂i − pη τ 2∂η � δf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) = −pτ τR δf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) + pµδuµ(x) τR � (feq − f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq � − pτ τR δT(x) T(τ) �T(τ) τR ∂τR ∂T (feq − f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq � + pτ τR � a δµa(x) � f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) qa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq + f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) qa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � (B1a) and � pτ∂τ + pi∂i − pη τ 2∂η � δfa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) = −pτ τR δfa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p) + pµδuµ(x) τR � (fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq − fa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � − pτ τR δT(x) T(τ) �T(τ) τR ∂τR ∂T (fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq − fa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p)) + pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � + pτ τR δµa(x)f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B1b) The perturbations of the rest-frame velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' δuµ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' the temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' δT(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' and the chem- ical potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' δµa(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' are obtained by the linearised Landau-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' As the velocity uµ is normalised to uµuµ = +1, we immediately find that uµδuµ = 0, from which δuτ = 0 (B2) directly follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The perturbed energy-momentum tensor and the perturbed charge current are given by δT µν = � d4p (2π)4 2π � −g(x) δ(p2)2θ(p0)pµpνδf(x, p) , (B3a) δN µ a = � d4p (2π)4 2π � −g(x) δ(p2)2θ(p0)pµδfa(x, p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B3b) 46 Using this, the perturbed eigenvalue problem for the energy-momentum tensor reads (uµ + δuµ)(T µν + δT µν) = (e + δe)(uν + δuν) , (B4a) while the one for the charge current is given by (uµ + δuµ)(N µ a + δN µ a ) = na + δna .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B5) By using the leading order solutions we can deduce the different components, namely δe = δT ττ , δuτ = 0 , δui = δT τi e + PT , δuη = δT τη e + PL , δna = δN τ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Evolution Equations for the Perturbed Moments The perturbation of the distribution functions are expanded in terms of spherical har- monics according to δE m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) = τ 1/3 � dpη (2π) � d2p (2π)2pτY m l (φpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' θp)δfk(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' |pη|) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B7a) δN m al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) = � dpη (2π) � d2p (2π)2Y m l (φpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' θp)δfa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' |pη|) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B7b) Similar to the background one can obtain the components of δT µν k as combinations of low order moments [43] τ 4/3δT ττ k = √ 4πδE 0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8a) δij iki |k|τ 4/3δT τj k = −i � 2π 3 � δE +1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k − δE −1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8b) ϵij iki |k|τ 4/3δT τj k = − � 2π 3 � δE +1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + δE −1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8c) τ 4/3(−τ)δT τη k = � 4π 3 δE 0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8d) δijτ 4/3δT ij k = � 16π 9 δE 0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k − � 16π 45 δE 0 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8e) kikj k2 τ 4/3δT ij k = � 4π 9 δE 0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k − � 4π 45 δE 0 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + � 2π 15 � δE +2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + δE −2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8f) ϵlj kikl k2 τ 4/3δT ij k = −i � 2π 15 � δE +2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k − δE −2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8g) 47 δij iki |k|τ 4/3(−τ)δT ηj k = −i � 2π 15 � δE +1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k − δE −1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8h) ϵij iki |k|τ 4/3(−τ)δT ηj k = − � 2π 15 � δE +1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + δE −1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8i) τ 4/3τ 2δT ηη k = � 16π 45 δE 0 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + � 4π 9 δE 0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B8j) It is also possible to obtain the components of δN µ a,k, which are given by τδN τ a,k = √ 4πδN 0 a0,k , (B9a) δij iki |k|τδN j a,k = −i � 2π 3 � δN +1 a1,k − δN −1 a1,k � , (B9b) ϵij iki |k|τδN j a,k = − � 2π 3 � δN +1 a1,k + δN −1 a1,k � , (B9c) τ(−τ)δN η a,k = � 4π 3 δN 0 a1,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B9d) Note that we decomposed transverse components parallel and perpendicular to the wave vector k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In order to shorten the notation in the following we define (∆E)m l ≡ (Eeq − E + E (1,0) eq )m l , (B10a) (∆Na)m l ≡ (Na,eq − Na + N (1,0) a,eq )m l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B10b) By direct application of the time derivative to the moments we can find their evolution equations to be τ∂τδE m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k = bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2δE m l−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0δE m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2δE m l+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k − i|k|τ 2 � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−δE m+1 l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+δE m+1 l+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−δE m−1 l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+δE m−1 l+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � − τ τR � δE m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + δTk T (E (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq )m l − � a δµa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � (E (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) qa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq + E (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) qa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq)m l �� − τ τR δTk T T(τ) τR ∂τR ∂T (Eeq − E)m l − τ τR δu∥ k 2 � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆E)m+1 l−1 + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆E)m+1 l+1 + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆E)m−1 l−1 + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆E)m−1 l+1 � − τ τR δu⊥ k 2i � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆E)m+1 l−1 + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆E)m+1 l+1 − dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆E)m−1 l−1 − dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆E)m−1 l+1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B11) 48 and τ∂τδN m al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k = Bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2δN m al−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + Bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0δN m al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + Bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2δN m al+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k − i|k|τ 2 � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−δN m+1 al−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+δN m+1 al+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−δN m−1 al−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+δN m−1 al+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � − τ τR � δN m al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + δTk T (N (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq )m l − δµa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(N (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq )m l � − τ τR δTk T T(τ) τR ∂τR ∂T (Na,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq − Na)m l − τ τR δu∥ k 2 � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆Na)m+1 l−1 + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆Na)m+1 l+1 + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆Na)m−1 l−1 + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆Na)m−1 l+1 � − τ τR δu⊥ k 2i � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆Na)m+1 l−1 + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆Na)m+1 l+1 − dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆Na)m−1 l−1 − dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆Na)m−1 l+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B12) where we used App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D in order to express the angle relations in terms of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The coefficients um l,± and dm l,± are given by um l,− = + � (l − m)(l − m − 1) 4l2 − 1 , um l,+ = − � (l + m + 1)(l + m + 2) 3 + 4l(l + 2) , (B13a) dm l,− = − � (l + m)(l + m − 1) 4l2 − 1 , dm l,+ = + � (l − m + 1)(l − m + 2) 3 + 4l(l + 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B13b) while the bm l and Bm l are the same as in the background evolution equations (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A7),(A8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The derivatives of the moments are defined to be (E (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq )m l (τ) = τ 1/3 � dpη (2π) � d2p (2π)2pτY m l (φpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' θp) pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq � pτ T(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' µ(τ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B14a) (E (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) eq )m l (τ) = τ 1/3 � dpη (2π) � d2p (2π)2pτY m l (φpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' θp)f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) eq � pτ T(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' µ(τ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B14b) (N (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq )m l (τ) = � dpη (2π) � d2p (2π)2Y m l (φpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' θp) pτ T(τ)f (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � pτ T(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' µ(τ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B14c) (N (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq )m l (τ) = � dpη (2π) � d2p (2π)2Y m l (φpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' θp)f (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq � pτ T(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' µ(τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B14d) A straightforward computation of the derivatives shows that we can relate them to (E (1,0) eq )m l (τ) = −4(Eeq)m l (τ) , (B15a) (E (0,1) eq )m l (τ) = 3τ 1/3 � a (Na,eq)m l (τ) , (B15b) 49 (N (1,0) a,eq )m l (τ) = −3(Na,eq)m l (τ) , (B15c) (N (0,1) a,eq )m l (τ) = τ √ 4πχaδl0δm0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B15d) However, as we are interested in perturbations around vanishing background density, the equations will simplify since (Na,eq)m l (τ) = 0 (B16) for µ(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Nevertheless, the susceptibilities χa = νq 6 �3µ2 a π2 + T 2 � (B17) are non-zero for zero density but reduce to χa(µa = 0) = νq 6 T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B18) We can also relate the perturbations of the intensive quantities to the perturbation of the extensive quantities for na = 0 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A15) δTk T = δek 4e , (B19a) δµa,k = 6 νq δna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B19b) Therefore we can replace δTk and δµa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k with δek and δna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' which is useful since we can express these quantities by low order moments again τ 4/3δek(τ) = √ 4πδE 0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B20a) τ 4/3(e + PT)δu∥ k(τ) = − � 2π 3 � δE +1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) − δE −1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B20b) τ 4/3(e + PT)δu⊥ k (τ) = i � 2π 3 � δE +1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) + δE −1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(τ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B20c) τδna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k = √ 4πδN 0 a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B20d) This results in a closed set of equations since all appearing perturbations can be written as linear combinations of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' At the level of the equations of motion we see that the dependence on the direction of the transverse wave-vector k has disappeared and the equations only depend on |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This is due to the decomposition of the distribution functions into spherical harmonics and represents the azimuthal rotation symmetry of the background in the transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 50 The equations of motion above (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B11) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B12)) are considered at a fixed value of the wave-number |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However, it is more convenient to rewrite the equation of motion in a mode where we consider it for a fixed value of the propagation phase κ = |k|(τ − τ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B21) By making this change of variable from |k| to κ, we also need to rewrite the time derivative according to τ∂τ �� k = τ∂τ �� |k|(τ−τ0) + τ τ − τ0 |k|(τ − τ0)∂|k|(τ−τ0) �� τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B22) This means, that we find an additional term resulting from the change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Furthermore, for conformal systems it is convenient to work again with the dimension- less variable x = τ/τR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Following the same procedure as for the background, we need to transform the derivative by making use of the scale factor (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' By introducing s(τ) = (τ − τ0)/τ with a(x)x∂xs(x) = 1 − s(x) (B23) we the finally find the equations we are using to compute the Green’s functions � s(x)a(x)x∂x + κ∂κ � δE m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k = s(x) � bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2δE m l−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0δE m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2δE m l+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � − iκ 2 � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−δE m+1 l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+δE m+1 l+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−δE m−1 l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+δE m−1 l+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � − xs(x) � δE m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + δTk T (E (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq )m l � − xs(x)δTk T T(τ) τR ∂τR ∂T (Eeq − E)m l − xs(x)δu∥ k 2 � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆E)m+1 l−1 + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆E)m+1 l+1 + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆E)m−1 l−1 + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆E)m−1 l+1 � − xs(x)δu⊥ k 2i � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆E)m+1 l−1 + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆E)m+1 l+1 − dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆E)m−1 l−1 − dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆E)m−1 l+1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B24a) 51 and � s(x)a(x)x∂x + κ∂κ � δN m al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k = s(x) � Bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2δN m al−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + Bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0δN m al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + Bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2δN m al+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � − iκ 2 � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−δN m+1 al−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+δN m+1 al+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−δN m−1 al−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+δN m−1 al+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k � − xs(x) � δN m al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k − δµa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='k(N (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq )m l � − xs(x)δTk T T(τ) τR ∂τR ∂T (Na,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq − Na)m l − xs(x)δu∥ k 2 � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆Na)m+1 l−1 + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆Na)m+1 l+1 + dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆Na)m−1 l+1 � − xs(x)δu⊥ k 2i � um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−(∆Na)m+1 l−1 + um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆Na)m−1 l−1 − dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+(∆Na)m−1 l+1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B24b) where (∆E)m l ≡ (Eeq − E + E (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0) eq )m l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B25a) (∆Na)m l ≡ (Na,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='eq − Na)m l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B25b) Note that N (1,0) a,eq = 0 since it is proportional to (Na,eq)m l , which is zero for vanishing back- ground density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Initial Energy Perturbations So far we considered the evolution of linearised perturbations on top of a background modeled as a Bjorken flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In order to describe the early time dynamics of heavy-ion collisions we need suitable initial conditions for the perturbations to solve the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Here we will consider initial energy and charge perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We follow the idea of [39], which means initial energy perturbations will be associated with an infinitesimal change of the energy scale of the background distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For the initial distribution function of the perturbations we will find therefore δfk(τ0, p, |pη|) = − �|p| 3 ∂|p|f (0) BG � e−ik· p |p| τ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B26) The factor e−ik· p |p| τ0 takes into account the free-streaming behaviour for times τ < τ0 ≪ τR, while f (0) BG is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We can insert Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B26) into the definition of δE m l,k to translate the initial condition to the moments according to δE m l,k(τ0) = τ 1/3 0 (−i)mJm(|k|τ0)ym l P m l (0)(eτ)0 (B27) 52 with (eτ)0 being the asymptotic energy density of the background Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A10a) and Jm(x) being the Bessel function of the first kind of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In agreement with [39] we find for the energy and velocity perturbations δek(τ0) e = J0(|k|τ0) , (B28a) e + PT e δu∥ k(τ0) = −iJ1(|k|τ0) , (B28b) e + PT e δu⊥ k (τ0) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B28c) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Initial Charge Perturbations The natural choice for the moments of conserved charges is an initial perturbation in terms of the number of quarks respectively the number of anti-quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Therefore we choose initial perturbations of the form δfa,k(τ0, p, |pη|) = δfqa,k(τ0, p, |pη|) − δf qa,k(τ0, p, |pη|) = � 1 + 1 2αa − 1 + 1 2αa � f (0) a,BGe−ik· p |p| τ0 = αaf (0) a,BGe−ik· p |p| τ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B29) In this particular case we choose αa = δna(τ0) na(τ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B30) Translated to the level of the charge moments the initial conditions are given by δN m al,k(τ0) = (−i)mJm(|k|τ0)ym l P m l (0)αa(naτ)0 , (B31) where (naτ)0 is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (A10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For the perturbation δna we find δna,k(τ0) na = αaJ0(|k|τ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (B32) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Numerics The procedure for finding the Green’s functions numerically is more or less the same as for the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However we will consider perturbations around zero densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' we 53 set the initial values for na to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' At a given lmax we truncate the equations of motions for the moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Regarding lmax our numerical studies have shown that we need take a relatively high value of l in order to find convergence for the charge moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' To check this, it is convenient to consider free-streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This is due to the fact that we are able to compute analytically the behaviour of the response functions in free-streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Based on this we can use free- streaming in order to check if the code runs correctly (at least without perturbation-terms in the equations of motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' It turns out that we find convergence towards the free-streaming behaviour for the energy moments a lot faster than for the charge moments in terms of lmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The results of this studies can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This justifies our choice of lmax = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0 5 10 15 20 25 30 35 40 45 50 lmax=64 G~ s s(w~,k∆τ) κ=k∆τ free streaming 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0 5 10 15 20 25 30 35 40 45 50 lmax=64 F~ as s(w~,k∆τ) κ=k∆τ free streaming 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0 5 10 15 20 25 30 35 40 45 50 lmax=128 G~ s s(w~,k∆τ) κ=k∆τ free streaming 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0 5 10 15 20 25 30 35 40 45 50 lmax=128 F~ as s(w~,k∆τ) κ=k∆τ free streaming 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0 5 10 15 20 25 30 35 40 45 50 lmax=256 G~ s s(w~,k∆τ) κ=k∆τ free streaming 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0 5 10 15 20 25 30 35 40 45 50 lmax=256 F~ as s(w~,k∆τ) κ=k∆τ free streaming 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0 5 10 15 20 25 30 35 40 45 50 lmax=512 G~ s s(w~,k∆τ) κ=k∆τ free streaming 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0 5 10 15 20 25 30 35 40 45 50 lmax=512 F~ as s(w~,k∆τ) κ=k∆τ free streaming Figure 14: Top row: ˜Gs s, bottom row: ˜F s s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' From left to right: Corresponding Green’s functions for lmax = 64,128, 256, 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The black curve corresponds to analytic free-streaming solutions while the blue curve corresponds to our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Clearly, for ˜Gs s we find convergence to to the free-streaming very fast (no significant improvement for lmax > 128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For ˜F s s we see acceptable convergence only for lmax ≥ 512, which justifies that we choose lmax = 512 for our numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 54 Appendix C: Non-Equilibrium Green’s Functions of Energy-Momentum Tensor and Current of Conserved Charges 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions of the Energy-Momentum Tensor We follow the construction of the response functions according to [24, 39] and express δT µν k (τ) as δT µν k (τ) e(τ) = 1 2 ˜Gµν αβ(k, τ, τ0)δT αβ k (τ0) e(τ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C1) We decompose the several response functions into a basis of scalars (s), vectors (v) and tensors (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For initial energy perturbations we thus have ˜Gττ ττ(k, τ) = ˜Gs s(κ, x) , (C2a) ˜Gτi ττ(k, τ) = −i ki |k| ˜Gv s(κ, x) , (C2b) ˜Gij ττ(k, τ) = δij ˜Gt,δ s (κ, x) + kikj |k|2 ˜Gt,k s (κ, x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C2c) Since the normalization of the linearized perturbation is arbitrary, we adopt the convention δe(τ0) e(τ0) = 1 (C3) such that we can express the decomposed response functions in terms of δT µν k (τ) (see [39]) according to ˜Gs s(κ, x) = δT ττ k (x) e(x) = δeκ(x) e(x) , (C4a) ˜Gv s(κ, x) = iδij ki |k| δT τj κ (x) e(x) , (C4b) ˜Gt,δ s (κ, x) = � δij − kikj |k|2 �δT ij κ (x) e(x) , (C4c) ˜Gt,k s (κ, x) = � 2kikj |k|2 − δij �δT ij κ (x) e(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C4d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions of the Current of Conserved Charges The Green’s functions corresponding to the conserved charges are defined by τδN µ k(τ) = ˜F µ α (k, τ, τ0) τ0δN α k (τ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C5) 55 Note that we dropped the flavor indices on ˜F µ α as we consider perturbations around vanishing background densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In such a setting the Green’s functions decouple in terms of the flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Following the same argumentation δN µ k does not depend on the flavor anymore neither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Like before, we will decompose ˜F µ α also in a scalar-vector-tensor basis according to ˜F τ τ (k, τ) = ˜F s s (κ, x) , (C6a) ˜F i τ (k, τ) = −i ki |k| ˜F v s (κ, x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C6b) Adapting the normalisation τ0δN τ k(τ0) = 1 (C7) we find ˜F s s (κ, x) = τδN τ κ(x) , (C8a) ˜F v s (κ, x) = i ki |k|τδN i κ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C8b) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Numerical Results for the Non-Equilibrium Green’s Functions of the Energy- Momentum Tensor The results for ˜Gs s an ˜F s s are presented in the main text in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 15 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 16 we will show the results for the other Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' In addition to the points discussed in the main part we can very clearly see the isotropy at later times in the figure for the pressure response ˜Gt,δ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' After scaling the response function, we see that at early times the longitudinal pressure is zero while at times when the system can be described by hydrodynamics ( ˜w ≥ 1), the longitudinal pressure is established and we find the effect of isotropy as the response function approaches one at zero propagation phase indicating e = 3P in the hydrodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions of the Energy-Momentum Tensor in Coordinate Space Similar to the decomposition in Fourier space, we can decompose the Green’s functions in coordinate space as well into a basis of scalars, vectors and tensors, such that we find Gττ ττ(r, τ) = Gs s(|r|, τ) , (C9a) 56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0 5 10 15 20 Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] Momentum response: G~ s v(w~,k∆τ) Wave number: κ=k∆τ free streaming 0 1 2 3 4 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 5 10 15 20 Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] Pressure response: 3G~ s t,δ(w~,k∆τ) Wave number: κ=k∆τ free streaming 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0 5 10 15 20 Evolution time: w~=τTeff(τ)/[4πη~Teff/(e+P))] Shear-stress response: G~ s t,k(w~,k∆τ) Wave number: κ=k∆τ free streaming 0 1 2 3 4 5 Figure 15: Evolution of the energy-momentum Green’s functions in response to initial energy per- turbations in the constant κ-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The different panels correspond to different response functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' different curves in each panel corresponds to different times ˜w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Gτi ττ(r, τ) = ri |r|Gv s(|r|, τ) , (C9b) Gij ττ(r, τ) = δijGt,δ s (|r|, τ) + rirj |r|2 Gt,r s (|r|, τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C9c) The relation to their counterparts in Fourier space is given by the following Fourier-Hankel transforms Gs s(|r|, τ) = 1 2π � d|k| |k|J0(|k||r|) ˜Gs s(|k|, τ) , (C10a) Gv s(|r|, τ) = 1 2π � d|k| |k|J1(|k||r|) ˜Gv s(|k|, τ) , (C10b) Gt,δ s (|r|, τ) = 1 2π � d|k| |k| � J0(|k||r|) ˜Gt,δ s (|k|, τ) + J1(|k||r|) |k||r| ˜Gt,k s (|k|, τ) � , (C10c) Gt,r s (|r|, τ) = −1 2π � d|k| |k|J2(|k||r|) ˜Gt,k s (|k|, τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C10d) 57 Figure 16: Evolution of the charge Green’s functions in response to initial charge perturbations in the constant κ-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The different panels correspond to different response functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' different curves in each panel corresponds to different times ˜w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Green’s Functions of the Current of Conserved Charges in Coordinate Space For the charge Green’s functions the decomposition in coordinate space is given by F τ τ (r, τ) = F s s (|r|, τ) , (C11a) F i τ (r, τ) = ri |r|F v s (|r|, τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C11b) The relation to their counterparts in Fourier space is given by the Fourier-Hankel transforms F s s (|r|, τ) = 1 2π � d|k| |k|J0(|k||r|) ˜F s s (|k|, τ) , (C12a) F v s (|r|, τ) = 1 2π � d|k| |k|J1(|k||r|) ˜F v s (|k|, τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (C12b) Appendix D: Identities For Spherical Harmonics and Associated Legendre Polyno- mials While deriving the equations of motion for E m l and N m al we used several identities for the associated Legendre polynomials and numerical coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' First we will list the appearing 58 freestreaming 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 Evolution time: W=TTefr(t)[4nTef/(e+P)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 3 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0 0 5 10 15 20 Wavenumber:k=k△t-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 Evolution time: w~=T(τ)τ / (4πη/s) Pressure response: 3∆τ2 Gs t,δ(∆x,∆τ,w~) Distance: ∆x/∆τ free streaming 0 1 2 3 4 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 Evolution time: w~=T(τ)τ / (4πη/s) Shear-stress response: ∆τ2 Gs t,r(∆x,∆τ,w~) Distance: ∆x/∆τ free streaming 0 1 2 3 4 5 Figure 17: Evolution of the energy Green’s functions in response to initial energy perturbations in coordinate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The different panels correspond to different response functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' different curves in each panel corresponds to different times ˜w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 59 2 free streaming 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 3 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 Distance:Ax/△tFigure 18: Evolution of the charge Green’s functions in response to initial charge perturbations in coordinate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The different panels correspond to different response functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' different curves in each panel corresponds to different times ˜w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' coefficients ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='− = + (l + 1)(l + m) 2l + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' ξm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='− = l + m 2l + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ = − l(l − m + 1) 2l + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' ξm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ =l − m + 1 2l + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2 = − ξ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2 − ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−ξm l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='− ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2 =am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2 ym l ym l−2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 =1 3 − ξ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 − ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−ξm l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ − ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ξm l+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='− ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 =am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2 = − ξ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2 − ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ξm l+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2 =am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2 ym l ym l+2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2 = − ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−ξm l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='− ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2 =Am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−2 ym l ym l−2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 = − ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='−ξm l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ − ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ξm l+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='− ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 =Am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2 = − ∆m l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ξm l+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Bm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2 =Am l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+2 ym l ym l+2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='− = + ym l (2l + 1)ym+1 l−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='− = + ym l (2l + 1)ym−1 l−1 σm l σ−m+1 l−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' um l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ = − ym l (2l + 1)ym+1 l+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' dm l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='+ = − ym l (2l + 1)ym−1 l+1 σm l σ−m+1 l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (D1a) 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 freestreaming 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 Evolution time: W=T(t)t / (4Tn/s) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4 Distance:Ax/△tand ξ(2),m l,−2 = ξm l,−ξm l−1,− , ξ(2),m l,0 = ξm l,−ξm l−1,+ + ξm l,+ξm l+1,− , ξ(2),m l,+2 = ξm l,+ξm l+1,+ , σm l = (−1)m(l − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (l + m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (D1b) These coefficients are now used to formulate the following identities in a compact way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For the background we use � 1 − x2� d dxP m l (x) = ∆m l,−P m l−1 (x) + ∆m l,+P m l+1 (x) (D2) and xP m l (x) = ξm l,−P m l−1 (x) + ξm l,+P m l+1 (x) (D3) together with x2P m l (x) = ξ(2),m l,−2 P m l−2 (x) + ξ(2),m l,0 P m l (x) + ξ(2),m l,+2 P m l+2 (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (D4) Combining these identities we find ��1 3 − x2 � − x � 1 − x2� d dx � P m l (x) = am l,−2P m l−2 (x) + am l,0P m l (x) + am l,+2P m l+2 (x) (D5) respectively −x � 1 − x2� d dxP m l (x) = Am l,−2P m l−2 (x) + Am l,0P m l (x) + Am l,+2P m l+2 (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (D6) Furthermore we make use of P −m l (x) = σm l P m l (x) (D7) and √ 1 − x2P m l (x) = 1 2l + 1 � P m+l l−1 (x) − P m+1 l+1 (x) � (D8) in order to find sin(θ)e+iφY m l (φ, θ) = um l,−Y m+1 l−1 (φ, θ) + um l,+Y m+1 l+1 (φ, θ) , (D9a) sin(θ)e−iφY m l (φ, θ) = dm l,−Y m−1 l−1 (φ, θ) + dm l,+Y m−1 l+1 (φ, θ) , (D9b) which are used to compute the additional terms in the equation of motion for the perturbed moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 61 Appendix E: Eccentricities, Cumulants, and Anisotropic Flow 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Standard Initial State Eccentricities Quantifying the geometry of the initial state is done using the standard definition of the complex eccentricity vector En given as En ≡ εn einψn ≡ − � rdrdφ rneinφ f(r, φ) � rdrdφ rn f(r, φ) , (E1) where f(r, φ) is an initial state distribution like the entropy or energy density which specifies the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' The magnitude of the eccentricity is εn and ψn is the complex (event-plane) angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' We can express this quantity in terms of the complex position vector r ≡ x + iy through rneinφ = rn: En ≡ − � d2r rn f(r) � d2r |r|n f(r) , (E2) where boldface is used to denote the complex vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Usually these definitions are specified as applying only in the center of mass frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This can be expressed in terms of a general coordinate system: En ≡ − � d2r (r − rCMS)n f(r) � d2r |r − rCMS|n f(r) (E3) with the center-of-mass vector rCMS ≡ � d2r r f(r) � d2r f(r) = 1 ftot � d2r r f(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (E4) A consequence of this definition is that the directed eccentricity E1 vanishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' This method for describing the initial state is well suited when the quantity f(r) being described is positive definite like the energy or entropy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' However if f(r) = ρ(r) is a charge density, particularly when total net charge is zero, it becomes impossible to define a corresponding frame such that E1 = 0 when the total charge vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Instead, there is always a nonzero E1 proportional to the dipole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Due to this inability to construct a center-of-charge frame and ensure that E1 vanishes for a conserved charge with qtot = 0, the usual definitions (E1) or (E2) must be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' For a conserved charge density ρX(r) (here we consider baryon number B, strangeness S, or electric charge Q for X), regions of positive charge with ρX(r) > 0 and negative charge 62 with ρX(r) < 0 will be treated separately by decomposing ρX ≡ ρ(X +) θ(ρX) + ρ(X −) θ(−ρX), (E5) where the position argument r is suppressed for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Then the eccentricities correspond- ing to the positive and negative charge densities are ε(X ±) n ≡ ����� � d2r (r − rCMS)n ρ(X ±)(r) � d2r |r − rCMS|n ρ(X ±)(r) ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (E6) A consequence of these choices is that when there is no charge density, then the eccentricity is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Cumulants To quantify the initial state geometry, we will use the cumulants for the initial eccentric- ities εn: εn{2} = � ⟨ε2 n⟩ (E7a) εn{4} = 4� 2 ⟨ε2 n⟩2 − ⟨ε4 n⟩ = εn{2} 4 � 1 − Var(ε2 n) ⟨ε2 n⟩2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (E7b) These have been shown to be good predictors of the final state flow harmonics, vn, due to a linear scaling relationship [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Aaron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' (H1, ZEUS), JHEP 01, 109 (2010), 0911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Luzum, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Ollitrault, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 85, 024908 (2012), 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Niemi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Denicol, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Holopainen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Huovinen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 87, 054901 (2013), 1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Teaney and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Yan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 86, 044908 (2012), 1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [12] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Qiu and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Heinz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 84, 024911 (2011), 1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='0650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Gardim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha-Hostler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Luzum, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Grassi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 91, 034902 (2015), 1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='2574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Betz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Gyulassy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Luzum, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha-Hostler, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Portillo, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Ratti, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Takahashi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 102, 064909 (2020), 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='13358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [16] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Giacalone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha-Hostler, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Ollitrault, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 95, 054910 (2017), 1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Sievert and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha-Hostler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 100, 024904 (2019), 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Sievert, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha-Hostler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 103, 034910 (2021), 1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='03677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Gardim, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Grassi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Hama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Luzum, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Ollitrault, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 83, 064901 (2011), 1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='4605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [20] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Gardim, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Grassi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Luzum, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Ollitrault, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 109, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Venugopalan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 110, 012302 (2013), 1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='6330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Schenke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Shen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Tribedy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' B 803, 135322 (2020), 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='06212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Shen, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Heinz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 91, 064906 (2015), [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='C 92, 049904 (2015)], 1504.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Teaney, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 122, 122302 (2019), 1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='01604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Werner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 232, 87 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [28] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Itakura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Kovchegov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' McLerran, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Teaney, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A 730, 160 (2004), hep-ph/0305332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [29] C.' metadata={'source': 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M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Asakawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Hirano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Kitazawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Morita, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Murase, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Nara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Martinez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Sievert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Wertepny, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha-Hostler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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Almaalol, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Dore, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Mroczek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Salinas san Martin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Spychalla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Carzon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Sievert, and J.' metadata={'source': 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Hoang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Radosz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 126, 222301 (2021), 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='11632.' metadata={'source': 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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 79, 759 (2019), 1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='04081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [38] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Ambrus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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Plaschke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Ochsenfeld, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Schlichting, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D 102, 056003 (2020), 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='06751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [44] W.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Gardim, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Ollitrault, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 93, 014909 65 (2016), 1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='03896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Shen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 98, 034916 (2018), 1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='10557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [47] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rocha, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Denicol, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 127, 042301 (2021), 2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='07489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [48] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Grad, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 2, 331 (1949).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [49] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Giacalone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Mazeliauskas, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Schlichting, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 123, 262301 (2019), 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='09068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [51] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Fotakis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Molnár, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Niemi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Greiner, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rischke, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D 106, 036009 (2022), 2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='11549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' D 104, 034014 (2021), 2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='08140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Greif, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Fotakis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Denicol, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Greiner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' 120, 242301 (2018), 1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='08680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Mantovani Sarti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha-Hostler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Parotto, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Portillo Vazquez, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Ratti, Phys.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Sievert, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Noronha-Hostler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' C 102, 054905 (2020), 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='00780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [57] J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Blaizot and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Yan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' B 780, 283 (2018), 1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content='03856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} +page_content=' [59] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfhgqt/content/2301.04572v1.pdf'} 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@@ -0,0 +1,468 @@ +Measuring Board Game Distance +Matthew Stephenson, Dennis J. N. J. Soemers, Éric Piette, and Cameron +Browne +Department of Advanced Computing Sciences, Maastricht University, +Paul-Henri Spaaklaan 1, 6229 EN, Maastricht, the Netherlands +{matthew.stephenson,dennis.soemers,eric.piette,cameron.browne} +@maastrichtuniversity.nl +Abstract. This paper presents a general approach for measuring dis- +tances between board games within the Ludii general game system. These +distances are calculated using a previously published set of general board +game concepts, each of which represents a common game idea or shared +property. Our results compare and contrast two different measures of dis- +tance, highlighting the subjective nature of such metrics and discussing +the different ways that they can be interpreted. +Keywords: Ludii · Concepts · Board Games · Distance. +1 +Introduction +Ludii is a relatively recent general game system that contains a large variety +of different board games [1]. This includes games with stochasticity and hidden +information, alternating and simultaneous move formats, between one and six- +teen players, piece stacking, team-based scoring, among many other features. +Games in Ludii are described using ludemes, which are specific keywords that +are defined within the Ludii Game Description Language (L-GDL). While indi- +vidually simple, these ludemes can be combined to express complex game rules +and mechanics. A previous study demonstrated that it is possible to use a game’s +ludemes to accurately predict the performance of various game-playing heuris- +tics [2]. However, representing a game solely as the set of ludemes within its +description can lead to issues. +Because these ludemes are often combined to express more complex rules +and mechanics, their specific order and arrangement can dramatically alter a +game’s behaviour. Just looking at the ludemes that are present within a game’s +description is often not enough to understand their wider context and intended +effect. For example, knowing that a game contains the move ludeme "hop" does +not tell us whether this type of move can be done over friendly or enemy pieces. +We are also not able to detect how frequently "hop" moves occur in typical play, +compared to other types of moves. To address these and other similar limitations, +a set of general board game concepts was proposed [3]. These concepts were +created as a way to identify and extract higher-level features within each game, +thus providing a more complete representation. +arXiv:2301.03913v1 [cs.AI] 10 Jan 2023 + +2 +M. Stephenson et al. +In this paper, we explore how these concepts can be used to calculate a mea- +sure of distance between any two games in Ludii. In addition to providing insight +into the types of games currently available within Ludii, being able to measure +the distance between two games has a variety of practical applications. One ex- +ample is the ability to improve the performance of general game playing agents +on unknown games, by identifying similar known games with pre-existing knowl- +edge and results. This application has already motivated prior investigations into +measuring game distance within other general game systems, including both the +Stanford GGP framework [4,5] and the General Video Game AI framework [6–8]. +Along with this, measures of game distance can be used by recommender systems +to suggest new games to users based on their prior preferences and ratings [9,10], +for transfer learning between similar games [11], to examine the variety of games +within a specific subset [12], or for game reconstruction purposes [13]. +The remainder of this paper is structured as follows. Section 2 describes +the games and concepts that will be used. Section 3 provides visualisations of +the overall distribution of concept values across all games within Ludii. Section +4 presents several different approaches for calculating distances between two +games using their concept values, and provides two specific examples based on +Cosine Similarity and Euclidean distance. Section 5 summarises and discusses +the results of these two distance measures when applied to all pairs of games. +Section 6 summarises our findings and suggests possibilities for future work. +2 +Datasets +This section describes the two datasets that were used for this study, that of +the games within Ludii and their associated concept values. These datasets were +obtained from v1.3.2 of the Ludii database, which is publicly available online.1 +2.1 +Games +As of the time of writing, Ludii version 1.3.2 includes 1059 fully playable games. +While some of these games also contain multiple options and rulesets for pro- +viding different variations of the same base rules, for the sake of simplicity we +will only be considering the default version for each game as provided by Ludii. +Due to the fact that Ludii was developed as part of the Digital Ludeme +Project [13], the majority of the board games it contains are traditional games +that date back many hundreds of years. Even though a large assortment of mod- +ern abstract games have also been implemented within Ludii, this set of games is +unlikely to be fully representative of the complete population of different games +that exist within the modern board game industry. For example, Ludii does not +currently include any card games, even though many modern board games often +use cards in some capacity. Nevertheless, Ludii still contains a substantial vari- +ety of different abstract games, and an analysis of its full game library is worth +performing. +1 www.ludii.games/downloads/database-1.3.2.zip + +Measuring Board Game Distance +3 +2.2 +Concepts +Each concept represents a specific property of a game as a single numerical value. +These concepts can be binary (e.g. if the game contains hidden information), +discrete (e.g. the number of players), or continuous (e.g. the likelihood of a game +ending in a draw). The Ludii database currently lists 499 distinct concepts with +computed values for every game. Each of these concepts is associated with one +of six categories based on what aspect of the game they represent, see Table 1. +Table 1. Concept Categories +Category +Examples +Count +Properties +Num Players, Stochastic, Asymmetric +21 +Equipment +Mancala Board, Hex Tiling, Dice, Hand +74 +Rules +Hop Capture, Turn Ko, Draw Frequency +302 +Math +Multiplication, Intersection, Union +33 +Visual +Go Style, Chess Component, Stack Type +42 +Implementation +Playouts Per Second, Moves Per Second +27 +Concepts also differ in the way they are computed. Compilation concepts can +be calculated from just the game’s description, and will be the same every time +they are computed. Playout concepts instead require one or more game traces +in order to compute them, and will often vary for the same game if different +traces are used (although using a large number of traces can reduce this vari- +ance). For this study, 87 of the concepts used were playout concepts. These were +computed for each game using 100 game traces generated from random play, +with a maximum limit of 2500 moves per game trace (after which the result is a +draw). +Due to the different value ranges that each concept can take, we decided +to normalise each concept to the same scale. However, one issue with these +concepts is that they are susceptible to having outlier values. For example, games +played on implied "boards" of potentially unbounded size (e.g. Dominoes) are +modelled in Ludii using extremely large static boards. Another example would +be the game Hermit which, due to the unique way in which each player’s score is +represented, produces an average score variance of over 100 million points. Due to +cases like these, directly applying Min-Max scaling to our concept values would +overemphasise these outliers and make all other values irrelevant. To mitigate +this problem, we first applied a bi-symmetric log transformation on all concept +values [14], see Equation (1). +f(x) = +�log2(x + 1) +x ≥ 0 +−log2(1 − x) +x < 0 +(1) +This transformation reduces the impact of the more extreme positive and +negative values, while ensuring that binary concepts are unaffected. These new + +4 +M. Stephenson et al. +transformed values are then normalised using Min-Max scaling, to give the final +concept values for each game. Thus our full dataset consists of 1059×499 matrix, +detailing the concept values for each game. +3 +Data Visualisation +Before calculating any game distances, we decided to first visualise the overall +distribution of concept values across all games within Ludii. To do this, we +applied t-distributed stochastic neighbor embedding (t-SNE) [15] to reduce our +concept dataset to two dimensions, see Figure 1. From this visualisation, we +identified four distinct clusters of games. The orange cluster contains 207 games, +the green cluster contains 147 games, the red cluster contains 105 games, and +the blue cluster contains 600 games. +Fig. 1. Game concept dataset reduced to two dimensions using t-SNE. Points are +coloured based on identified game clusters. +Analysing these clusters closer reveals some general trends within each group. +– The orange cluster contains all games with a Mancala Board, such as Oware +or Kalah. These games are highly separated with a very distinct way of +playing, leading to a lot of concepts that are unique to them such as the +mechanic of sowing stones. It therefore makes sense that these games would +form their own distinct cluster. +– The green cluster contains all games with both dice and a track for pieces to +move along (e.g. Backgammon or Snakes and Ladders), as well as a few other +games such as EinStein Würfelt Nicht and So Long Sucker. Despite the clear +separation of the games in this cluster from the rest, this distinction is not the + +40 +20 +0 +-20 +-40 +-40 +-20 +0 +20 +40 +60Measuring Board Game Distance +5 +sole result of any one game element. Instead, it seems that such games would +be better characterised as those with a high degree of uncertainty, either +because stochastic elements have a large impact on the game or because of +other player’s decisions. +– The red cluster contains all games with a "Threat" mechanism, predomi- +nantly used in Chess-like games when seeing if the king is in Check, as well +as some other similar games such as Ploy and Chaturaji. Like the green clus- +ter, this group of games is not characterised by a single concept. The key +aspects that group these games together, seem to be the combination of a +large number of pieces for each side, as well as complex movement rules for +each individual piece. +– The blue cluster contains all other games which do not fall into the previous +clusters. +From this, we can first see that the blue cluster is considerably larger than +the others, making up more than half the total number of games. While this +cluster could probably be split up further into sub-clusters, the separation is not +as clear as for the clusters identified. Admittedly, the separation between the +blue and red clusters is also not as distinct as the others, but is clear enough +that we felt it worth mentioning. With the exception of the orange cluster, which +is uniquely defined by the existence of the "Mancala Board" concept, there is +no single concept that is responsible for any one cluster. Each cluster is instead +defined by a combination of multiple concept values, making previous attempts +to categorise games based on singular properties incapable of creating such a +distinction. Based on these findings, it is clear that the recorded concept values +provide significant information about a game’s mechanics and properties, and +are likely to be an effective basis for measuring the distance between two games. +4 +Game Distance +When it comes to calculating the distances between games, each game is rep- +resented as a vector of 499 normalised concept values. Comparing two games +can therefore be done using a variety of different vector distance/similarity mea- +sures. This includes Euclidean distance, Manhattan distance, Cosine similarity, +Jaccard index, Jenson-Shannon divergence, among many other options. +Additional pre-processing can also be applied to adjust the importance of +each concept. For example, Inverse Document Frequency (IDF) could be applied +to all binary concepts, increasing the weights for concepts that occur in fewer +games while decreasing the weights for those that occur in many. Each concept +category could also be adjusted as a whole. For example, each concept could +be scaled relative to the size of its category, resulting in each category carrying +equal collective weighting. Some categories could even be excluded completely, +as the importance of each category will likely vary based on the intended appli- +cation. For example the "Visual" category of concepts has no bearing on actual +gameplay, and would provide no benefit for the application of training a general +game-playing agent. + +6 +M. Stephenson et al. +Unfortunately, due the lack of any concrete benchmarks for measuring game +distance, the effectiveness of these different measures and weight adjustments +cannot be objectively evaluated without a specific application in mind. Due to +the open-ended nature of this exploratory study, we decided to only compare +the Cosine similarity and Euclidean distance measures, two of the most popular +vector distance measures, without any additional pre-processing or weight ad- +justments. We encourage other researchers who wish to use this dataset for their +own work, to experiment with these different distance measure approaches and +identify what works best for their desired application. +The normalised Cosine distance between two games G1 and G2, with concept +vectors denoted by ⃗c1 and ⃗c2 respectively, is given by Equation (2). +CosineDistance(G1, G2) = 1 +2 +� +1 − +⃗c1 · ⃗c2 +∥⃗c1∥∥⃗c2∥ +� +(2) +The normalised Euclidean distance between two games, using the same terms +in the previous equation and n denoting the total number of concepts, is given +by Equation (3). +EuclideanDistance(G1, G2) = ∥⃗c1 − ⃗c2∥ +√n +(3) +Both of these distance measures are normalised within the zero to one range, +with one representing maximal difference and zero representing maximal simi- +larity. +5 +Results +Both Euclidean and Cosine distances were calculated between each possible pair +of our 1059 games, giving a total of 560,211 unique game pairs for each distance +measure. +Fig. 2 presents a box plot visualisation of each distance measure across all +game pairs. From this, we can see that the Euclidean distance is typically larger +than the Cosine distance. The inter-quartile range for Cosine distance is situ- +ated almost exactly equally between its minimum and maximum values, while +the inter-quartile range for Euclidean distance is skewed much closer to the max- +imum value. The difference between the median values of each distance measure +(0.1358) is also much larger than the difference between their maximum values +(0.0572). +Fig. 3 provides further details on this observation, showing the general trend +for each distance measure across all game pairs when ordered from smallest to +largest along the x-axis. Looking at the 10% - 90% interpercentile range, repre- +sented by the area between the two green dashed lines, shows that the gradient +of each distance measure is approximately equal. The most significant difference +between the trends of these two distance measures is instead located at the ex- +tremities. While both distance measures begin at zero, the Euclidean distance +initially increases far more rapidly than the Cosine distance. The opposite is true + +Measuring Board Game Distance +7 +Fig. 2. Box plot for Cosine and Euclidean distances across all game pairs. +Fig. 3. Cosine and Euclidean distance trends, ordered by size along the x-axis. +at the upper percentile end, with the Cosine distance taking a sharp increase to +raise itself closer to the Euclidean distance. +Fig. 4 visualises the differences between the Cosine and Euclidean distances +for each individual game pair. From this we can see that there is a strong positive +relationship between both distance measures, with a Pearson correlation coeffi- +cient of 0.7574. The overall upward curve of the points also reiterates our prior +observation, that the rate at which the Euclidean distance increases is initially +much higher, but then gradually falls to be more in line with that of the Cosine +distance. +5.1 +Discussion +While these statistical results provide a broad overview of how these distance +measures compare across all games, we also explore how they differ with re- +gard to specific game pairs. The game pair with the greatest difference in their +Cosine and Euclidean distances (with a higher Cosine distance) was between +Magic Square and Rock-Paper-Scissors, with a Cosine distance of 0.4438 and a +Euclidean distance of 0.3036. Both of these games are relatively simple and share + +Cosine +Euclidean +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Distance0.6 +0.5 +Distance +0.4 +■ +0.3 +！！ +0.2 +0.1 +！！ +0 +- Cosine +- Euclidean8 +M. Stephenson et al. +Fig. 4. The Cosine and Euclidean distance for all 560,211 game pairs. +very little in common, with Magic Square being a logic puzzle and Rock-Paper- +Scissors being a two-player game with simultaneous moves. From the alternative +perspective, the game pair with the greatest difference in their Euclidean and +Cosine distances (with a higher Euclidean distance) was between Tenjiku Shogi +and Chex, with a Cosine distance of 0.1429 and a Euclidean distance of 0.3792. +Both of these games are very complex, featuring large boards along with many +different pieces and rules. +Based on these two contrasting game pair examples, it initially seems that +Cosine distance is greatest between simpler games with relatively few high-value +concepts, while Euclidean distance is greatest between more complex games with +a larger number of high-value concepts. +To dive deeper into how each distance measure compares across multiple +game pairs, we looked at which game pairs received the largest values from each +measure. One immediate observation was that the majority of the largest Cosine +and Euclidean distances were between a logic puzzle, such as Sudoku, Kakuro +or Hoshi, and non-puzzle game. This significant logic puzzle presence makes +intuitive sense, as they are a unique type of game that would likely produce very +distinct concept values. +For the Cosine distance measure, all of the 20 largest game pair distances in- +volved either Rock-Paper-Scissors, Morra or Aksadyuta. All three of these games +are very simple in terms of their rules, and are essentially purely random in terms +of their outcome. These games contain no boards or pieces (at least in the tra- +ditional sense), and typically last for only a few moves. It would therefore make +sense for these games to be highly distant from most other games, and further +backs up our theory that the Cosine distance measure gives the greatest distance +values to game pairs that include a simpler game with less common concepts. +For the Euclidean distance measure, all of the 20 largest game pair distances +involved either Beirut Chess, Ultimate Chess or Chex. These games are all Chess + +0.5 +0.4 +Cosine Distance +0.3 +0.2 +0.1 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Euclidean DistanceMeasuring Board Game Distance +9 +variants with some unique twist on their rules. All three of these games would +probably be considered more complex than the majority of other games in our +dataset. This likely results in a substantial number of large value concepts for +each game, again supporting the idea that the greatest Euclidean distances are +typically given to game pairs that involve at least one high complexity game. +Based on these further comparisons, it appears that neither distance measure +is inherently better than the other. It instead seems likely that each approach, as +well as the other suggested measures that we did not explore deeper, has its own +strengths, weaknesses and biases. The choice of which distance measure is most +suitable is a highly subjective decision, and would depend on the intended appli- +cation. We therefore reiterate our previous statement that multiple approaches +should be tested and evaluated for each specific use-case, rather than attempting +to develop a single correct measure of game distance. +6 +Conclusion +In this paper, we have investigated the use of general board game concepts +to measure the distance between pairs of games. Based on the same original +concept dataset, two different measurements were proposed based on Cosine +and Euclidean distance. Our results highlight the differences between these ap- +proaches. Cosine distance tended to give its highest values to pairs of relatively +simple games with very few shared concepts. Euclidean distance on the other +hand, appeared to include larger and more complex games in its highest value +pairs, where any similarities between the games were outweighed by their differ- +ences. While we are unable to conclude which distance measure might be better +or worse for any specific application, the contrasting outputs between these two +distance measures illustrates the importance of experimenting with multiple dis- +tance measurement approaches. +Possible future work could involve a more complete analysis and summary of +a larger range of distance measurements, as well as the effect that different pre- +processing and weight adjustment techniques has on their outputs. Additional +concepts could also be added to fill knowledge gaps within the existing dataset. +One addition could be the inclusion of playout concepts based on alternative +game traces, such as those produced from different game-playing agents or hu- +man players. Rather than adding more concepts, a more nuanced and critical +look at the existing corpus may instead lead to the removal or weight reduction +of certain items. It may not make sense to treat meaningful concepts, such as +whether a game involves hidden or stochastic information, with as much impor- +tance as overly niche concepts, such as whether the game includes pieces that +move backwards to the left. However, such alterations to the concept dataset are +likely to be application specific, as adding or removing concepts for one purpose +may inadvertently affect another. + +10 +M. Stephenson et al. +Acknowledgements +This research is funded by the European Research Council as part of the Digital +Ludeme Project (ERC Consolidator Grant #771292) led by Cameron Browne +at Maastricht University’s Department of Advanced Computing Sciences. +References +1. É. Piette, D. J. N. J. Soemers, M. Stephenson, C. F. Sironi, M. H. M. Winands, +and C. Browne, “Ludii – the ludemic general game system,” in Proceedings of the +24th European Conference on Artificial Intelligence (ECAI 2020), vol. 325, 2020, +pp. 411–418. +2. M. Stephenson, D. J. N. J. Soemers, É. Piette, and C. Browne, “General game +heuristic prediction based on ludeme descriptions,” in Proceedings of the 2021 IEEE +Conference on Games. +IEEE, 2021, pp. 878–881. +3. É. Piette, M. Stephenson, D. J. N. J. Soemers, and C. Browne, “General board +game concepts,” in Proceedings of the 2021 IEEE Conference on Games (CoG). +IEEE, 2021, pp. 932–939. +4. J. D. A. Jung and J. Hoey, “Distance-based mapping for general game playing,” in +2021 IEEE Conference on Games (CoG), 2021, pp. 445–452. +5. D. Michulke and S. Schiffel, “Distance features for general game playing agents,” +in Proceedings of the 4th International Conference on Agents and Artificial Intel- +ligence, 2012, pp. 127–136. +6. P. Bontrager, A. Khalifa, A. Mendes, and J. Togelius, “Matching games and algo- +rithms for general video game playing,” in AIIDE, vol. 12, no. 1, 2016, pp. 122–128. +7. A. Mendes, J. Togelius, and A. Nealen, “Hyper-heuristic general video game play- +ing,” 2016 IEEE Conference on Computational Intelligence and Games (CIG), pp. +94–101, 2016. +8. H. Horn, V. Volz, D. Pérez-Liébana, and M. Preuss, “MCTS/EA hybrid GVGAI +players and game difficulty estimation,” in 2016 IEEE Conference on Computa- +tional Intelligence and Games (CIG). +IEEE Press, 2016, pp. 459–466. +9. J. Kim, J. Wi, S. Jang, and Y. Kim, “Sequential recommendations on board-game +platforms,” Symmetry, vol. 12, no. 2, 2020. +10. J. Zalewski, M. Ganzha, and M. Paprzycki, “Recommender system for board +games,” in 2019 23rd International Conference on System Theory, Control and +Computing (ICSTCC), 2019, pp. 249–254. +11. D. J. N. J. Soemers, V. Mella, É. Piette, M. Stephenson, C. Browne, and O. Tey- +taud, “Transfer of fully convolutional policy-value networks between games and +game variants,” https://arxiv.org/abs/2102.12375, 2021. +12. M. Stephenson, D. Anderson, A. Khalifa, J. Levine, J. Renz, J. Togelius, and +C. Salge, “A continuous information gain measure to find the most discriminatory +problems for AI benchmarking,” in 2020 IEEE Congress on Evolutionary Compu- +tation (CEC), 2020, pp. 92–99. +13. C. Browne, “Modern techniques for ancient games,” in IEEE Conference on Com- +putational Intelligence and Games. +Maastricht: IEEE Press, 2018, pp. 490–497. +14. J. B. W. Webber, “A bi-symmetric log transformation for wide-range data,” Mea- +surement Science and Technology, vol. 24, no. 2, p. 027001, 2012. +15. L. van der Maaten and G. Hinton, “Visualizing data using t-sne,” Journal of Ma- +chine Learning Research, vol. 9, no. 86, pp. 2579–2605, 2008. + diff --git a/l9E2T4oBgHgl3EQfeAfg/content/tmp_files/load_file.txt b/l9E2T4oBgHgl3EQfeAfg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..23d6fac7917abfb85789e5914c569f3399480b58 --- /dev/null +++ b/l9E2T4oBgHgl3EQfeAfg/content/tmp_files/load_file.txt @@ -0,0 +1,360 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf,len=359 +page_content='Measuring Board Game Distance Matthew Stephenson, Dennis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Soemers, Éric Piette, and Cameron Browne Department of Advanced Computing Sciences, Maastricht University, Paul-Henri Spaaklaan 1, 6229 EN, Maastricht, the Netherlands {matthew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='stephenson,dennis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='soemers,eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='piette,cameron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='browne} @maastrichtuniversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='nl Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' This paper presents a general approach for measuring dis- tances between board games within the Ludii general game system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' These distances are calculated using a previously published set of general board game concepts, each of which represents a common game idea or shared property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Our results compare and contrast two different measures of dis- tance, highlighting the subjective nature of such metrics and discussing the different ways that they can be interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Keywords: Ludii · Concepts · Board Games · Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 1 Introduction Ludii is a relatively recent general game system that contains a large variety of different board games [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' This includes games with stochasticity and hidden information, alternating and simultaneous move formats, between one and six- teen players, piece stacking, team-based scoring, among many other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Games in Ludii are described using ludemes, which are specific keywords that are defined within the Ludii Game Description Language (L-GDL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' While indi- vidually simple, these ludemes can be combined to express complex game rules and mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' A previous study demonstrated that it is possible to use a game’s ludemes to accurately predict the performance of various game-playing heuris- tics [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' However, representing a game solely as the set of ludemes within its description can lead to issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Because these ludemes are often combined to express more complex rules and mechanics, their specific order and arrangement can dramatically alter a game’s behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Just looking at the ludemes that are present within a game’s description is often not enough to understand their wider context and intended effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' For example, knowing that a game contains the move ludeme "hop" does not tell us whether this type of move can be done over friendly or enemy pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' We are also not able to detect how frequently "hop" moves occur in typical play, compared to other types of moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' To address these and other similar limitations, a set of general board game concepts was proposed [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' These concepts were created as a way to identify and extract higher-level features within each game, thus providing a more complete representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='03913v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='AI] 10 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Stephenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' In this paper, we explore how these concepts can be used to calculate a mea- sure of distance between any two games in Ludii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' In addition to providing insight into the types of games currently available within Ludii, being able to measure the distance between two games has a variety of practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' One ex- ample is the ability to improve the performance of general game playing agents on unknown games, by identifying similar known games with pre-existing knowl- edge and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' This application has already motivated prior investigations into measuring game distance within other general game systems, including both the Stanford GGP framework [4,5] and the General Video Game AI framework [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Along with this, measures of game distance can be used by recommender systems to suggest new games to users based on their prior preferences and ratings [9,10], for transfer learning between similar games [11], to examine the variety of games within a specific subset [12], or for game reconstruction purposes [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Section 2 describes the games and concepts that will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Section 3 provides visualisations of the overall distribution of concept values across all games within Ludii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Section 4 presents several different approaches for calculating distances between two games using their concept values, and provides two specific examples based on Cosine Similarity and Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Section 5 summarises and discusses the results of these two distance measures when applied to all pairs of games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Section 6 summarises our findings and suggests possibilities for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 2 Datasets This section describes the two datasets that were used for this study, that of the games within Ludii and their associated concept values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' These datasets were obtained from v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='2 of the Ludii database, which is publicly available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='1 Games As of the time of writing, Ludii version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='2 includes 1059 fully playable games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' While some of these games also contain multiple options and rulesets for pro- viding different variations of the same base rules, for the sake of simplicity we will only be considering the default version for each game as provided by Ludii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Due to the fact that Ludii was developed as part of the Digital Ludeme Project [13], the majority of the board games it contains are traditional games that date back many hundreds of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Even though a large assortment of mod- ern abstract games have also been implemented within Ludii, this set of games is unlikely to be fully representative of the complete population of different games that exist within the modern board game industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' For example, Ludii does not currently include any card games, even though many modern board games often use cards in some capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Nevertheless, Ludii still contains a substantial vari- ety of different abstract games, and an analysis of its full game library is worth performing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 1 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='ludii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='games/downloads/database-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='zip Measuring Board Game Distance 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='2 Concepts Each concept represents a specific property of a game as a single numerical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' These concepts can be binary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' if the game contains hidden information), discrete (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' the number of players), or continuous (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' the likelihood of a game ending in a draw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The Ludii database currently lists 499 distinct concepts with computed values for every game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Each of these concepts is associated with one of six categories based on what aspect of the game they represent, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Concept Categories Category Examples Count Properties Num Players, Stochastic, Asymmetric 21 Equipment Mancala Board, Hex Tiling, Dice, Hand 74 Rules Hop Capture, Turn Ko, Draw Frequency 302 Math Multiplication, Intersection, Union 33 Visual Go Style, Chess Component, Stack Type 42 Implementation Playouts Per Second, Moves Per Second 27 Concepts also differ in the way they are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Compilation concepts can be calculated from just the game’s description, and will be the same every time they are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Playout concepts instead require one or more game traces in order to compute them, and will often vary for the same game if different traces are used (although using a large number of traces can reduce this vari- ance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' For this study, 87 of the concepts used were playout concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' These were computed for each game using 100 game traces generated from random play, with a maximum limit of 2500 moves per game trace (after which the result is a draw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Due to the different value ranges that each concept can take, we decided to normalise each concept to the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' However, one issue with these concepts is that they are susceptible to having outlier values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' For example, games played on implied "boards" of potentially unbounded size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Dominoes) are modelled in Ludii using extremely large static boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Another example would be the game Hermit which, due to the unique way in which each player’s score is represented, produces an average score variance of over 100 million points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Due to cases like these, directly applying Min-Max scaling to our concept values would overemphasise these outliers and make all other values irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' To mitigate this problem, we first applied a bi-symmetric log transformation on all concept values [14], see Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' f(x) = �log2(x + 1) x ≥ 0 −log2(1 − x) x < 0 (1) This transformation reduces the impact of the more extreme positive and negative values, while ensuring that binary concepts are unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' These new 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Stephenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' transformed values are then normalised using Min-Max scaling, to give the final concept values for each game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Thus our full dataset consists of 1059×499 matrix, detailing the concept values for each game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 3 Data Visualisation Before calculating any game distances, we decided to first visualise the overall distribution of concept values across all games within Ludii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' To do this, we applied t-distributed stochastic neighbor embedding (t-SNE) [15] to reduce our concept dataset to two dimensions, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' From this visualisation, we identified four distinct clusters of games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The orange cluster contains 207 games, the green cluster contains 147 games, the red cluster contains 105 games, and the blue cluster contains 600 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Game concept dataset reduced to two dimensions using t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Points are coloured based on identified game clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Analysing these clusters closer reveals some general trends within each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' – The orange cluster contains all games with a Mancala Board, such as Oware or Kalah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' These games are highly separated with a very distinct way of playing, leading to a lot of concepts that are unique to them such as the mechanic of sowing stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' It therefore makes sense that these games would form their own distinct cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' – The green cluster contains all games with both dice and a track for pieces to move along (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Backgammon or Snakes and Ladders), as well as a few other games such as EinStein Würfelt Nicht and So Long Sucker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Despite the clear separation of the games in this cluster from the rest, this distinction is not the 40 20 0 20 40 40 20 0 20 40 60Measuring Board Game Distance 5 sole result of any one game element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Instead, it seems that such games would be better characterised as those with a high degree of uncertainty, either because stochastic elements have a large impact on the game or because of other player’s decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' – The red cluster contains all games with a "Threat" mechanism, predomi- nantly used in Chess-like games when seeing if the king is in Check, as well as some other similar games such as Ploy and Chaturaji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Like the green clus- ter, this group of games is not characterised by a single concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The key aspects that group these games together, seem to be the combination of a large number of pieces for each side, as well as complex movement rules for each individual piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' – The blue cluster contains all other games which do not fall into the previous clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' From this, we can first see that the blue cluster is considerably larger than the others, making up more than half the total number of games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' While this cluster could probably be split up further into sub-clusters, the separation is not as clear as for the clusters identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Admittedly, the separation between the blue and red clusters is also not as distinct as the others, but is clear enough that we felt it worth mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' With the exception of the orange cluster, which is uniquely defined by the existence of the "Mancala Board" concept, there is no single concept that is responsible for any one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Each cluster is instead defined by a combination of multiple concept values, making previous attempts to categorise games based on singular properties incapable of creating such a distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Based on these findings, it is clear that the recorded concept values provide significant information about a game’s mechanics and properties, and are likely to be an effective basis for measuring the distance between two games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 4 Game Distance When it comes to calculating the distances between games, each game is rep- resented as a vector of 499 normalised concept values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Comparing two games can therefore be done using a variety of different vector distance/similarity mea- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' This includes Euclidean distance, Manhattan distance, Cosine similarity, Jaccard index, Jenson-Shannon divergence, among many other options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Additional pre-processing can also be applied to adjust the importance of each concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' For example, Inverse Document Frequency (IDF) could be applied to all binary concepts, increasing the weights for concepts that occur in fewer games while decreasing the weights for those that occur in many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Each concept category could also be adjusted as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' For example, each concept could be scaled relative to the size of its category, resulting in each category carrying equal collective weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Some categories could even be excluded completely, as the importance of each category will likely vary based on the intended appli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' For example the "Visual" category of concepts has no bearing on actual gameplay, and would provide no benefit for the application of training a general game-playing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Stephenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Unfortunately, due the lack of any concrete benchmarks for measuring game distance, the effectiveness of these different measures and weight adjustments cannot be objectively evaluated without a specific application in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Due to the open-ended nature of this exploratory study, we decided to only compare the Cosine similarity and Euclidean distance measures, two of the most popular vector distance measures, without any additional pre-processing or weight ad- justments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' We encourage other researchers who wish to use this dataset for their own work, to experiment with these different distance measure approaches and identify what works best for their desired application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The normalised Cosine distance between two games G1 and G2, with concept vectors denoted by ⃗c1 and ⃗c2 respectively, is given by Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' CosineDistance(G1, G2) = 1 2 � 1 − ⃗c1 · ⃗c2 ∥⃗c1∥∥⃗c2∥ � (2) The normalised Euclidean distance between two games, using the same terms in the previous equation and n denoting the total number of concepts, is given by Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' EuclideanDistance(G1, G2) = ∥⃗c1 − ⃗c2∥ √n (3) Both of these distance measures are normalised within the zero to one range, with one representing maximal difference and zero representing maximal simi- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 5 Results Both Euclidean and Cosine distances were calculated between each possible pair of our 1059 games, giving a total of 560,211 unique game pairs for each distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 2 presents a box plot visualisation of each distance measure across all game pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' From this, we can see that the Euclidean distance is typically larger than the Cosine distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The inter-quartile range for Cosine distance is situ- ated almost exactly equally between its minimum and maximum values, while the inter-quartile range for Euclidean distance is skewed much closer to the max- imum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The difference between the median values of each distance measure (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='1358) is also much larger than the difference between their maximum values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='0572).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 3 provides further details on this observation, showing the general trend for each distance measure across all game pairs when ordered from smallest to largest along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Looking at the 10% - 90% interpercentile range, repre- sented by the area between the two green dashed lines, shows that the gradient of each distance measure is approximately equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The most significant difference between the trends of these two distance measures is instead located at the ex- tremities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' While both distance measures begin at zero, the Euclidean distance initially increases far more rapidly than the Cosine distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The opposite is true Measuring Board Game Distance 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Box plot for Cosine and Euclidean distances across all game pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Cosine and Euclidean distance trends, ordered by size along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' at the upper percentile end, with the Cosine distance taking a sharp increase to raise itself closer to the Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 4 visualises the differences between the Cosine and Euclidean distances for each individual game pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' From this we can see that there is a strong positive relationship between both distance measures, with a Pearson correlation coeffi- cient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='7574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The overall upward curve of the points also reiterates our prior observation, that the rate at which the Euclidean distance increases is initially much higher, but then gradually falls to be more in line with that of the Cosine distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='1 Discussion While these statistical results provide a broad overview of how these distance measures compare across all games, we also explore how they differ with re- gard to specific game pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The game pair with the greatest difference in their Cosine and Euclidean distances (with a higher Cosine distance) was between Magic Square and Rock-Paper-Scissors, with a Cosine distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='4438 and a Euclidean distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='3036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Both of these games are relatively simple and share Cosine Euclidean 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='5 Distance0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='5 Distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='4 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='3 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='1 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 0 Cosine Euclidean8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Stephenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The Cosine and Euclidean distance for all 560,211 game pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' very little in common, with Magic Square being a logic puzzle and Rock-Paper- Scissors being a two-player game with simultaneous moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' From the alternative perspective, the game pair with the greatest difference in their Euclidean and Cosine distances (with a higher Euclidean distance) was between Tenjiku Shogi and Chex, with a Cosine distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='1429 and a Euclidean distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='3792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Both of these games are very complex, featuring large boards along with many different pieces and rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Based on these two contrasting game pair examples, it initially seems that Cosine distance is greatest between simpler games with relatively few high-value concepts, while Euclidean distance is greatest between more complex games with a larger number of high-value concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' To dive deeper into how each distance measure compares across multiple game pairs, we looked at which game pairs received the largest values from each measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' One immediate observation was that the majority of the largest Cosine and Euclidean distances were between a logic puzzle, such as Sudoku, Kakuro or Hoshi, and non-puzzle game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' This significant logic puzzle presence makes intuitive sense, as they are a unique type of game that would likely produce very distinct concept values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' For the Cosine distance measure, all of the 20 largest game pair distances in- volved either Rock-Paper-Scissors, Morra or Aksadyuta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' All three of these games are very simple in terms of their rules, and are essentially purely random in terms of their outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' These games contain no boards or pieces (at least in the tra- ditional sense), and typically last for only a few moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' It would therefore make sense for these games to be highly distant from most other games, and further backs up our theory that the Cosine distance measure gives the greatest distance values to game pairs that include a simpler game with less common concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' For the Euclidean distance measure, all of the 20 largest game pair distances involved either Beirut Chess, Ultimate Chess or Chex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' These games are all Chess 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='4 Cosine Distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content='6 Euclidean DistanceMeasuring Board Game Distance 9 variants with some unique twist on their rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' All three of these games would probably be considered more complex than the majority of other games in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' This likely results in a substantial number of large value concepts for each game, again supporting the idea that the greatest Euclidean distances are typically given to game pairs that involve at least one high complexity game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Based on these further comparisons, it appears that neither distance measure is inherently better than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' It instead seems likely that each approach, as well as the other suggested measures that we did not explore deeper, has its own strengths, weaknesses and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' The choice of which distance measure is most suitable is a highly subjective decision, and would depend on the intended appli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' We therefore reiterate our previous statement that multiple approaches should be tested and evaluated for each specific use-case, rather than attempting to develop a single correct measure of game distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 6 Conclusion In this paper, we have investigated the use of general board game concepts to measure the distance between pairs of games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Based on the same original concept dataset, two different measurements were proposed based on Cosine and Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Our results highlight the differences between these ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Cosine distance tended to give its highest values to pairs of relatively simple games with very few shared concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Euclidean distance on the other hand, appeared to include larger and more complex games in its highest value pairs, where any similarities between the games were outweighed by their differ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' While we are unable to conclude which distance measure might be better or worse for any specific application, the contrasting outputs between these two distance measures illustrates the importance of experimenting with multiple dis- tance measurement approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Possible future work could involve a more complete analysis and summary of a larger range of distance measurements, as well as the effect that different pre- processing and weight adjustment techniques has on their outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Additional concepts could also be added to fill knowledge gaps within the existing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' One addition could be the inclusion of playout concepts based on alternative game traces, such as those produced from different game-playing agents or hu- man players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Rather than adding more concepts, a more nuanced and critical look at the existing corpus may instead lead to the removal or weight reduction of certain items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' It may not make sense to treat meaningful concepts, such as whether a game involves hidden or stochastic information, with as much impor- tance as overly niche concepts, such as whether the game includes pieces that move backwards to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' However, such alterations to the concept dataset are likely to be application specific, as adding or removing concepts for one purpose may inadvertently affect another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Stephenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' Acknowledgements This research is funded by the European Research Council as part of the Digital Ludeme Project (ERC Consolidator Grant #771292) led by Cameron Browne at Maastricht University’s Department of Advanced Computing Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E2T4oBgHgl3EQfeAfg/content/2301.03913v1.pdf'} +page_content=' É.' metadata={'source': 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index 0000000000000000000000000000000000000000..b8936e4d62d50cd8a806cba272cf2c8727bc665d --- /dev/null +++ b/ntE1T4oBgHgl3EQf1gVK/content/tmp_files/2301.03468v1.pdf.txt @@ -0,0 +1,1388 @@ +1 +Knowledge-Aware Semantic Communication +System Design and Data Allocation +Sachin Kadam and Dong In Kim, Fellow, IEEE +Abstract +The recent emergence of 6G raises the challenge of increasing the transmission data rate even +further in order to overcome the Shannon limit. Traditional communication methods fall short of the +6G goals, paving the way for Semantic Communication (SemCom) systems that have applications in +the metaverse, healthcare, economics, etc. In SemCom systems, only the relevant keywords from the +data are extracted and used for transmission. In this paper, we design an auto-encoder and auto-decoder +that only transmit these keywords and, respectively, recover the data using the received keywords and +the shared knowledge. This SemCom system is used in a setup in which the receiver allocates various +categories of the same dataset collected from the transmitter, which differ in size and accuracy, to a +number of users. This scenario is formulated using an optimization problem called the data allocation +problem (DAP). We show that it is NP-complete and propose a greedy algorithm to solve it. Using +simulations, we show that the proposed methods for SemCom system design outperform state-of-the-art +methods in terms of average number of words per sentence for a given accuracy, and that the proposed +greedy algorithm solution of the DAP performs significantly close to the optimal solution. +Index Terms +Semantic Communications, Knowledge Base, 6G, Data Allocation, Wireless Communications +I. INTRODUCTION +As per the prediction in [2], semantic communication (SemCom) technology is identified +as one of the key ingredients in 6G due to the requirement of low latency and high data +rate transmissions. The recent emergence of SemCom technologies finds applications in wide +A preliminary version of this paper is submitted to a conference and it is currently under review [1]. +S. Kadam and D. I. Kim are with the Department of Electrical and Computer Engineering, Sungkyunkwan University (SKKU), +Suwon 16419, Republic of Korea (e-mail: sachinkadam@skku.edu, dikim@skku.ac.kr). +arXiv:2301.03468v1 [eess.SP] 30 Dec 2022 + +2 +range of fields such as economics [3], metaverse [4], autonomous transportation systems [5], +healthcare [6], smart factories [7], and so on. In SemCom, we only transmit useful and necessary +information to the recipients. The semantic extraction (SE) is a process wherein the useful +and necessary features are extracted from the original raw data. For example, the essential +speech features are extracted using an attention-based mechanism in [8]–[10], image features +are extracted using ResNet-50 [11] in [12], etc. +During critical applications such as military operations, search operations by forest personnel +in a dense forest, medical emergencies in remote areas, fire incidents in a remote agricultural +land, the release of water from a nearby dam, etc., only the essential information needs to be +communicated on an urgent basis. The messages could be in the form of text or audio and +they come from a limited dataset. In a non-critical application, such as broadcasting a text/audio +summary of commentary provided by live football commentators. Among all the words spoken +by them, only a limited set of useful or important words are relevant to the game. These words are +drawn from a limited dataset such as football vocabulary [13] which includes words such as goal, +player names, red card, football, score, assist, half-time, etc. This limited dataset provides an +opportunity, in the context of SemCom design, for a significant overhead reduction by extracting +and processing only the relevant keywords. For example, an uttered commentary sentence in +2022 FIFA world cup final game is: ‘Messi shoots the ball into the right-bottom of the net +and it’s a goal!’ The extracted keywords in this example are Messi, shoots, ball, right-bottom, +net, goal. Only these keywords are transmitted in place of the entire sentence, and the receiver +reconstructs a meaningful sentence. The reconstructed sentence in this case is: ‘Messi shoots the +ball into the right-bottom of the net to score a goal.’ This sentence is not exactly the same as +the original sentence, but it conveys the same meaning. +The first goal of this paper is to use SemCom technology to reduce communication overhead +while maintaining a certain minimum accuracy in wireless communication systems. The over- +head reduction is performed with high accuracy in the literature [14], [15]. However, in some +applications, high data rates are preferred over high accuracy. As a result, we present the results +of the trade-off between overhead reduction and accuracy. Model parameters are chosen based +on the context. Instead of transmitting raw data, the transmitter is designed to transmit semantic +data, which significantly reduces network data traffic. A knowledge graph (KG) is a knowledge +base that integrates data using a graph-structured topology. They are used to store interconnected +event descriptions. These are used to predict the missing words in the received data (keywords) + +3 +to construct a meaningful sentence. +Next, we apply the designed SemCom system to a realistic problem in which the transmitter +and receiver are assumed to be a cloud server and a data center, respectively. A cloud server +located on a remote cloud platform has access to a large raw dataset that is stored in the cloud. +A data center is a centralized information storage facility that can store, process, and distribute +massive amounts of data to its users [16]. A data center requests a portion of the raw dataset +in various categories. These dataset categories are based on their size and accuracy levels. The +proposed SemCom technology-based communication system is used to transmit these datasets +to the data center. Since the data center also has access to the shared KG, these datasets can be +easily decoded. The cloud server determines the price of each category dataset based on its size +and quality (in terms of accuracy). +In our case, the data center is assumed to store the different categories of the same portion of +the dataset in its storage facility. The users can access these datasets directly from the storage +facility by paying a certain price. The different-quality datasets are priced differently based on +their quality and sizes. For example, a highly compressed dataset is small in size but poor in +accuracy, so it is less expensive, whereas a lightly compressed dataset is large in size but superior +in accuracy, so it is more expensive. Every user has a budget constraint, just as the data center has +a storage constraint. The data center replicates these datasets based on the needs and budgets of +the users. It strives to provide the highest-quality dataset possible to every user with a sufficient +budget. It is a highly desirable scenario for the following two reasons: (a) the data center can +maximize profits, and (b) the user is extremely satisfied with the service and can provide a high +rating as feedback. +Unfortunately, this is not always possible due to resource constraints, such as data storage +capacity at the data center and the need to serve all subscribed users. This problem is most +noticeable when there are a large number of subscribers. This scenario prompts us to formulate +an optimization problem in which the data center attempts to maximize its profit given the +constraint on its storage capacity in order to serve all the subscribed users. We denote this +problem as the Data Allocation Problem (DAP). +The organization of the paper is as follows: A brief literature review on SemCom technologies +and KGs is provided in Section II. We introduce our proposed SemCom system model and show +that the overall semantic distortion function has an upper bound in Section III. Then we define +our DAP, show that it is an NP-complete problem, and propose a greedy algorithm to solve it in + +4 +Section IV. We provide simulation results related to the proposed SemCom system model and +the solutions of the DAP in Section V. Finally, we conclude the paper along with a few future +research directions in Section VI. +II. RELATED WORK +The study on SemCom technologies started very recently. The following state-of-the-art survey +papers provide in-depth discussions on various SemCom technologies and their applications [17]– +[22]. Deep learning based SemCom technologies are proposed in [14], [15]. A brief tutorial on +the framework of SemCom and a method to calculate a bound on semantic data compression is +provided in [23]. The SemCom technology wherein both transmitter and receiver are empowered +with the capability of contextual reasoning is proposed in [24]. The SemCom technology for a +system where transmitter and receiver speak different languages is designed in [25]. A multiuser +task-oriented SemCom system for multi-modal data transmission is proposed in [26]. A nonlinear +transform based source-channel coding approach for SemCom is proposed in [27], wherein a +nonlinear transform mechanism is used to extract the source semantic features. The work in [28] +introduced a reinforcement learning (RL)-powered SemCom paradigm that gives a system the +ability to express semantics. In [29], a SemCom framework for textual data transmission is +proposed. In this framework, semantic information is represented by a KG made up of a set +of semantic triples and the receiver recovers the original text using a graph-to-text generation +model. Another SemCom system based on the KG is proposed in [30]. In this system, transmitted +sentences are converted into triplets using the KG, which are seen as fundamental semantic sym- +bols for semantic extraction and restoration, and they are ordered based on semantic relevance. +All of these works are focused on achieving an overhead reduction without compromising the +accuracy of the received data. None of these works investigated the possibility of further overhead +reduction, thereby improving transmission data rate, while sacrificing a little accuracy. This issue +is addressed in this paper using a shared knowledge base. +A significant research on the usage of KGs is carried out in the field of natural language +processing (NLP). The survey work in [31] provides a comprehensive study of KGs, which +leverage large-scale data collections for usage in a variety of industry and academic applications. +A survey paper based on KG is presented in [32]. Similarly, another survey paper on KG +text generation is presented in [33]. A method to generate a summary of sentences by using +a given set of keywords is proposed in [34]. Similarly, a method to generate a summary of + +5 +Auto-Decoder +Auto-Encoder +Semantic +Encoder +Channel +Encoder +Channel +Decoder +Channel Noise +Semantic +Decoder +Destination +Sentence +Generator +Source +Shared Knowledge +(K) +Context, +keywords, +events, etc. +Input +Text +Output +Text +Keyword +Extraction +𝑋𝑖 +Ω𝑖 +෩Ω𝑖 +ഥΩ𝑖 +෡Ω𝑖 +෠𝑌𝑖 +෠𝑌𝑖𝑗 +෠𝑋𝑖 +Sentence +Generator +෠𝑋𝑖𝑗 +argmax +෠𝑋𝑖𝑗,𝑗=1,…,𝑀 +BLEU( ෠𝑋𝑖𝑗; 𝑋𝑖) +argmax +෠𝑌𝑖𝑗,𝑗=1,…,𝑀 +BLEU(෠𝑌𝑖𝑗; ෠𝑋𝑖) +𝑋𝑖 +Auto-Decoder +Auto-Encoder +Semantic +Encoder +Channel +Encoder +Channel +Decoder +Channel Noise +Semantic +Decoder +Sentence +Generator +Source +Input +Text +Output +Text +Keyword +Extraction +෩Ω𝑖 +ഥΩ𝑖 +෡Ω𝑖 +෠𝑌𝑖 +Shared Knowledge +Context, +keywords, +events, etc. +(a) +(b) +Destination +𝑖 ∈ {1, … , 𝑁} +𝑋𝑖 +𝑖 ∈ {1, … , 𝑁} +Ω𝑖 +Fig. 1: The block diagram of our proposed SemCom system model. The model in Fig. (a) is +used for training the system parameters and the model in Fig. (b) is used for evaluating the +SemCom system. +sentences by using a knowledge base is shown in [35]. Recently, KGs are utlized in the context +of SemCom design [29], [30], [36], [37]. But these works do not focus on the issue presented +in this paper, which is to design a SemCom system with a significant overhead reduction with +a little compromise on accuracy. +III. SYSTEM MODEL +The system model of the proposed SemCom system is shown in Fig. 1. Let X be the input +text dataset with N sentences, Xi be the ith, i ∈ {1, . . . , N}, sentence of X, and K be the shared +knowledge base. First, we extract the keywords from X using K. Let the total set of keywords +be Ω = �N +i=1 Ωi, where Ωi denotes the set of keywords present in Xi. The keyword extraction +process is executed by multiplying the input sentence Xi, i ∈ {1, . . . , N}, with a binary vector + +6 +bi = [bi(ℓ), ℓ = 1, . . . , |Xi|],1 which is defined as follows: +bi(ℓ) = +� +� +� +� +� +1, +if ℓth word of sentence Xi, Xi(ℓ), is a keyword in K +0, +else. +(1) +Hence, Ω is obtained by collecting the non-zero elements from X ⊙ b, where ⊙ is a word-wise +multiplication operator. +Let ξ be a function which generates a set of M (say) sentences from a given set of keywords +with the help of a given knowledge base. Using the keywords in Ωi, i ∈ {1, . . . , N}, and the +knowledge K, for a given sentence Xi, the sentence generator at the transmitter generates a +set of M sentences using the function ξλ, where λ is a parameter. Let that set of sentences be +� +Xij, j = {1, . . . , M}. So, +� +Xij = ξλ(Ωi, K), j = {1, . . . , M}. +(2) +Next, out of these M sentences we choose the most semantically equivalent sentence based on +the BLEU scores [38]2 compared with input sentence Xi, i.e., +� +Xi = arg max +� +Xij,j=1,...,M +BLEU( � +Xij; Xi), i ∈ {1, . . . , N}. +(3) +Note that the set of sentences � +X = { � +Xi, i ∈ {1, . . . , N}}, is generated at the transmitter during +the training process only and it is shared with the receiver a priori. Next, ith keyword set Ωi +is encoded using the auto-encoder which consists of semantic and channel encoders. Let us +denote Sθe and Cφe as the semantic and channel encoders with θe and φe as the parameter sets, +respectively. After encoding Ωi, we get the following set of symbols: +�Ωi = Cφe(Sθe(Ωi)), i ∈ {1, . . . , N}. +(4) +The encoded set of symbols �Ωi is transmitted via the AWGN (additive white Gaussion noise) +channel. Let h be the channel gain and η be the noise which gets added to �Ωi during transmission. +So, the set of received symbols at the receiver is Ωi = h�Ωi + η. After receiving, this set of +symbols is decoded using the auto-decoder which consists of channel and semantic decoders. +Let us denote Cφd and Sθd as the channel and semantic decoders with φd and θd as the parameter +sets, respectively. After decoding Ωi, we get the following set of keywords: +�Ωi = Sθd(Cφd(Ωi)), i ∈ {1, . . . , N}. +(5) +1|A| denotes the cardinality of set A. +2We define the BLEU score in Section V. + +7 +From the decoded set of keywords and the shared knowledge K, the sentence generator at +the receiver generates a set of M sentences using the function ξµ, where µ is a parameter, and +let that set of sentences be �Yij, j = {1, . . . , M}, i.e., +�Yij = ξµ(�Ωi, K), j = {1, . . . , M}. +(6) +To select the most desired sentence among these sentences, we compute the BLEU scores +between �Yij, j = {1, . . . , M} and the sentence at the transmitter � +Xi, and choose the one which +maximizes the BLEU score. That is: +�Yi = arg max +�Yij,j=1,...,M +BLEU(�Yij; � +Xi), i = {1, . . . , N}, +(7) +where �Y = {�Yi, i ∈ {1, . . . , N}} denote the desired set of sentences generated at the receiver. +A. Training the System Model +Let X be the set of all possible sentences and pX(x), pλ(�x), and pµ(�y) denote the probability +distributions of sentences X ∈ X, �X ∈ +� +X, and �Y ∈ �Y , respectively, for all x, �x, �y ∈ X. +The sentences �X and �Y are generated by the sentence generators in transmitter and receiver, +respectively, parameterized by λ and µ, respectively, with the help of shared knowledge K (see +Fig. 1(a)). Note that, since �X and �Y are generated with the help of knowledge K, throughout +the paper, the distributions pλ(.) and pµ(.) are conditioned on the event K = k, ∀k ∈ K. +The overall cross entropy (CE) loss measures the difference between the actual probability +distribution at the input and the estimated probability distribution at the output and it can be +minimized using the stochastic gradient descent (SGD) methods [39]. So we define the overall +cross entropy (CE) loss as: +LCE(µ) = − +� +x∈X +pX(x) log pµ(�y/k). +(8) +Let DKL(p||q) represent the KL divergence between the probability distributions p and q and is +defined as follows: +DKL(p||q) = +� +x∈X +p(x) log p(x) +q(x). +(9) +From Fig. 1, we observe that there are information losses at auto-encoder, auto-decoder, and +sentence generation blocks both in transmitter and receiver. We aim to minimize the summation +of all these losses. The characterization of these losses are as follows. + +8 +• The loss of information between sentences X and �X at the transmitter is measured using +the CE loss. i.e., +LCE +1 (λ) = H(X) + DKL(pX(x)||pλ(�x/k)) +(10) += − +� +x∈X +pX(x) log pλ(�x/k). +(11) +• Similarly, the loss of information between sentences �X and �Y at the receiver is also measured +using the CE loss. i.e., +LCE +2 (µ, λ) = H(�X/K) + DKL(pλ(�x/k)||pµ(�y/k)) +(12) += − +� +�x∈X +pλ(�x/k) log pµ(�y/k). +(13) +• Lastly, the loss of information in the channel is measured in terms of mutual information +(MI) between transmitted symbols and received symbols. i.e., +LMI +3 (θe, φe, θd, φd) = I(�Ω; Ω). +(14) +Now, we define the overall semantic distortion function as follows: +L(θe, φe, θd, φd, λ, µ) = LCE +1 (.) + LCE +2 (.) − γLMI +3 (.), +(15) +where γ ≥ 0 is a hyper-parameter. In Theorem 1, we show that the overall semantic distortion +L(.)3 attains an upper bound which can be optimized. We aim to compute the optimal parameters +of auto-encoder, auto-decoder, and sentence generation blocks. These blocks are characterized +by the parameters θe, φe, θd, φd, λ, µ and are obtained by training the system model, shown in +Fig. 1(a), by minimizing the overall loss function defined in (15). +Theorem 1. The overall semantic distortion function attains the following upper-bound: +L(θe, φe, θd, φd, λ, µ) ≤ LCE(µ) − γLMI +3 (θe, φe, θd, φd) = B +(16) +Proof. The proof is given in Appendix A. +Remark 1. The upper-bound B, provided in Theorem 1, is proved to be optimized using the +SGD algorithms [39]. Also, in the published works [14], [25], the semantic distortion of the +overall system is minimized using a similar expression as that of B. +3We use (.) in place of parameters, wherever convenient, for ease of representation. + +9 +Hence, to minimize the overall semantic distortion defined in (15), we seek to minimize the +upper-bound B provided in Theorem 1. The loss due to mutual information LMI +3 (.) can be +estimated using state-of-the-art mutual information neural estimator (MINE) [40]. +B. Accuracy versus Overhead Reduction Trade-off +Given the limited size of knowledge base, though the accuracy of the reconstructed sentences +in �Y may not be sufficiently high, the useful content in those sentences is summarized and +conveyed to the receiver. This novel approach saves a significant amount of overhead. +There exists a trade-off between overhead reduction and the accuracy that depends on the size +of the knowledge base K. For example, if the size of the set K is small, then on an average only +a few keywords are extracted from the given input sentences in X, encoded and transmitted, +which implies higher amount of average overhead reduction. This creates a large amount of +missing information on an average, due to which accuracy of the reconstructed sentences in �Y +is expected to be low. On the other side, if the size of the set K is large, then on an average +a significant number of keywords are extracted from the given sentences in X, encoded and +transmitted, which implies lower average overhead reduction. This creates a small amount of +missing information on an average, due to which accuracy of the reconstructed sentences in �Y +is expected to be high. This phenomenon is numerically shown in Section V-A. +So, in this paper we aim at minimizing the transmission of average number of words per sen- +tence (equivalent to maximizing the average overhead reduction) by keeping a certain minimum +accuracy information τ in the received sentence, i.e., +min 1 +N +N +� +i=1 +|Ωi| +(17) +BLEU(�Yi; Xi) ≥ τ, i = {1, . . . , N}, +(18) +where |Ωi| denotes the number of keywords in Ωi that corresponds to sentence Xi. +C. Shared Knowledge Base +In this paper, we generate the shared knowledge base K by using the keywords from a limited +dataset Ω which consists of only the relevant words of a particular event, like that of a football +game in our case. We assume that both transmitter and receiver have access to K. During the +feature extraction process, in every sentence, only the words w ∈ Ω are encoded and transmitted + +10 +to the receiver in their corresponding time slots. At other time slots, nothing is transmitted. By +utilizing K, the receiver reconstructs the sentence based on the received words in that sentence. +To improve the accuracy of the reconstructed sentences, we can increase the size of K by adding +more keywords from the vocabulary generated using X. This result is shown using simulations +in Section V-A. +IV. DATA ALLOCATION PROBLEM +Let us assume that the transmitter and receiver shown in Fig. 1 are a cloud server and a data +center, respectively. The original copy of the dataset X is stored in the cloud, and a data center +requests a portion of it from the cloud server, say X ⊂ X. The cloud server uses SemCom +technology to communicate the requested portion of data to the data center, as described in +Section III. The data center obtains each copy of Xτ, for some specific values of τ ∈ [0, 1], by +tuning the parameter τ. It can only make a limited number of copies of these data due to storage +constraints. This data center serves a number of users, each of whom has a budget constraint. +Assume that the users do not have sufficient memory to store the data. They use data stored in +the data center. Once data portions are allocated, users can access them whenever they need. +Next, we formulate an optimization problem, which we call Data Allocation Problem (DAP), to +maximize the profit for this data center given its storage constraint and the budget constraints +of its associated users. +The setup used to describe the DAP is shown in Fig. 2. Let G be the number of data categories +that the data center has based on the accuracy τ. They are indexed by i = 1, . . . , G, such that +τ ∈ [τmin, τmax], τmin < τmax, is one-to-one corresponding to i ∈ {1, . . . , G}. The data center can +produce mi copies of Xi, i = 1, . . . , G. Each copy of data is zi in size, and its selling cost is ci +and they are related by zi1 < zi2 and ci1 < ci2 for i1 < i2, ∀i1 ∈ {1, . . . , G−1}, ∀i2 ∈ {2, . . . , G}. +These constraints indicate that the sizes and costs of different categories of data increase as τ +increases, which is shown using the indices i ∈ {1, . . . , G}. Now, we would carefully incorporate +the results obtained from (17) and (18) in terms of size and accuracy, so as to make the DAP +meaningful from the perspective of SemCom developed in this paper. +Assume the data center serves J users, with each user having a budget constraint bj ≥ c1, ∀j ∈ +{1, . . . , J}.4 Let Ui,j represent an indicator variable that returns 1 when bj ≥ ci, which means +4This constraint ensures that every user is eligible to access at least one category of data. + +11 +Fig. 2: This figure shows the setup used to describe the data allocation problem (DAP). +that the ith category data can be provided to user j when its cost is not more than the user +budget and 0 otherwise, i.e., for a given i ∈ {1, . . . , G}, j ∈ {1, . . . , J}, +Ui,j = +� +� +� +� +� +1, +if bj ≥ ci, +0, +else. +(19) +Similarly, let Vi,j represent an indicator random variable that returns 1 when the ith category +data is actually provided to user j and 0 otherwise, i.e., +Vi,j = +� +� +� +� +� +1, if category data i is actually provided to user j, +0, else. +(20) +So, the value of mi can be obtained as follows: +mi = +J +� +j=1 +Vi,j, ∀i ∈ {1, . . . , G}. +(21) +The value of mi computed using (21) shows that it also denotes the number of users who are +provided with ith, i ∈ {1, . . . , G}, category data. + +Shared Knowledge Base +Cloud Platform +Cloud +Server +Users +0 +Data Center +0 +O +O +O +O12 +Now we consider the purchase price of the data from the cloud server. The limited backhaul +capacity between the cloud server and data center constrains the rate of data transfer between +them. Due to this, the size of the data, and hence the use of SemCom technology, plays an +important role. The purchase prices d(zi), i ∈ {1, . . . , G}, of different categories of data from the +cloud server are based on their sizes, i.e., d(zi1) < d(zi2) for i1 < i2, ∀i1 ∈ {1, . . . , G−1}, ∀i2 ∈ +{2, . . . , G}. Based on this information, an optimization problem, which we call DAP, to maximize +the profit of the data center is formulated as follows: +max +Vi,j, +i∈{1,...,G}, +j∈{1,...,J} +G +� +i=1 +� +ci +J +� +j=1 +Vi,j − d(zi) +� +(22a) +G +� +i=1 +� +zi +J +� +j=1 +Vi,j +� +≤ Z, +(22b) +G +� +i=1 +J +� +j=1 +Ui,jVi,j = J, +(22c) +G +� +i=1 +Vi,j = 1, j ∈ {1, . . . , J}, +(22d) +Vi,j ∈ {0, 1}, ∀i ∈ {1, . . . , G}, j ∈ {1, . . . , J}. +(22e) +The constraint (22b) ensures that the total size of all copies of allocated data is within the limit +of the maximum permissible size Z at the data center. Similarly, the constraint (22c) indicates +that all data portions are assigned to users while making sure that the cost of every allocated +data portion is not more than the user budget (see (19)), whereas the constraint (22d) indicates +that each user j ∈ {1, . . . , J} is allocated exactly one category of data. The last constraint (22e) +indicates that the variables Vi,j, ∀i ∈ {1, . . . , G}, j ∈ {1, . . . , J}, are binary. This makes our +optimization problem, DAP, defined in (22a)-(22e) as a type of binary integer programming. +In general, the integer programming problems are shown to be NP-complete [41]. We show +that the DAP belongs to the class of NP-complete problems by reducing the well known knapsack +problem (KP) [42] to it. +Theorem 2. The DAP is NP-complete. +Proof. The proof is given in Appendix B + +13 +A. Greedy Algorithm +In Theorem 2, we have shown that the DAP belongs to the class of NP-complete problems. +Now, we present a greedy algorithm to solve the DAP. First, we identify the condition under +which the solution is feasible. From the DAP formulation, it is clear that there is a limit to the +number of users that the data center can serve. In the worst-case scenario, all users could be +assigned the least desirable data category, i = 1. The total data size in this case is z1 times J +and is limited by Z. Hence the condition for the solution to exist is: +J ≤ Z +z1 +. +(23) +Given that the primary goal of the DAP is to maximize profit for the data center while ensuring +data allocations to all users, the proposed algorithm allocates the best possible category data to +each user based on their budget. This is accomplished by determining k(j) ∈ {1, . . . , G}, ∀j ∈ +{1, . . . , J}, such that ck(j) ≤ bj < ck(j)+1 (which is same as finding i such that Ui,j = 1 and +Ui+1,j = 0), and allocating the data category i = k(j) for jth user. This gives Vi,j = Vk(j),j = +1, ∀j ∈ {1, . . . , J}. Next, the algorithm computes the total size due to this allocation policy, i.e., +Z = �J +j=1 zk(j)Vk(j),j. If Z ≤ Z, then we have found the solution, V ⋆, of the DAP and it is as +follows: +V ⋆ +i,j = +� +� +� +� +� +1, +if i = k(j), +0, +else. +(24) +And the profit is P = �G +i=1 +� +ci +�J +j=1 V ⋆ +i,j − d(zi) +� +. But if Z > Z, which implies the violation of +the constraint (22b), then the algorithm updates the data allocation policy in the following way. It +finds the smallest argument k(j′) which minimizes the ratio rk(j) = (ck(j)/zk(j)), ∀j ∈ {1, . . . , J}, +and does the following updates using it: Vk(j′),j′ = 0, Vk(j′)−1,j′ = 1, Z → Z − zk(j′) + zk(j′)−1, +k(j′) → k(j′)−1.5 The algorithm again compares Z, computed with updated value of k(j′), and +Z. This process continues until it encounters Z ≤ Z and the solution is V ⋆ = V . The detailed +algorithm is provided in Algorithm 1. +5This approach ensures the smallest possible reduction in selling cost from P to (P − ck(j′) + ck(j′)−1) and/or the largest +possible data size reduction from Z to (Z − zk(j′) + zk(j′)−1), which aids in satisfying the constraint (22b). If only the selling +cost is considered in place of the ratio, which is the case in most greedy algorithms, the algorithm ignores the impact of data +sizes on the DAP. We call this algorithm as greedy-cost algorithm and show, using simulations, in Section V-B that the proposed +greedy algorithm outperforms the greedy-cost algorithm. + +14 +Algorithm 1 Greedy Algorithm +1: Input: ci, zi, d(zi), Ui,j, i ∈ {1, . . . , G}, j ∈ {1, . . . , J}, G, J, Z +2: if J ≤ Z/z(1) then +3: +Initialize j = 1, r(1) = ∞, i = 2, V = 0G×J. +4: +while i ≤ G do +5: +ri ← ci/zi, +6: +i ← i + 1. +7: +end while +8: +while j ≤ J do +9: +Find i such that Ui,j = 1 and Ui+1,j = 0. +10: +k(j) ← i +11: +Vk(j),j ← 1 +12: +j ← j + 1. +13: +end while +14: +Compute Z = �J +j=1 zk(j) (Note: Vk(j),j = 1, ∀j ∈ {1, . . . , J}). +15: +while (1) do +16: +if Z ≤ Z then +17: +End the algorithm and output V ⋆ = V . +18: +else +19: +Compute k(j′) = arg mink(j),j∈{1,...,J} rk(j). +20: +Vk(j′),j′ ← 0, Vk(j′)−1,j′ ← 1, +21: +Z ← Z − zk(j′) + zk(j′)−1, +22: +k(j′) ← k(j′) − 1. +23: +end if +24: +end while +25: else +26: +End the algorithm and display ‘No feasible solution’. +27: end if +28: Output: V ⋆ of size G × J, and the profit: P = �G +i=1 +� +ci +�J +j=1 V ⋆ +i,j − d(zi) +� +. + +15 +TABLE I: Simulation parameters +Number of matches used in training +1580 +Number of matches used in evaluation +340 +Number of epochs during training +10 +SNR +6 dB +Learning rate +0.001 +Batch Size +64 +Channel +AWGN +B. Computational Complexity of the Greedy Algorithm +Now, we find the computational complexity of the proposed greedy algorithm, if the solution +exists. First, we compute the values of ri, ∀i ∈ {1, . . . , G}, using a loop described in lines 4–7 +of Algorithm 1. This computation results in the time complexity of O(G). Similarly, we find +that the computational complexity of the loop described in lines 8–13 is O(J). Next, in the loop +described in lines 15–24, we compute the argument minimizer in line 19 whose computational +complexity is O(J), and this loop, in the worst case, executes till all k(j), j ∈ {1, . . . , J}, become +1. This happens after O(G) times execution of the loop. Thus the computational complexity of +the loop described in lines 15–24 is O(GJ). Hence, the total computational complexity of the +proposed greedy algorithm in Algorithm 1 is O(G + J + GJ). +The computational complexity of finding the solution for the DAP using the brute-force search +method is O(2GJ), since it uses all the possible binary matrices of size G × J, sequentially, to +compute the solution. +Remark 2. The proposed greedy algorithm is highly efficient in terms of the computational +complexity w.r.t. the brute-force search method, i.e., O(G + J + GJ) << O(2GJ). +The comparison study of the numerical solutions of the DAP using the proposed greedy +algorithm and Gurobi software [43] as a solver is shown in Section V-B. +V. SIMULATION RESULTS +In this section, we first provide the simulation results related to the designed SemCom system +in Section V-A and then provide the results related to the DAP in Section V-B. + +16 +A. The performance of SemCom System Model +First, we evaluate the performance of the text data transmission in terms of accuracy using +BLEU score [38]. The BLEU(s, ˆs) ∈ [0, 1] score between transmitted sentence s and recon- +structed sentence ˆs is computed as follows: +BLEU(s, ˆs) = BP(s, ˆs) exp +� W +� +n=1 +wn ln pn(s, ˆs) +� +, +(25) +where pn denote the modified n-gram precision function up to length W, wn denote the weights, +and brevity penalty (BP) is given by the following expression: +BP(s, ˆs) = +� +� +� +� +� +1 +ℓc > ℓr +e1−ℓr/ℓc +ℓc ≤ ℓr, +(26) +where ℓc is the length of the candidate translation and ℓr is the effective reference corpus +length [38]. +In our work, we use the dataset provided in [44]. We parse the football commentary data of +1920 matches from the website goal.com. The considered football matches are from Union of +European Football Associations (UEFA) Champions League, UEFA Europa League, and Premier +League between 2016 and 2020. The simulation parameters used for plots in this section are +shown in Table I. The simulations are performed in a computer with NVIDIA GeForce RTX +3090 GPU and Intel Core i9-10980XE CPU with 256GB RAM. +Let ρ be the fraction of the total vocabulary V , which contains all the dataset words, to be +added to K. ρ = 0 indicates that no additional vocabulary is added and the system is evaluated +only with the initial keyword set Ω0. Based on the way of adding the vocabulary words to +Ω0, we propose two types of schemes. In the first type, ρ|V | vocabulary words are uniformly +chosen at random from V and added to K. In the second type, the words in V are first arranged +in the decreasing order of the frequency of appearances in the dataset, and then the first ρ|V | +vocabulary words are added to K. We call these schemes as ‘Proposed Method (random)’ and +‘Proposed Method (order)’, respectively. +The accuracy performance of both the schemes in terms of BLEU score vs. ρ is shown in +Fig. 3. From the plot we can infer that even with ρ = 0, the initial keyword set can produce a +BLEU score of 0.55 (for 1-gram). This shows that the context-related keywords produce good +results. Also, we see that as we add more vocabulary words to Ω0, the BLEU score increases. +For the same value of ρ and n, the ‘order’ scheme performs better than the ‘random’ scheme + +17 +because of the addition of high frequency words. And, in terms of different n-grams, BLEU +score decreases as n increases, which is an expected result. +Next, we evaluate the performance of the proposed schemes, in terms of the transmission of +average number of words per sentence, with respect to the deep learning based SemCom system +method named DeepSC [14] and the results are shown in Fig. 4a. Let W denote the average +number of words per sentence. From the plot we observe that both the schemes outperform +DeepSC. Among the proposed schemes, for a given ρ the ‘random’ scheme outperforms the +‘order’ scheme. This is because, in the ‘order’ scheme high frequency words are added which +increases the number of words to be encoded in the input data as compared with the ‘random’ +scheme. +Now, we solve the problem presented in (17) and (18) using both the proposed schemes. +For this purpose, we evaluate W vs. τ and the results are shown in Fig. 4b. From the plot we +observe that both the schemes outperform DeepSC. Also, we see that the performance of both +the schemes is same for a given accuracy threshold τ. Hence, we can choose any one of the +proposed methods to solve the problem given in (17) and (18). +Now, we compare the performance of our scheme with that of the DeepSC and the scheme +proposed in [25]. In the DeepSC scheme [14] and the proposed scheme, an average n0 (say) +number of symbols are used for every word during encoding, and the scheme proposed in [25] , +named Adaptive Scheme, uses an adaptive method for choosing the number of symbols for every +word that depends on the size of that word. Let Xi, i = {1, . . . , N}, denote the ith sentence in +the dataset X and ωiℓ, ℓ = {1, . . . , |Xi|}, denote the ℓth word in the sentence Xi, then we can +write, +Xi = {ωiℓ| ℓ = 1, . . . , |Xi|}, ∀i ∈ {1, . . . , N}. +(27) +Hence, the total number of words present in the dataset are N0 = �N +i=1 |Xi|. In our proposed +scheme, as described in the system model (see Section III), we only transmit the extracted +keywords before encoding. So the total number of keywords to be transmitted is: +Nτ = +N +� +i=1 +|Ωi(τ)|, +(28) +where Ωi(τ), i = {1, . . . , N}, is the set of keywords present in the sentence Xi for a given +accuracy τ. The total number of symbols used for communicating the data in the DeepSC + +18 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +BLEU Score +1-gram (random) +2-gram (random) +3-gram (random) +4-gram (random) +1-gram (order) +2-gram (order) +3-gram (order) +4-gram (order) +Fig. 3: This plot shows the BLEU score vs. ρ for different values of n-grams, where n = +{1, 2, 3, 4}. +0 +0.2 +0.4 +0.6 +0.8 +1 +5 +10 +15 +20 +Proposed Method (random) +Proposed Method (order) +DeepSC +(a) +0.5 +0.6 +0.7 +0.8 +0.9 +1 +5 +10 +15 +20 +Proposed Method (random) +Proposed Method (order) +DeepSC +(b) +Fig. 4: The plot in left (respectively, right) shows the average number of words per sentence vs. +ρ (respectively, τ) for the proposed schemes and the DeepSC scheme [14]. + +19 +scheme, Ψ0, and the proposed scheme, Ψτ, respectively, are as following: +Ψ0 = n0N0, +(29) +Ψτ = n0Nτ. +(30) +Let piℓ denote the probability of occurrence of the word ωiℓ, i.e., +piℓ = +|ωiℓ| +�N +i=1 +�|Xi| +ℓ=1 |ωiℓ| +, ∀ℓ ∈ {1, . . . , |Xi|}, ∀i ∈ {1, . . . , N}. +(31) +Now, based on the value piℓ, in the adaptive scheme [25] the number of symbols used to encode +the word ωiℓ is chosen using the following equation: +qiℓ = min (max(nmin, ⌊n0N0piℓ + 0.5⌋), n0) , +(32) +where nmin < n0 denote the minimum number of possible symbols. Hence, the total number of +symbols used in the adaptive scheme [25] is: +�Ψ = +N +� +i=1 +|Xi| +� +ℓ=1 +qiℓ. +(33) +We show the comparisons among the proposed method and the schemes proposed in [14], [25] +in terms of �Ψ, Ψ0, and Ψτ, in Fig. 5a. +Now, we compare the performance of our scheme in terms of the accuracy parameter τ. Let +ατ represent the product of the average number of symbols used for each word and the fraction +of total words transmitted required to achieve the accuracy τ. Since all words are transmitted +in both DeepSC [14] and adaptive schemes [25], the ατ values for these schemes are n0(τ) +and �Ψ(τ)/N0, respectively. In case of the proposed scheme, only a fraction of all the words are +transmitted. It uses same number of symbols as that of the DeepSC scheme, that is n0(τ), but +still it is able to achieve better results due to the transmission of only the keywords. The value +of ατ for the proposed scheme is n0(τ) Nτ +N0 . These comparisons are shown in Fig. 5b. +B. Solutions of the Data Allocation Problem +Now, we present the simulation results related to the solution of the DAP defined in (22a)– +(22e). We solve the DAP using three methods: Optimal, Greedy, and Greedy-cost. We refer to +the solutions obtained by the Gurobi software [43] and the greedy algorithm (see Algorithm 1) +as Optimal and Greedy, respectively. Similarly, the solution obtained by the algorithm, which is +the same as that of the greedy algorithm, except that the argument minimizer minimizes the cost + +20 +0 +0.2 +0.4 +0.6 +0.8 +1 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +106 +Proposed Scheme +Adaptive Scheme +DeepSC Scheme +(a) +0.5 +0.6 +0.7 +0.8 +0.9 +1 +100 +Proposed Scheme +Adaptive Scheme +DeepSC Scheme +(b) +Fig. 5: The plot in left shows the total number of symbols used in each of the schemes with +respect to ρ. From this plot, we observe that the proposed scheme transmits a significantly smaller +number of symbols compared with both schemes, from ρ = 0 to ρ = 0.6. We use nmin = 1, +n0 = 4. The plot in right shows the ατ values in each of the schemes with respect to the given +accuracy τ. From this plot, we observe that the proposed scheme outperforms both schemes for +accuracy levels upto 82%. +ci instead of the ratio ci/zi, ∀i ∈ {1, . . . , G}, in line 19 of the proposed Algorithm 1 is called +greedy-cost. The primary goal of using the greedy-cost algorithm is to demonstrate numerically +that maximizing profit by greedily changing only the costs does not produce better results than +the proposed greedy algorithm. In our simulations, we have assumed that the data center has +purchased a standard persistent disk (PD) from Google cloud [45] and has a memory capacity of +Z = 64TB, minimum and maximum data sizes are 10GB and 100GB, respectively, and G = 20. +The costs and sizes are chosen uniformly at random from [0, 1] and [10, 100], respectively, and +the number of iterations in our simulations is 25. +First, we compute the total profit gained by the data center with respect to the number of users +it serves, and the results obtained by all three methods are shown in Fig. 6a. From the plot, we +can observe that for a set of a smaller number of users, in particular from J = 500 to J = 1200 +in our case, the profit computed by all three methods is the same. This is because every method +is successful in allocating the best possible category data to every user without violating the +size constraint (22b), due to the small number of users. As the number of users increases the + +21 +1000 +1500 +2000 +2500 +3000 +Number of Users +200 +300 +400 +500 +600 +700 +Profit +Optimal +Greedy +Greedy-cost +(a) +0 +10% +20% +30% +40% +50% +60% +Discount Factor +550 +560 +570 +580 +590 +600 +610 +Profit +Optimal +Greedy +Greedy-cost +(b) +Fig. 6: The plot in left shows the total profit gained using all three algorithms with respect +to the number of subscribed users J. The maximum profit is observed for J = 2100, and the +profit computed by the proposed greedy algorithm (respectively, greedy-cost algorithm) at the +same value of J is 90.54% (respectively, 77.76%) of the optimal maximum profit. The plot in +right shows the total profit gained using all three algorithms with respect to the discount factor. +The average fall of the profit with each discount factor for the optimal, greedy, and greedy-cost +algorithms is 0.653%, 0.666%, and 0.706%, respectively. This shows that the discount factor +does not affect the profit significantly. Hence, there is a room to attract more subscribers without +loosing the significant profit. We use J = 1500 in this case. +profit obtained by Optimal starts outperforming both greedy algorithms. The proposed greedy +algorithm solution closely follows the optimal solution, but the greedy-cost algorithm solution +starts moving away significantly from the optimal solution. This is because the greedy-cost +algorithm only accounts for the cost maximization without bothering about the data sizes, which +results in violation of the size constraint (22b) more often than the proposed greedy algorithm, +which accounts for both the costs and the data sizes. The plot in Fig. 6a also shows that the +profit increases initially for all three algorithms, then reaches its maximum value and begins to +decrease again. +Next, we evaluate the performance of the algorithms in terms of profit gained with respect +to the discount factor, which is the percentage discount given by the data center to its users +in comparison to the purchase price of the data to attract more new subscribers. The results + +22 +500 +1000 +1500 +2000 +2500 +3000 +Number of Users +2 +2.5 +3 +3.5 +4 +4.5 +5 +Average Rating +Optimal +Greedy +Greedy-cost +Fig. 7: This plot shows the average rating given by users for the data center’s service using each +of the algorithms w.r.t. the number of users. For J = 2100 users, where profit is maximized, +the average ratings are 3.27, 3.18, and 3.02 for services provided using the optimal, greedy, and +greedy-cost algorithm solutions, respectively. This result demonstrates that the average rating +provided for the optimal solution is not significantly higher compared to that of the proposed +greedy algorithm solution. +are shown in Fig. 6b. As expected, the optimal solution outperforms both greedy algorithms, +and also, the proposed greedy algorithm outperforms the greedy-cost algorithm, for all discount +factors. Also, as the amount of discount increases, the profit for a fixed number of users reduces, +which is also along expected lines. +Now, we evaluate the users’ satisfaction with the service provided by the data center by using +their ratings. The ratings are provided by users using one of the numbers between 1 and 5, where +1 and 5 signify the worst and best user experiences, respectively. Let ¯i(j),˜i(j) ∈ {1, . . . , G}, +be the quantities such that U¯i(j),j = 1 and U¯i(j)+1,j = 0, and V ⋆ +˜i(j),j = 1, j ∈ {1, . . . , J}. Let + +23 +TABLE II: Relation between the satisfaction levels and user ratings +Satisfaction Level +0 +1-2 +3-5 +6-10 +11-20 +User Rating +5 +4 +3 +2 +1 +SL(j) denote the satisfaction level of the users j ∈ {1, . . . , J}. We define satisfaction level as +SL(j) = ¯i(j) −˜i(j), j ∈ {1, . . . , J}. For the purpose of evaluation, we assume that the ratings +and satisfaction levels are related as shown in Table II. The plot in Fig. 7 shows the average rating +provided by the subscribed users for the services provided by all three algorithms. From the plot +we observe that all users provide the rating of 5 to the service when number of subscribers +is low, i.e., 500 ≤ J ≤ 1100 in our example. This is due to the lower number of subscribers, +which resulted in the best possible category of data allocation based on each subscriber’s budget. +However, as J increases beyond 1100, every algorithm starts allocating lower level categories +of data to some of the subscribers so that the size constraint (22b) is not violated. Hence, there +is a fall in the average rating. +The histograms of the ratings provided by the users are shown in Fig. 8. These plots support +the observation that when J is small, the size constraint (22b) is easily satisfied, and thus the +best possible category of data is allocated to each user. Conversely, when J is large, to satisfy +the size constraint (22b) algorithms tend to allocate a lower category of data to users, resulting +in lower ratings. +VI. CONCLUSIONS AND FUTURE WORK +In this paper, we first extracted relevant keywords from the dataset using the shared knowledge +base. Then, using the received keywords and the shared knowledge, we designed an auto-encoder +and auto-decoder that only transmit these keywords and, respectively, recover the data. We +proved that the overall semantic distortion function has an upper bound that is shown to be +optimized using the SGD algorithms. We computed the accuracy of the reconstructed sentences +at the receiver quantitatively. We demonstrated through simulations that the proposed methods +outperform a state-of-the-art method in terms of the average number of words per sentence. The +designed SemCom system is then applied to a realistic scenario in which a cloud server and a data +center serve as transmitter and receiver, respectively. We formulated a data allocation problem +(DAP), in which the data center optimally allocates various categories of datasets received from + +24 +1 +2 +3 +4 +5 +Rating +0 +200 +400 +600 +800 +1000 +1200 +Number of Users +Optimal +Greedy +Greedy-cost +1 +2 +3 +4 +5 +Rating +0 +500 +1000 +1500 +Number of Users +Optimal +Greedy +Greedy-cost +Fig. 8: These plots show the histogram of the ratings provided by the users for each of the +algorithms. The left figure shows the results for a small number of users, J = 1200, whereas +the right figure shows the results for a large, J = 4000, number of users. +the cloud server to its subscribers. We proved that the DAP belongs to a class of NP-complete +problems and proposed a greedy algorithm for solving it. Furthermore, we have numerically +demonstrated that the solutions of the proposed greedy algorithm, in terms of profits, are 90% +of the optimal solutions. In this paper, we focused solely on the text dataset; however, similar +approaches can be used in the future for other types of datasets such as image, audio, and video. +Also, a real-time DAP can be formulated by considering the dynamic storage facility using cache +memories at the data center in place of a static storage facility. +APPENDIX A +PROOF OF THEOREM 1 +Let us define the following term, parameterized by λ, µ, and k ∈ K: +δ(λ, µ, k) ≜ log +�pµ(�y/k) +pλ(�x/k) +� +, ∀�x, �y ∈ X. +(34) + +25 +From (15), we know that +L(.) = LCE +1 (.) + LCE +2 (.) − γLMI +3 (.) +(35a) += − +� +x∈X +pX(x) log pλ(�x/k) − +� +�x∈X +pλ(�x/k) log pµ(�y/k) − γI(�Ω; Ω) +(35b) += − +� +x∈X +pX(x) log +� +pλ(�x/k)pµ(�y/k) +pµ(�y/k) +� +− +� +�x∈X +pλ(�x/k) log +� +pµ(�y/k)pλ(�x/k) +pλ(�x/k) +� +−γI(�Ω; Ω) +(35c) += − +� +x∈X +pX(x) log pµ(�y/k) + δ(λ, µ, k)− +� +�x∈X +pλ(�x/k) log pλ(�x/k)−δ(λ, µ, k)−γI(�Ω; Ω) +(35d) += LCE(.) + H(�X/K) − γI(�Ω; Ω) +(35e) +≤ LCE(.) − γI(�Ω; Ω). +(35f) +In (35b), we expand the loss function expressions using their respective definitions provided +in (11), (13), and (14), respectively. In (35c), we multiply and divide pµ(�y/k) and pλ(�x/k) in +the first and second terms, respectively. Using (34) and algebraic simplifications we get (35d). +By using (8) and the definition of entropy, we write (35e). And, finally the inequality in (35f) +is due to H(�X/K) ≥ 0. +APPENDIX B +PROOF OF THEOREM 2 +The decision version of the DAP is as follows: “Given a number L, does there exist a binary +matrix V , of size G×J, that satisfy the constraints (22b)-(22d) such that �G +i=1 +� +ci +�J +j=1Vi,j−d(zi) +� +≥ L”? Given V , we can check in polynomial time whether it satisfies (22b)-(22d) and whether +�G +i=1 +� +ci +�J +j=1 Vi,j − d(zi) +� +≥ L. Thus the DAP is in class NP [46]. By using (21), we simplify +the expressions (22a) and (22b) as following: +max +mi,i∈{1,...,G} +G +� +i=1 +cimi − d(zi) +(36a) +G +� +i=1 +zimi ≤ Z. +(36b) +We now show that the DAP is NP-complete by reducing the knapsack problem (KP), which has +been shown to be NP-complete [46], to it. + +26 +Let us consider the following KP: We want to pack n different types of items in a knapsack +which can withstand a maximum weight of W. Each item of type i ∈ {1, . . . , n} is categorised +with two parameters: a weight wi and a value vi. The goal of the KP is to find a set of items +that produce the maximum possible value, with the restriction that the total weight of the set +should not exceed W. It is also written as follows: +max +xi,i∈{1,...,n} +n +� +i=1 +vixi +(37a) +n +� +i=1 +wixi ≤ W, +(37b) +xi ∈ {0, 1, . . . , }, +(37c) +where xi, i ∈ {1, . . . , n}, denote the number of items of type i. The decision version of the KP +is as follows: “Given a number L, is it possible to achieve �n +i=1 vixi ≥ L without exceeding the +total weight constraint W”? Let us denote the decision version inequalities of both the problems +as following: +D1 : +G +� +i=1 +cimi − d(zi) ≥ L, +(38a) +D2 : +n +� +i=1 +vixi ≥ L. +(38b) +Now, we show that the KP is polynomial-time reducible to DAP, i.e., KP

, Lior Wolf . +noisy (Ballance et al., 2016; Huang et al., 2019; Foxen et al., +2020). However, adapting existing classical ECC meth- +ods to the quantum domain (QECC) is not straightforward +(Raimond & Haroche, 1996). +The first difficulty in applying ECC-based knowledge to +QECC arises from the no-cloning theorem for quantum +states (Wootters & Zurek, 1982), which asserts that it is +impossible to clone a quantum state and thus add arbitrarily +redundant parity information, as done in classical ECC. The +second challenge is the need to detect and correct quan- +tum continuous bit-flips, as well as phase-flips (Cory et al., +1998; Schindler et al., 2011), while classical ECC addresses +only bit-flip errors. A third major challenge is the wave +function collapse phenomenon: any direct measurements +(while being standard in ECC) of the qubits would cause the +wave function to collapse and erase the encoded quantum +information (Neumann et al., 1955). +Shor (1995) proposed the first quantum error correction +scheme, demonstrating that these challenges can be over- +come. Subsequently, threshold 1 theorems have shown that +increasing the distance of a code will result in a correspond- +ing reduction in the logical error rate, signifying that quan- +tum error correction codes can arbitrarily suppress the log- +ical error rate (Aharonov & Ben-Or, 1997; Kitaev, 1997a; +Preskill, 1998). This distance increase is obtained by de- +veloping encoding schemes that reliably store and process +information in a logical set of qubits, by encoding it redun- +dantly on top of a larger set of less reliable physical qubits +(Nielsen & Chuang, 2002). +Most current QECC methods fall in the category of stabilizer +codes, which can be seen as a generalization of the classical +linear codes (Gottesman, 1997). Similarly to the classical +parity check constraints, a group of stabilizer operators can +provide a syndrome, while preserving the logical quantum +states and enabling error detection. +Optimal decoding is defined by the unfeasible NP-hard max- +imum likelihood rule (Dennis et al., 2002; Kuo & Lu, 2020). +Despite considerable research dedicated to the design of +codes with some additional algebraic structure, which can +support efficient decoding, such QECC decoders remain in- +1In this context, a threshold is the minimal error rate such that +adding more physical qubits in the code results in fewer logical +errors. +arXiv:2301.11930v1 [quant-ph] 27 Jan 2023 + +Deep Quantum Error Correction +efficient or impractical (Calderbank & Shor, 1996; Schindler +et al., 2011). One of the most promising categories of +codes is that of topological codes, particularly surface codes, +which originated from Toric codes (Bravyi & Kitaev, 1998; +Freedman & Meyer, 2001; Kitaev, 1997a;c; Dennis et al., +2002; Kitaev, 2003). Given boundary conditions, the idea +is to encode every logical qubit in a 2D lattice of physical +qubits. This local design of the code via nearest-neighbors +coupled qubits allows the correction of a wide range of er- +rors. Under certain assumptions, surface codes provide an +exponential reduction in the error rate (Kitaev, 1997a;b). +In this work, we present a novel neural quantum decoding +algorithm that can be applied to any stabilizer code. We first +predict an approximation of the system noise, turning the +quantum syndrome decoding task into a form that is compli- +ant with the classical ECC Transformer decoder (Choukroun +& Wolf, 2022), which is based on the parity-check matrix. +This initial estimate of the noise is reminiscent of Monte +Carlo Markov Chains methods (Wootton & Loss, 2012), +which start the decoding process with a random error that is +compatible with the syndrome, and then refine it iteratively. +Second, in order to support logical error optimization, we +develop a novel differentiable way of learning under the +Boolean algebra constraints of the logical error rate met- +ric. Finally, we propose a new architecture that is capable +of performing quantum error correction under faulty syn- +drome measurements. As far as we can ascertain, this is +the first time that (i) a Transformer architecture (Vaswani +et al., 2017) is applied to the quantum syndrome decoding +setting, (ii) a decoding algorithm is optimized directly over +a finite field metric, (iii) a deep neural network is applied to +the faulty syndrome decoding task. +Applied to a variety of code lengths, noise and measurement +errors, our method outperforms the state-of-the-art method. +This performance gain holds even when employing shallow +architectures, providing a sizable speedup over the existing +neural decoders, which are known to be computationally +inefficient (Edmonds, 1965; Dennis et al., 2002). +2. Related Work +Maximum Likelihood quantum decoding is an NP-hard +problem (Kuo & Lu, 2020) and several approximation meth- +ods have been developed as practical alternatives, trading off +the accuracy for greater complexity. The Minimum-Weight +Perfect Matching (MWPM) algorithm runs in polynomial +time and is known to achieve nearly the optimal thresh- +old for independent noise models (Bose & Dowling, 1969; +Micali & Vazirani, 1980). It is formulated as a problem +of pairing excitation and can be solved using Blossom’s +algorithm (Kolmogorov, 2009). Approximations of this +algorithm have been developed by parallelizing the group +processing or removing edges that are unlikely to contribute +to the matching process. However, MWPM implementa- +tions and modifications generally induce degradation in +accuracy or are too slow even for the current generation of +superconducting qubits (Fowler et al., 2012; Fowler, 2013; +Meinerz et al., 2022). +Among other approaches, methods based on Monte Carlo +Markov Chains (MCMC) (Wootton & Loss, 2012; Hutter +et al., 2014) iteratively modify the error estimate, to in- +crease its likelihood with respect to the received syndrome. +Renormalization Group methods (Duclos-Cianci & Poulin, +2010; 2013) perform decoding by dividing the lattice into +small cells of qubits. Union-Find decoders (Huang et al., +2020; Delfosse & Nickerson, 2021) iteratively turn Pauli +errors into losses that are corrected by the Union-Find data +structure and are more suited to other types of noise. +Multiple neural networks-based quantum decoders have +emerged in the last few years (Varsamopoulos et al., 2017; +Torlai & Melko, 2017; Krastanov & Jiang, 2017; Cham- +berland & Ronagh, 2018; Andreasson et al., 2019; Wagner +et al., 2020; Sweke et al., 2020; Varona & Martin-Delgado, +2020; Meinerz et al., 2022). These methods are amendable +to parallelization and can offer a high degree of adaptability. +Current contributions make use of multi-layer perceptrons +(Varsamopoulos et al., 2017; Torlai & Melko, 2017; Kras- +tanov & Jiang, 2017; Wagner et al., 2020) or relatively shal- +low convolutional neural networks (Andreasson et al., 2019; +Sweke et al., 2020), or couple the decoding with classical +methods for boosting the decoding accuracy (Meinerz et al., +2022). +In parallel, deep learning methods have been improving +steadily for classical ECC, reaching state-of-the-art results +for a broad range of codes. Many of these methods rely on +augmenting the Belief-propagation algorithm with learnable +parameters (Pearl, 1988; Nachmani et al., 2016; Lugosch +& Gross, 2017; Nachmani & Wolf, 2019; Buchberger et al., +2020), while others make use of more general neural net- +work architectures (Cammerer et al., 2017; Gruber et al., +2017; Kim et al., 2018; Bennatan et al., 2018; Choukroun & +Wolf, 2023). +Recently, Choukroun & Wolf (2022) have proposed a +transformer-based architecture (Vaswani et al., 2017) that +is currently the state of the art in neural decoders for clas- +sical codes. We choose to address QECC by expanding +the ECCT architecture to account for the challenges arising +from the transition from classical to quantum neural decod- +ing. Analogous expansions can be applied to other neural +decoders. +3. Background +We provide the necessary background on classical and quan- +tum error correction coding and a description of the state-of- + +Deep Quantum Error Correction +the-art Error Correction Code Transformer (ECCT) decoder. +3.1. Classical Error Correction Code +A linear code C is defined by a binary generator matrix G +of size k × n and a binary parity check matrix H of size +(n − k) × n defined such that GHT = 0 over the order 2 +Galois field GF(2). +The input message m ∈ {0, 1}k is encoded by G to a code- +word x ∈ C ⊂ {0, 1}n satisfying Hx = 0 and transmitted +via a symmetric (potentially binary) channel,e.g., an additive +white Gaussian noise (AWGN) channel. Let y denote the +channel output represented as y = xs +ε ∈ S ⊆ Rn, where +xs denotes the modulation of x, e.g. Binary Phase Shift Key- +ing (BPSK), and ε is random noise independent of the trans- +mitted x. The main goal of the decoder f : S → {0, 1}n is +to provide an approximation of the codeword ˆx = f(y). +An important notion in ECC is the syndrome, which is +obtained by multiplying the binary mapping of y with the +parity check matrix over GF(2) such that +s := s(y) = Hyb := H(x ⊕ εb) = Hεb, +(1) +where ⊕ denotes the XOR operator, and yb and εb denote +the hard-decision vectors of y and ε, respectively. An il- +lustration of the classical ECC framework is presented in +Figure 1(a). +3.2. Quantum Error Correction Code +The fundamental transition to the quantum realm is defined +by the shift from the classical bit to the quantum bit (qubit), +whose quantum state |ψ⟩ is defined by +|ψ⟩ = α|0⟩ + β|1⟩, s.t. α, β ∈ C, |α|2 + |β|2 = 1 (2) +A coherent quantum error process E can be decomposed +into a sum of operators from the Pauli set {I, X, Z, XZ}, +where the Pauli basis is defined by the identity mapping I, +the quantum bit-flip X and the phase-flip Z, such that the +single-qubit errors are defined as +I|ψ⟩ =|ψ⟩ +X|ψ⟩ =αX|0⟩ + βX|1⟩ = α|1⟩ + β|0⟩ +Z|ψ⟩ =αZ|0⟩ + βZ|1⟩ = α|0⟩ − β|1⟩ +E|ψ⟩ =αI|ψ⟩ + αXX|ψ⟩ + αZZ|ψ⟩ + αXZXZ|ψ⟩ +(3) +with αI, αX, αZ, αXZ ∈ C being the expansion coeffi- +cients of the noise process. +According to the no-cloning theorem, a quantum state +|ψ⟩ cannot be copied redundantly (i.e. |ψ⟩ ⊗ · · · ⊗ |ψ⟩), +where ⊗ · · · ⊗ +� �� � +n +denotes the n-fold tensor product. How- +ever, quantum information redundancy is possible through +a logical state encoding |ψ⟩n of a given state |ψ⟩ via +quantum entanglement and a unitary operator U such that +|ψ⟩n = U +� +|ψ⟩ ⊗ |0⟩ ⊗ . . . |0⟩) +� +. An example of such a +unitary operator is the GHZ state (Greenberger et al., 1989), +which is generated with CNOT gates. In |ψ⟩n, the logical +state is defined within a subspace of the expanded Hilbert +space, which determines both the codespace C and its or- +thogonal error space F defined such that E|ψ⟩n ∈ F. +The orthogonality of C and F makes it possible to deter- +mine the subspace occupied by the logical qubit through +projective measurement, without compromising the encoded +quantum information. In the context of quantum coding, +the set P of non-destructive measurements of this type are +called stabilizer measurements and are performed via ad- +ditional qubits (ancilla bits). The result of all of the stabi- +lizer measurements on a given state is called the syndrome, +such that for a given stabilizer generator P ∈ P we have +P|ψ⟩n = |ψ⟩n, and PE|ψ⟩n = −E|ψ⟩n, ∀|ψ⟩n ∈ C given +an anti-commuting (i.e. −1 eigenvalue) and thus detectable +error E. If the syndrome measurement is faulty, it might be +necessary to repeat it to improve confidence in the outcome +(Dennis et al., 2002, Section IV.B). +An important class of Pauli operators is the class of logical +operators. These operators are not elements of the stabilizer +group, but commute with every stabilizer. While stabilizers +operators act trivially in the code space, i.e. P|ψ⟩n = |ψ⟩n, +logical operators ℓ ∈ L act non-trivially in it, i.e. ∃|φ⟩n ∈ +C s.t. ℓ|ψ⟩n = |φ⟩n. Such operators commute with the +stabilizers but can also represent undetectable errors (Brun, +2020). Thus, QECC benchmarks generally adopt logical +error metrics, which measure the discrepancy between the +predicted projected noise Lˆε and the real one Lε, where L +is the discrete logical operators’ matrix. +QECC from the ECC perspective: +Another way to rep- +resent stabilizer codes is to split the stabilizer operators into +two independent parity check matrices, defining the block +parity check matrix H such that H = +� HZ +0 +0 +HX +� +, sep- +arating phase-flip checks HZ and bit-flip checks HX. The +syndrome s is then computed as s = Hε, H being the +check-matrix defined according to the code stabilizers in +P, and ε the binary noise. The main goal of the quantum +decoder f : {0, 1}|P| → {0, 1}|L| is to provide a noise +approximation given only the syndrome. +Therefore, the quantum setting can be reduced to its classical +counterpart as follows. The k logical qubits are similar to +the classical k information bits, and the n physical qubits +are similar to the classical codeword. The syndrome of +the quantum state can be computed or simulated similarly +to the classical way, by defining the binary parity check +matrix built upon the code quantum stabilizers. The main +differences from classical ECC are: (i) no access to the + +Deep Quantum Error Correction +(a) +(b) +Figure 1. Illustration of the (a) classical and (b) quantum ECC +system. Our work focuses on the design and training of the param- +eterized quantum decoder. +current state is possible, while arbitrary measurement of +y is standard in the classical world; (ii) we are interested +in the logical qubits, predicting the code up to the logical +operators mapping L; and finally, (iii) repetitive sampling +of the syndrome due to the syndrome measurement error. +These differences are at the core of our contributions. +An illustration of the quantum coding and decoding frame- +work is presented in Figure 1(b). The goal of our method +is to learn a decoder that is parameterized by a vector of +weights θ such that ˆε = fθ(s). +3.3. Error Correction Code Transformer +The state-of-the-art ECCT by Choukroun & Wolf (2022) +has been recently proposed for classical error decoding. It +consists of a transformer architecture (Vaswani et al., 2017) +with several modifications. +Following Bennatan et al. (2018), the model’s input +is defined by the concatenation of the codeword- +independent +magnitude +and +syndrome, +such +that +h(y) := [|y|, 1 − 2s(y)] ∈ R2n−k +where +[·, ·] +denotes +vector concatenation. +Each element is then embedded +into a high-dimensional space for more expressivity, +such that the initial positional embedding Φ is given +by Φ = +� +h(y) · 1T +d +� +⊙ W, where W ∈ R2n−k×d is the +learnable embedding matrix and ⊙ is the Hadamard +product. +The interaction between the bits is performed naturally via +the self-attention modules coupled with a binary mask de- +rived from the parity-check matrix +AH(Q, K, V ) = Softmax +�QKT + g(H) +√ +d +� +V +(4) +where g(H) is a binary masking function designed accord- +ing to the parity-check matrix H, and Q, K, V are the clas- +sical self-attention projection matrices. Masking enables +the incorporation of sparse and efficient information about +the code while avoiding the loop vulnerability of belief +propagation-based decoders (Pearl, 1988). Finally, the trans- +formed embedding is projected onto a one-dimensional vec- +tor for noise prediction. +4. Method +We present in this Section the elements of the proposed +decoding framework, the complete architecture, and the +training procedure. From now on, the binary block parity +check matrix is denoted by H, the binary noise by ε, the +syndrome by s = Hε, and the logical operators’ binary +matrix by L. +4.1. Overcoming Measurement Collapse by Prediction +While syndrome decoding is a well-known procedure in +ECC, most popular decoders, and especially neural de- +coders, assume the availability of arbitrary measurements +of the channel’s output. In the QECC setting, only the syn- +drome is available, since classical measurements are not +allowed due to the wave function collapse phenomenon. +We thus propose to extend ECCT, by replacing the mag- +nitude of the channel output y with an initial estimate +of the noise. In other words, we replace the channel’s +output magnitude measurement h(y) = [|y|, 1 − 2s] by +hq(s) = [gω(s), s]. +Denoting the ECCT decoder by fθ we have +ˆz = fθ +� +hq(s) +� += fθ +� +[gω(s), s] +� +, +(5) +where gω : {0, 1}ns → Rn is the initial noise estimator, +given by a shallow neural network parameterized by ω. This +way, the ECCT can process the estimated input and per- +form decoding by analyzing the input-syndrome interac- +tions via the masking. As a non-linear transformation of +the syndrome, gω(s) is also independent of the quantum +state/codeword and thus robust to overfitting. The estimator +gω(s) is trained via the following objective +Lg = BCE +� +gω(s), ε +� +, +(6) +where BCE is the binary cross entropy loss. This shift from +magnitude to initial error estimation is crucial for overcom- +ing quantum measurement collapse and, as is explored in +Section 6.2, leads to markedly better performance. +4.2. Logical Decoding +Contrary to classical ECC, quantum error correction aims +to restore the noise up to a logical generator of the code, +such that several solutions can be valid. Accordingly, a +commonly used metric is the logical error rate (LER), which +provides valuable information on the practical decoding + +Encoder +Decoder +x E (0,1]n +xs E(±1)n +yERn +m E (0,1]k +G +BPSK +Xs +ε +fe() +xSyndrome +Decoding +Encoding +Measurement +E|Φ)n +syndrome +山 +→PE|> +fe() +L2 +EDeep Quantum Error Correction +performance (Lidar & Brun, 2013). Given the code’s logical +operator in its matrix form L ∈ {0, 1}k×n, we wish to +minimize the following LER objective +LLER = BCE +� +Lfθ(s), Lε +� +. +(7) +where the multiplications are performed over GF(2) and +are thus highly non-differentiable. +We propose to optimize the objective using a differ- +entiable equivalence mapping of the XOR (i.e., sum +over GF(2)) operation as follows. +Defining the +bipolar mapping φ : {0, 1} → {±1} over GF(2) as +φ(u) = 1 − 2u, u ∈ {0, 1}, we obtain the following prop- +erty φ(u ⊕ v) = φ(u)φ(v), ∀u, v ∈ {0, 1}. Thus, with Li +the i-th row of L and x a binary vector, we have ∀i ∈ +{1 . . . k} +� +Λ(L, x) +� +i := Li ⊕ x = φ−1 +� +Πjφ +� +(L)ij · xj +�� +. +(8) +Thus, as a composition of differentiable functions Λ(L, x) +is differentiable and we can redefine our training objective +as folllows +LLER = BCE +� +Λ +� +L, bin(fθ(s)) +� +, Lε +� +, +(9) +where bin denotes the binarization of the soft prediction of +the trained model. While many existing works make use of +the straight-through estimator (STE) (Bengio et al., 2013) +for the binary quantization of the activations, we opt for its +differentiable approximation with the sigmoid function (i.e., +bin(x) = σ(x) = (1 + e−x)−1). As shown in our ablation +analysis (Sec. 6.2), the performance of the STE is slightly +inferior to the sigmoid approach. +In addition to directly minimizing the LER metric, we are +interested in noise prediction solutions that are close to +the real system noise. We, therefore, suggest regularizing +the objective with the classical and popular Bit Error Rate +(BER) objective defined as +LBER = BCE +� +fθ(s), ε +� +. +(10) +We further discuss the importance of this regularization in +Section 6.2. +Combining the loss terms, the overall objective is given by +L = λBERLBER + λLERLLER + λgLg +(11) +where λBER,LER,g denote the weights of each objective. +4.3. Noisy Syndrome Measurements +In the presence of measurement errors, each syndrome mea- +surement is repeated T times. This gives the decoder input +an additional time dimension. Formally, given binary sys- +tem noises {εt}T +t=1 and binary measurement noises {˜εt}T +t=1, +we have the syndrome st at a given time t ∈ N+ defined as +st = +� +H(x ⊕ ε1 ⊕ · · · ⊕ εt) +� +⊕ ˜εt +(12) +In order to remain invariant to the number of measurements, +we propose to first analyze each measurement separately +and then perform global decoding by applying a symmetric +pooling function, e.g. an average, in the middle of the neural +decoder. +Given a neural network decoder with N layers and the +hidden activation tensor ϕ ∈ RT ×n×dl at layer l = ⌊N/2⌋, +the new pooled embedding is given by summation along the +first dimension ˜ϕ = 1 +T +�T +t=1 ϕt. +The loss Lg is thus defined as the distance between the +pooled embedding and the noise +Lg = BCE +�� +t +gω(st)/T, ε +� +(13) +Since we are extending the ECCT architecture to its quan- +tum counterpart (QECCT), the hidden activation tensor is +now of shape ϕ ∈ RT ×h×n×dl with h the number of self- +attention heads, and pooling is performed at the ⌊N/2⌋ +Transformer block. +4.4. Architecture and Training +The initial encoding is defined as a d dimensional one-hot +encoding of the n+ns input elements where n is the number +of physical qubits and ns the length of the syndrome. The +network gω is defined as a shallow network with two fully +connected layers of hidden dimensions equal to 5ns and +with a GELU non-linearity. The decoder is defined as a con- +catenation of N decoding layers composed of self-attention +and feed-forward layers interleaved with normalization lay- +ers. In case of faulty syndrome measurements, the ⌊N/2⌋th +layer performs average pooling over the time dimension. +The output is obtained via two fully connected layers. The +first layer reduces the element-wise embedding to a one- +dimensional n + ns vector and the second to an n dimen- +sional vector representing the soft decoded noise, trained +over the objective given Eq. 11. An illustration of the pro- +posed QECCT is given in Figure 2. The complexity of the +network is linear with the code length n and quadratic with +the embedding dimension d and is defined by O(Nd2n) +for sparse (e.g. topological) codes. The acceleration of the +proposed method (e.g. pruning, quantization, distillation) +(Wang et al., 2020; Lin et al., 2021) is left for future work. +The Adam optimizer (Kingma & Ba, 2014) is used with +128 samples per minibatch, for 200 to 400 epochs depend- +ing on the code length, with 5000 minibatches per epoch. + +Deep Quantum Error Correction +Figure 2. Illustration of the proposed Quantum Error Correction +Code Transformer architecture (QECCT). The main modifications +are indicated in green. The pooling layer is performed only after +the middle self-attention block of the model and M(H) denotes +the adapted mask from the parity-check matrix. +The noise is sampled randomly per batch in the physical +error rate testing range. The default weight parameters are +λg = 0.5, λLER = 1, λBER = 0.5. Other configurations and +longer training can be beneficial but were not tested due to +a lack of computational resources. The default architecture +is defined by N = 6, d = 128. +We initialized the learning rate to 5 · 10−4 coupled with a +cosine decay scheduler down to 5·10−7 at the end of training. +No warmup (Xiong et al., 2020) was employed. Training +and experiments were performed on a 12GB Titan V GPU. +The training time is in the range of 153 to 345 seconds per +epoch for the 8 to 128 code lengths, respectively with the +default architecture. Test time is in the range of 0.1 to 0.6 +milliseconds per sample. The number of testing samples +is set to 105. No optimization of the sparse self-attention +mechanism was employed. +5. Application to Topological Codes +While our framework is universal in terms of the code, we +focus on the popular surface codes and, more specifically, +on Toric codes (Kitaev, 1997a;c; 2003), which are their +variant with periodic boundary conditions. These codes are +among the most attractive candidates for quantum comput- +ing experimental realization, as they can be implemented on +a two-dimensional grid of qubits with local check operators +(Bravyi et al., 2018). The physical qubits are placed on the +edges of a two-dimensional lattice of length L, such that +the stabilizers are defined with respect to the code lattice +architecture that defines the codespace, where n = 2L2. +The stabilizers are defined in two groups: vertex operators +are defined on each vertex as the product of X operators on +the adjacent qubits and plaquette operators are defined on +each face as the product of Z operators on the bordering +qubits. Therefore, there exist a total of 2L2 stabilizers, L2 +for each stabilizer group. Assuming that a qubit is associated +Figure 3. The lattice representation of the (k=3) Toric code and its +embedding into a three-dimensional torus. Gray edges represent +periodic boundaries. +(a) +(b) +(c) +(d) +Figure 4. (a) The parity-check matrix and (b) the induced masking +for the 2-Toric code. (c,d) the corresponding matrix and mask for +the 4-Toric code. The parity check matrix comprises two block +matrices for the X and Z stabilizers. We can observe the high +locality and sparsity of the Toric code. +with every edge of the lattice, for a given vertex v we have +the vertex operator defined as Xv = Πi∈vXi, and for a +given plaquette p, the plaquette operator defined as Zp = +Πi∈pZi. An illustration is given in Figure 3. +The mask is defined such that the self-attention mechanism +only takes into consideration bits related to each other in +terms of the stabilizers (i.e., the parity-check matrix). The +parity-check matrices and their corresponding masks for +several Toric codes are provided in Figure 4, where one can +observe the high locality induced by the code architecture. +6. Experiments +We evaluate our method on various toric codes, consid- +ering the two common noise models: independent and +depolarization. In independent (uncorrelated) noise, X +and Z errors occur independently and with equal prob- +abilities; therefore, decoding can be performed on the + +[N/2 ] +(i-1 +Pooling over T +Initial Embedding +Decoding +h(L, o(z)) +L +N× +Z + Feed Fwd +(stT-1 +[] +Norm +FC +2n-k→n +ga() +M(H) +MH-SA +FC +2n - k Bits +d→1 +Embedding +4 +Norm +Norm +d +(pil-Edge +Plaquette +VertexPCMatrix +0 +1 +2 +3 +4 +5 +6 +7 +0 +2 +4 +6 +8 +10 +12 +14Mask +5 +10 +15 +20 +0 +5 +10 +15 +20PC Matrix +10 +15 +20 +25 +30 +0 +10 +20 +30 +40 +50 +60Mask +20 +40 +60 +80 +0 +20 +40 +60 +80Deep Quantum Error Correction +X or Z stabilizers separately. Depolarization noise as- +signs equal probability p/3 to all three Pauli operators +P(X) = P(Z) = P(Y ) = p/3, P(I) = 1 − p, where Y is +the Pauli operator defined as Y = iXZ. +In the experiments where measurement errors are incorpo- +rated, each syndrome measurement is repeated T = n times +and the probability of the measurement error is the same as +the probability of the syndrome error, i.e., the distributions +of ε and ˜ε in Eq. 12 are the same. +The implementation of the Toric codes is taken from (Kras- +tanov & Jiang, 2017). +As a baseline, we consider the +Minimum-Weight Perfect Matching (MWPM) algorithm +with the complexity of O(n3), also known as the Edmond +or Blossom algorithm, which is the most popular decoder +for topological codes. Its implementation is taken from +(Higgott & Gidney, 2022; Higgott, 2022). In the experi- +ments, we employ code lengths similar to those for which +the existing neural decoders were tested (Varsamopoulos +et al., 2017; Torlai & Melko, 2017; Krastanov & Jiang, +2017; Chamberland & Ronagh, 2018; Andreasson et al., +2019; Wagner et al., 2020). It is worth noting that none of +these previous methods outperform MWPM. The physical +error ranges are taken around the thresholds of the different +settings, as reported in (Dennis et al., 2002; Wang et al., +2003; Krastanov & Jiang, 2017). On the tested codes and +settings, the Union-find decoder (Park & Meinerz, 2022) +was not better than the MWPM algorithm. +As metrics, we present both the bit error rates (BER) and +the logical error rate (LER), see Sec. 4.2. The LER metric +here is a word-level error metric, meaning there is an error if +at least one qubit is different from the ground truth. Unless +stated otherwise, we use the N = 6, d = 128 architecture. +Larger architectures that can be used for larger codes would +be possible and beneficial in terms of accuracy, given the +necessary computational resources. For example, GPT-3 +(Brown et al., 2020) successfully operates on 2K inputs with +a similar Transformer model but with N = 96, d = 12K. +6.1. Results +Figure 5 depicts the performance of the proposed method +and the MWPM algorithm for different Toric code lengths +under the independent noise model and without noisy mea- +surements. Figure 6 presents a similar comparison with +noisy measurements with T = L and uniformly distributed +syndrome error. Figure 7 and 8 compare our method with +MWPM for the depolarized noise model, with and without +noisy measurements, respectively. +As can be seen, the proposed QECCT outperforms the state- +of-the-art MWPM algorithm by a large margin. However, as +mentioned above, these experiments do not fully expose the +full potential of the method: our runs are heavily restricted +Figure 5. LER and BER performance for various physical error +rates and lattice length on the Toric code with independent noise +and without faulty syndrome measurements. +Figure 6. LER and BER performance for various physical error +rates and lattice length on the Toric code with independent noise +and with faulty syndrome measurements. +by computational resources, and as the code length increases +it would be beneficial to use models with larger capacity +and to employ longer training sessions. +6.2. Exploratory experiments +Impact of the noise estimator gω +Figure 9(a) explores +the impact of gω on performance. Since in order to use +the QECCT the initial scaling estimate needs to be defined, +we compare our approach to constant scaling with ones, +meaning gω := 1n. We further explore two architectures +for gω: one with a single hidden layer and one with three +hidden layers. Evidently, the initial noise estimator is critical +for performance. However, the network architecture is less +impactful. Given more resources, one can check the impact +of additional architectures. +Impact of the various objectives +Figure 10(a) depicts +the impact of the different loss terms in Eq. 11. First, we +can observe that training solely with the LLER objective +converges rapidly to a bad local minimum. This is illustrated +in Figure 10(b) where we display the gradient norm average +over the network layers. +We hypothesize that this issue may be due to the fact that +the dimension of the LER objective is rather small and the +gradients can collapse in the product of the code elements +in Eq 8, since σ(x)n −−−−→ +n→∞ 0. +Evidently, training with LBER only produces far worse +results than combining it with the LLER objective. For + +Toric Code +10-1 +10-2 +LER +L=2 MWPM +10-3 +L=4.MWPM +L=6 MWPM +=8MWPM +L=2 QECCT +10-4 +L=4 QECCT +L=6 QECCT +L=8OECCT +10-5 +0.0100 +0.0435 +0.07710.11100.1440 +0.1780 +Physical Error RateToric Code +10-1 +8 +L=2 MWPM +L=4MWPM +=6.MWPM +=8MWPM +=2QECCT +L=4 QECCT +10-3 +L=6QECCT +L=8OECCT +0.0100 +0.0435 +0.0771 +0.11100.1440 +0.1780 +Physical Error RateToric Code +LER +10-1 +L=2 MWPM +L=4MWPM +L=6 MWPM +L=2 OECCT +=4 QECCT +L=6 QECCT +0.02000.02350.02710.03060.03410.0376 +Physical ErrorRateToric Code +10-1 +BER +B +=2 MWPM +=4MWPM +L=6 MWPM +L=2 QECCT +=4QECCT +L=6 QECCT +0.02000.0235 +0.02710.03060.0341 +0.0376 +PhysicalErrorRateDeep Quantum Error Correction +Figure 7. LER and BER performance for various error physical +rates and lattice length on the Toric code with depolarization noise +and without faulty syndrome measurements. +Figure 8. LER and BER performance for various error physical +rates and lattice length on the Toric code with depolarization noise +and with faulty syndrome measurements. +high SNR, a model trained with LBER objective yields a +27 times higher LER than the model trained with combined +objectives. Moreover, optimizing with the noise estimator +objective Lg results, for high SNR, in a 46% improvement +over not employing regularization. +Impact of Pooling +Figure 9(b) depicts the impact of dif- +ferent pooling (averaging) scenarios. Pooling is performed +either after the initial embedding, in the middle of the net- +work, or over the final embedding. Evidently, performing +pooling in the middle layer allows better decoding. Pooling +the final embedding (similarly to voting) is not effective. +Impact of the Mask and the Architecture +Finally, we +present the impact of masking in Figure 11(a) and the impact +of the model’s capacity and the performance of the STE as +bin function from Eq 9 in Figure 11(b). We can observe that +while less important than with classical codes (Choukroun +& Wolf, 2022), the mask still has a substantial impact on +performance. Also, we note that increasing the capacity of +the network enables better representation and decoding. +7. Conclusion +We present a novel Transformer-based framework for de- +coding quantum codes. The proposed approach enables +effective representation and training under the QECC con- +straints. The framework helps overcome the measurement +collapse phenomenon, such that a Transformer architecture +can be employed, combined with efficient and size-invariant +(a) +(b) +Figure 9. Impact of the gω on LER performance during training +with an N = 6, d = 128 architecture (a). ”No gω” refers to +constant mapping, i.e., gω = 1n. Impact of pooling (averaging) +performed at different levels of the network for a N = 6, d = 128 +model (b). The average final testing LER of mid-level pooling is +5% lower than the initial embedding pooling. +(a) +(b) +Figure 10. Impact of the different loss terms on performance (a). +Impact of regularization on the training dynamics (b) with a N = +6, d = 128 model. +(a) +(b) +Figure 11. (a) +Impact +of +masking +on +accuracy +with +an +N = 2, d = 128 model. (b) Impact of the model size on accuracy +with an N = 6, d = 128 model. We also provide the performance +of the STE as the bin function. +pooling for faulty measurement scenarios. A major contribu- +tion is the differentiable training of highly non-differentiable +functions present in every error correction setting. The pro- +posed framework is the first neural decoder to outperform +the state-of-the-art classical methods. Since the lack of effec- +tive and efficient error correction is a well-known limiting +factor for the development of quantum computers, our con- +tribution can play a role in using machine learning tools for +overcoming the current technological limitations of many +qubit systems. + +Toric Code +LER +L=2 MWPM +L=4MWPM +L=6 MWPM +=8 MWPM +L=2 QECCT +L=4 OECCT +10-1 +L=6 QECCT +L=8 QECCT +0.1030 +0.1290 +0.1560 +0.1820 +Physical Error RateToric Code +10-1 +ER +=2MWPM +B +=4MWPM +L=6 MWPM +=8 MWPM +=2QECCT +L=4QECCT +=6QECCT +L=8.QECCT +0.1030 +0.1290 +0.1560 +0.1820 +Physical Error RateToric Code +10-1 +LER +L=2MWPM +L=4-MWPM +L=6 MWPM +L=2 QECCT +L=4 QECCT +10-2 +L=6.OECCT +0.0200 +0.0275 +0.0350 +Physical Error RateToric Code +10-1 +BER +B +=2 MWPM +L=4 MWPM +10-2 +L=6_MWPM +L=2-QECCT +L=4 OECCT +L=6-OECCT +0.0200 +0.0275 +0.0350 +Physical Error RateToric Code (L=4) +3.6 × 10-1 +gw#Layers=1 +gw#Layers=3 +3.5 × 101 +No gw +3.4 × 101 +R +3.3 × 10-1 +3.2 × 10-1 +3.1 × 101 +3 × 10-1 +0 +25 +50 +75 +100125 +150 +175 +200 +EpochToric Code (L=4) +0.40 +Pooledinitialemb +Pooled mid emb. +Pooledfinal emb. +0.35 +0.30 +LER +0.25 +0.20 +0.15 +0 +25 +50 +75 +100 +125 +150 +175 +200 +EpochToric Code (L=6) +入BER,入LER,入g=[0.5,1.0,0.0] +入BER,入LER,入g=[0.5,1.0,0.5] +5 ×10-1 +BER,ALER,入g=[0.0,1.0,0.5] +入BER,ALER,Ag=[1.0,0.0,0.0] +4× 10-1 +0 +25 +50 +75 +100 +125 +150 +175 +200 +EpochToric Code (L=4) +10-2. +10-4 +10-6 +10-8 +入BER入LER,入q=[0.0,1.0,0.0] +入BER,入LER,入q=[0.5,1.0,0.5] +10-12 +10-14 +10-16 +0 +25 +50 +75 +100 +125 +150 +175 +200 +EpochToric Code (L=4) +0.36 +Masked +UNmasked +0.35 +0.34 +R +0.33 +0.32 +0.31 +0.30 +0 +25 +50 +75 +100 +125 +150 +175 +200 +EpochToric Code (L=4) +0.38 +N=2,d=32 +0.37 +N=2,d=128 +N=6,d=32 +0.36 +N=6,d=128 +N=6,d=128 with STE +0.35 +0.33 +0.32 +0.31 +0.30 +0 +25 +50 +75 +100 +125 +150 +175 +200 +EpochDeep Quantum Error Correction +References +Aharonov, D. and Ben-Or, M. 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On layer normalization in the transformer +architecture. arXiv:2002.04745, 2020. + diff --git a/u9FKT4oBgHgl3EQf3y5P/content/tmp_files/load_file.txt b/u9FKT4oBgHgl3EQf3y5P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..13cea77321192160a65ec2429f7c6533a22b1d40 --- /dev/null +++ b/u9FKT4oBgHgl3EQf3y5P/content/tmp_files/load_file.txt @@ -0,0 +1,965 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf,len=964 +page_content='Deep Quantum Error Correction Yoni Choukroun 1 Lior Wolf 1 Abstract Quantum error correction codes (QECC) are a key component for realizing the potential of quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' QECC, as its classical counterpart (ECC), enables the reduction of error rates, by distributing quantum logical information across redundant physical qubits, such that errors can be detected and corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' In this work, we efficiently train novel deep quantum error decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' We resolve the quantum measurement collapse by augmenting syndrome decoding to predict an initial estimate of the system noise, which is then refined iteratively through a deep neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' The logical error rates calculated over finite fields are directly optimized via a differentiable objective, enabling efficient decoding under the constraints imposed by the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' Finally, our architecture is extended to support faulty syndrome measurement, to allow efficient decoding over repeated syndrome sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' The proposed method demonstrates the power of neural decoders for QECC by achieving state-of-the-art accuracy, outperforming, for a broad range of topological codes, the existing neural and classical decoders, which are often computationally prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' Introduction Error Correcting Codes (ECC) are required in order to over- come computation and transmission corruption in almost every computation device (Shannon, 1948;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' Lidar & Brun, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' The demands of high-performance hardware and communication networks have led to the rapid development of classical ECC theory and algorithms (MacKay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' Quantum systems are known for being extremely 1The Blavatnik School of Computer Science, Tel Aviv University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FKT4oBgHgl3EQf3y5P/content/2301.11930v1.pdf'} +page_content=' Correspondence to: Yoni Choukroun xpt[1]+sin(t),t->xpt[2]+cos(t), ,2n,color=:lightgrey,labe +label=" +end +#plot the disks and their sites +label="disks" +forxpt inX +plot!(t->xpt[1]+0.5sin(t),t->xpt[2]+.5cos(t),,2n,color=:lightblue +label="" +end +xl = [X[i][1] for i in l:LX1] +x2 = [X[i] [2] for i in 1:LX1] +scatter!(xl,x2,label="sites",color=:red, markersize=3,linecolor=:match) +#plot the connectivity graph +label="connected" +for e in CE +plot!([e[1][1],e[2][1]],[e[1][2],e[2][2]],label=label,color=:green] +label=" +end +#plot the boundary graph +label ="boundary" +for e in DXE +plot!([e[1][1],e[2][1]],[e[1][2],e[2][2]],label=label,color=:red] +label=" +end +plot!(title="level 1") +#displayplot +: [6]3.n0 +level 1 +disks +sites +connected +boundary +0 +1 +3 +2 +-1 +0 +2 +3 +3 of 18 +15/4/21,4:48pm +3of18 +15/11/22.12:59pm2 +Mathematical modelling +5 +Packing subintervals of [0, 6) then corresponds to packing unit diameter disks +touching a central unit diameter disks. Thus the centers of the touching disks +are on the unit (radius) circle S2. One can thus apply R´enyi’s point processes +to this packing problem. +We now introduce real and vector valued features of the point process +which are polynomial functions of the random variables v(Y1), . . . , v(YK) in- +variant under permutations of the Yi. In particular we will consider features +invariant under any translation of the parameters a and b. Clearly, K(a, b) is +an example as K(a + t, b + t) = K(a, b) and is K(a, b) is a constant function +of the Yi. A linear vector-valued polynomial is +F(a, b) = +K(a,b) +� +i=1 +v(Yi). +The splitting property of F(a, b) follows from the definition and is +F(a, b; Y1 = y) = F(a, y) + F(y + 1, b) + v(y). +(4) +But F is not invariant under translations as +F(a + t, b + t) = Q(t)F(a, b). +(5) +Note that F(a, b)/K(a, b) is the centre of mass of the circular polygon defined +by the Yi. An important feature is +L(a, b)2 = ∥F(a, b)∥2. +This second-degree feature is invariant under translations. Another second +degree feature is +E2(a, b) = +� +Yi 1, +where M(x, y) = I + Q(x − y) for the vector case u(x) = EF(0, x) and +M(x, y) = 2 for the scalar cases where u(x) = EK(0, x) or u(x) = EE2(0, x). +We are interested in computing u(x) for x ∈ [0, 5]. +The solution of the integral equation 8 is then obtained for x ∈ [0, 2] as +u(x) = 0.0, +x ∈ [0, 1) +u(x) = g(x), +x ∈ [1, 2)]. +Now if one interprets the function u(x) as a block vector where each block +corresponds to a function on an interval [k − 1, k] then the integral equa- +tion (8) is of block lower triangular structure and can be solved by repeated +substitution. Specifically, one computes u(x) for x ∈ [k, k + 1] by +u(x) = +1 +x − 1 +� x−1 +0 +M(x, y)u(y) + g(x), +x ∈ [k, k + 1) +which uses u(x) for x ∈ [k − 1, k]. It then follows directly that u is C∞ on +each interval [j − 1, j] for j = 1, 2, . . . and is continuous on [1, ∞). +The approach was implemented in the Julia language [1] using the Ap- +proxFun package [6, 5]. The code is available on request from the first author. +So far we have only considered the random part but the method starts +by introducing one disk at Y0 = 0 and then running the code for x ∈ [1, 6]. +For the rotationally invariant expectations of K and E2 we run the code for +x ∈ [0, 5] and then add an extra disk at YK+1 = 5. One can then directly +compute the expectations with this added disk using only the expectations +from the random disks with Y ∈ [0, 5]. +Acknowledgements +The code was developed and tested using the cloud +service for VSCode on juliahub.com. + +References +10 +References +[1] +Jeff Bezanson et al. “Julia: A Fresh Approach to Numerical +Computing”. In: SIAM Review 59.1 (2017), pp. 65–98. doi: +10.1137/141000671 (cit. on p. 9). +[2] +Matthew P. Clay and N´andor J. Sim´anyi. “R´enyi’s parking problem +revisited”. In: Stoch. Dyn. 16.2 (2016), pp. 1660006, 11. issn: +0219-4937. doi: 10.1142/S0219493716600066. (Cit. on p. 2). +[3] +M. Lal and P. Gillard. “Evaluation of a constant associated with a +parking problem”. In: Math. Comp. 28 (1974), pp. 561–564. issn: +0025-5718. doi: 10.2307/2005927 (cit. on p. 2). +[4] +George Marsaglia, Arif Zaman, and John C. W. Marsaglia. “Numerical +solution of some classical differential-difference equations”. In: Math. +Comp. 53.187 (1989), pp. 191–201. issn: 0025-5718. doi: +10.2307/2008355. (Cit. on p. 2). +[5] +S. Olver. ApproxFun.jl v0.13.14. 2022. url: +https://github.com/JuliaApproximation/ApproxFun.jl (cit. on +p. 9). +[6] +S. Olver and A Townsend. “A practical framework for +infinite-dimensional linear algebra”. In: Proceedings of the 1st First +Workshop for High Performance Technical Computing in Dynamic +Languages. 2014, pp. 57–62 (cit. on pp. 2, 9). +[7] +Alfr´ed R´enyi. “On a one-dimensional problem concerning random +space filling”. In: Magyar Tud. Akad. Mat. Kutat´o Int. K¨ozl. 3.1-2 +(1958), pp. 109–127. issn: 0541-9514 (cit. on p. 2). +Author addresses +1. +M. Hegland, Mathematical Sciences Institute, The Australian +National University, Australian Capital Territory 2601, Australia. +mailto:markus.hegland@anu.edu.au +orcid:0000-0002-5136-2883 +2. +Z. Stachurski, School of Engineering, The Australian National +University, Australian Capital Territory 2601, Australia. +3. +C.J. Burden, Mathematical Sciences Institute, The Australian +National University, Australian Capital Territory 2601, Australia. + diff --git a/v9FLT4oBgHgl3EQfkS__/content/tmp_files/load_file.txt b/v9FLT4oBgHgl3EQfkS__/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..09946bf55429aa68109963f643f54228ecd76462 --- /dev/null +++ b/v9FLT4oBgHgl3EQfkS__/content/tmp_files/load_file.txt @@ -0,0 +1,355 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf,len=354 +page_content='Computing expected moments of the R´enyi parking problem on the circle M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Hegland1 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Stachurski2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Burden3 28 January 2023 Abstract A highly accurate and efficient method to compute the expected values of the count and sum of the centre vectors of a random maximal sized collection of non-overlapping unit diameter disks touching a fixed unit-diameter disk is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' This extends earlier work on R´enyi’s parking problem [Magyar Tud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Kutat´o Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' K¨ozl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' 3 (1-2), 1958, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' 109-127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Underlying the method is a splitting of the the problem conditional on the value of the first disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' This splitting is proven and then used to derive integral equations for the expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' These equations take a lower block triangular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' They are solved using substitution and approximation of the integrals to very high accuracy using a high degree polynomial approximation within the blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Mathematical modelling 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='1 R´enyi Point Process for [a, b] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='2 Randomly packing disks touching a central disk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' 4 3 Integral equations 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='1 Equation for EK(0, x) .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' 8 4 Numerical solution 9 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='12115v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='NA] 28 Jan 2023 1 Introduction 2 1 Introduction R´enyi [7] analysed a stochastic process arising when randomly packing unit intervals [Yi, Yi + 1) into an interval [a, b] such that the unit intervals do not overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' The packing is completed when all the remaining gaps are less than one and let K(a, b) be the (random) number of unit intervals packed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' R´enyi’s stochastic process defines a random set α(a, b) = {Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' , YK(a,b)} ⊂ [a, b − 1] which is a translational invariant point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Lemma 1 (Translational Invariance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' α(a + t, b + t) = t + α(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Proof idea: The stochastic process for the interval [a+t, b+t] generates a sequence of random variables Y1 + t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' , YK + t where the Yi are the random variables generated by the process for the interval [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' ♠ It follows that K(a, a + x) = K(0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' R´enyi did show that the expected value EK(0, x) satisfies an integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' He established a formula for the limit of the expected packing density limx→∞ EK(0, x)/x ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' While the EK(0, x) can be computed explicitly for small x the formulas get very complicated even for moderate x, take a long time to compute and often evaluate poorly due to rounding errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Thus numerical approaches were established for computing the expectation, see [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' These numeri- cal approaches do give highly accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' The approach taken here is similar in that it also uses a highly accurate piecewise polynomial approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' In addition to computing EK(0, a) we discuss the computation of the expectation of a random variable L2 which arises in the problem of packing disks attached to a central disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' This requires the solution of a system of integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' We review the underlying splitting property in Section 2 and derive the system integral equations in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' The method is then implemented using the new framework developed by Olver and Townsend [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' The algorithm is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' In recent years, there has also been some interest in discrete versions of the R´enyi process [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' 2 Mathematical modelling 3 Figure 1: Samples of the packing numbers K(0, x) and their expectations as a function of the interval size x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' 2 Mathematical modelling 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='1 R´enyi Point Process for [a, b] A consequence of the one-dimensionality of the R´enyi point process is a splitting property of the random set conditional on the first point Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' This conditional set is denoted by α(a, b | Y1 = y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' The splitting property gives rise to the integral equations derived in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Lemma 2 (Splitting Property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' α(a, b | Y1 = y) = {y} ∪ α(a, y) ∪ α(y + 1, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Proof: When the random variables Y2, Y3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' are generated, each element is either in [a, y − 1] or in [y + 1, b − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Now Yi, Yj are independent if Yi ∈ [y + 1, b − 1] and Yj ∈ [a, y − 1] as they cannot overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Consequently, the points in the two subsets generated for the conditional process produce two independent random sets α(a, y) and α(y + 1, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' ♠ The R´enyi point process gives rise to a random counting measure with parameters a, b defined by N(a, b) = � y∈α(a,b) δy where δy is the atomic measure for which δy(S) = � 1, if y ∈ S 0, otherwise 2 Mathematical modelling 4 Figure 2: randomly positioned disks of diameter 1 touching a central disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' The areas shaded grey are the exclusion areas, any disk centres in those areas results in overlap with other disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' for any S ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' It follows that the support of this measure is α(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' This measure is defined for any set M ⊂ R by N(a, b)(M) = |α(a, b) ∩ M|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' (1) In particular, one has K = K(a, b) = N(a, b)([a, b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Note that K is transla- tional invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=', K(a + t, b + t) = K(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' The conditional counting measure N(a, b | Y1 = y) satisfies a splitting property N(a, b | Y1 = y) = δy + N(a, y) + N(y + 1, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' (2) This is a direct consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='2 Randomly packing disks touching a central disk We map the interval [0, 6) bijectively onto the unit circle S2 by v(y) = Q(y) �1 0 � , y ∈ [0, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' where Q(y) = �cos(πy/3) − sin(πy/3) sin(πy/3) cos(πy/3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' (3) Firefox about:blank RSAQ-V4 about:srcdoc : [6] uI #plotting plot(ratio=:equal)#initiateplot #plottheexclusionzone label="exclusionzone" forxptinX plot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' (t->xpt[1]+sin(t),t->xpt[2]+cos(t), ,2n,color=:lightgrey,labe label=" end #plot the disks and their sites label="disks" forxpt inX plot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' (t->xpt[1]+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='5sin(t),t->xpt[2]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='5cos(t),,2n,color=:lightblue label="" end xl = [X[i][1] for i in l:LX1] x2 = [X[i] [2] for i in 1:LX1] scatter!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' (xl,x2,label="sites",color=:red, markersize=3,linecolor=:match) #plot the connectivity graph label="connected" for e in CE plot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' ([e[1][1],e[2][1]],[e[1][2],e[2][2]],label=label,color=:green] label=" end #plot the boundary graph label ="boundary" for e in DXE plot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' ([e[1][1],e[2][1]],[e[1][2],e[2][2]],label=label,color=:red] label=" end plot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' (title="level 1") #displayplot : [6]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='n0 level 1 disks sites connected boundary 0 1 3 2 1 0 2 3 3 of 18 15/4/21,4:48pm 3of18 15/11/22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content='12:59pm2 Mathematical modelling 5 Packing subintervals of [0, 6) then corresponds to packing unit diameter disks touching a central unit diameter disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Thus the centers of the touching disks are on the unit (radius) circle S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' One can thus apply R´enyi’s point processes to this packing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' We now introduce real and vector valued features of the point process which are polynomial functions of the random variables v(Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' , v(YK) in- variant under permutations of the Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' In particular we will consider features invariant under any translation of the parameters a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Clearly, K(a, b) is an example as K(a + t, b + t) = K(a, b) and is K(a, b) is a constant function of the Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' A linear vector-valued polynomial is F(a, b) = K(a,b) � i=1 v(Yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' The splitting property of F(a, b) follows from the definition and is F(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Y1 = y) = F(a, y) + F(y + 1, b) + v(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' (4) But F is not invariant under translations as F(a + t, b + t) = Q(t)F(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' (5) Note that F(a, b)/K(a, b) is the centre of mass of the circular polygon defined by the Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' An important feature is L(a, b)2 = ∥F(a, b)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' This second-degree feature is invariant under translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FLT4oBgHgl3EQfkS__/content/2301.12115v1.pdf'} +page_content=' Another second degree feature is E2(a, b) = � Yi0,η=± +2πiD +β +2 +(2ipn+ηΩ)3 , +(21) +λp = +� +ipn>0,η=± +k2 +F +3m +2πiD +βΩ2 +� +4 +2ipn+ηΩ − 1 +ipn +− +1 +ipn+ηΩ +� +, (22) +with parameters: +αp = D +� ωD +−ωD +dξk +tanh( βcξk +2 )−tanh( βξk +2 ) +2ξk +=D ln T +Tc +,(23) +γp = +� +ipn>0,η=± +πiD +βΩ2 +� +4 +2ipn+ηΩ − 1 +ipn +− +1 +ipn+ηΩ +� +. +(24) +Here, D and ωD denote the density of states and Debye +frequency, respectively; f(x) stands for the Fermi dis- +tribution; β = kBT and βc = kBTc with kB being the +Boltzmann constant. +At low frequency (Ω < kBTc), one finds the specific +parameters: +γp = βp ≈ +� +n>0 +πD +βp3n += 7Dζ(3) +8(πT )2 , +(25) +λp ≈ +� +n>0 +k2 +F +3m +2πD +4β +4 +p3n += k2 +F +3m +7Dζ(3) +(2πT )2 , +(26) +with ζ(x) being the Riemann zeta function. +Then, +through the transformation into time-space coordinate, +the time-dependent Ginzburg-Landau Lagrangian of su- +perconductors is obtained: +L = γp|i∂t∆|2 +2 +− +� +αp|∆|2+ βp|∆|4 +2 ++ λp|(∇−2ieA)∆|2 +4m +� +, +(27) +which is exactly same as the one obtained by Pekker and +Varma through symmetry analysis and Lorentz invari- +ance from the general Ginzburg-Landau free energy6 as +it should be. Moreover, the parameters αp [Eq. (23)] and +βp [Eq. (25)] as well as λp [Eq. (26)] here are exactly same +as the obtained Landau parameters in the previous work +by Gorkov18 in which the Ginzburg-Landau equation is +derived from Gorkov equation. +It is noted that in the derivation of the gap dynam- +ics within the Matsubara formalism in the present work, +we take the optical frequency Ω as continuous variable +rather than bosonic Matsubara frequency iΩm, and only +with continuous Ω in this circumstance, can one re- +cover the Ginzburg-Landau Lagrangian as demonstrated +above. Actually, as seen from Eq. (22), if Ω is taken as +iΩm = 2miπT , different irrational response coefficients +are obtained at cases with odd and even m. +For odd +m, a divergent pole (2n + 1 = m and η = −1) which is +unable to circumvent emerges in the formulation of the +first term on the right-hand side of Eq. (22). As for even +m, through the frequency displacement in the Matsubara +frequency summation, the response coefficient λp directly +vanishes for nonzero m, but due to the prefactor 1/Ω2 in +Eq. (22), there exists an undefined singularity at zero +frequency. +Then, an unphysical discontinuity between +Ω = 0 and Ω → 0 emerges. The difference between cases +with odd and even m in bosonic Matsubara frequency +iΩm = 2miπT is totally unreasonable within the Mat- +subara formalism, and neither of them can recover the +finite Landau parameter at low frequency. Clearly, any +treatment that leads to consequence strongly against the +fundamental model of symmetry analysis and Lorentz in- +variance as well as Landau phase-transition theory can +not be correct. All of the irrationalities here simply sug- +gest that the treatment of taking Ω as iΩm can not be +correct in the derivation of gap dynamics in supercon- +ductors. +Furthermore, from the Lagrangian above, with the +equilibrium gap ∆0 = +� +−αp/βp, by considering the gap +fluctuation δ|∆| through |∆| = ∆0 +δ|∆|, its equation of +motion at long-wavelength limit is directly obtained: +� +(2∆0)2 − ∂2 +t +� +δ|∆| = −λp +γp +e2A2 +m 2∆0. +(28) +Then, one immediately finds the Higgs-mode energy +spectrum (i.e., resonance pole) ωH = 2∆0 on the left- +hand side of the equation above, and in particular, a +finite second-order response of Higgs mode at clean limit +on the right-hand side of the equation above. It is pointed +out that although the derivation of the equation of mo- +tion of the Higgs mode here is based on the small gap and +only holds near Tc, the serious derivation in regime ex- +tending to T = 0 through three different microscopic ap- +proaches including the gauge-invariant kinetic equation, +Eilenberger equation as well as path-integral approach +also obtains the similar equation of motion as a conse- +quence of the renormalization group (scaling) theory or +basic local Abelian U(1) model (complex scalar field cou- +pled to an electromagnetic potential) in the field theory. +IV. +DERIVATION OF HIGGS-MODE +RESPONSE THROUGH EILENBERGER +EQUATION +To eliminate the confusion of the Matsubara frequency +from a separate framework in addition to the physical +justification above, in this section, we perform the for- +mulation of the gap dynamics within the Keldysh for- +malism, which is totally irrelevant to Matsubara space. +Specifically, we apply the Eilenberger equation within +the Keldysh formalism to derive the second-order optical +response of the Higgs mode at clean limit. The Eilen- +berger equation42,58 is derived from the basic Gorkov + +6 +equation of τ3-Green function through the quasiclassical +approximation49: +gR/K/A +R,kF +(t, t′) = i +π +� +dξk +� +drτ3GR/K/A(x, x′)e−ik·(x−x′). +(29) +Here, R = (x + x′)/2 represents the center-of-mass spa- +tial coordinate; the retarded (R), advanced (A) and +Keldysh (K) Green functions are defined by49,58 +GR(x, x′) = −i⟨{ψ(x), ψ†(x′)}⟩θ(t − t′), +(30) +GA(x, x′) = i⟨{ψ(x), ψ†(x′)}⟩θ(t′ − t), +(31) +GK(x, x′) = −i⟨[ψ(x), ψ†(x′)]. +(32) +The Eilenberger equation within the Keldysh formal- +ism at clean limit reads42,58: +i{τ3∂t, ˆg}t−[(∆0+δ|∆|)τ1τ3, ˆg]t+[eA·vFτ3, ˆg]t =0, (33) +where the green function matrices ˆg is defined as +ˆg = +� +gR gK +0 +gA +� +. +(34) +Here, the operators [X, ˆg]t = X(t1)ˆg(t1, t2) − ˆg(t1, t2)X(t2) +and {X, ˆg}t = X(t1)ˆg(t1, t2) + ˆg(t1, t2)X(t2). +Moreover, to guarantee the unique solution, the Eilen- +berger equation must be supplemented by the normaliza- +tion condition42,58: +ˆg ◦ ˆg = 1, +(35) +where the operator ◦ is defined by relation A ◦ B = +� +dtA(t1, t)B(t, t2). +The corresponding gap equation is written as +∆0 + δ|∆| = −iUTr[⟨gK +R,kF(t, t)⟩F τ2/2], +(36) +with ⟨...⟩F denoting the angular average over the Fermi +surface. +Considering an external optical field with A(t) = +A0e−iΩt, by self-consistently solving Eqs. (33)-(36), one +can formulate the Higgs-mode generation at clean limit. +Specifically, in this circumstance, one can expand the +quasiclassical Green function matrices as +ˆg = ˆg(0) + δˆg(1) + δˆg(2), +(37) +with the m-th order response δˆg(m) on the initial state +ˆg(0). Correspondingly, the Higgs-mode generation δ|∆| = +δ|∆|(1)e−iΩt + δ|∆|(2)e−2iΩt from Eq. (36) with δ|∆|(1) +and δ|∆|(2) being the excitations in first- and second- +order optical responses, respectively. +Particularly, the +first-order response of Keldysh Green function must be +anisotropic in momentum space, leading to a vanishing +δ|∆|(1) after the angular average over the Fermi surface +in the gap equation. We then directly take δ|∆|(1) = 0 +for convenience. +Consequently, the Eilenberger equation in Eq. (33) be- +comes a chain of equations: +{τ3∂t, δˆg(1)}t+[∆0τ2, δˆg(1)]t−i[eA·vF τ3, ˆg(0)]t =0, (38) +{τ3∂t, δˆg(2)}t + [∆0τ2, δˆg(2)]t − i[eA · vF τ3, δˆg(1)]t ++ [δ|∆|(2)e−2iΩtτ2, ˆg(0)]t =0, +(39) +and one can solve δˆg(1) and δˆg(2) in sequence with the +given initial state ˆg(0). Then, with the obtained solution +of the Keldysh Green function, one can further derive the +response of the Higgs mode from the gap equation. +A. +Solution of retarded Green function +In this part, we first solve the retarded Green functions +from Eq. (33). From the expansion of Green function ma- +trices in Eq. (37), the retarded Green function is written +as +gR(t, t′) = gR(0)(t, t′) + δgR(1)(t, t′) + δgR(2)(t, t′) += +� dE +2π eiE(t′−t)[gR(0)(E)+δgR(1)(E)e−iΩt ++ δgR(2)(E)e−2iΩt]. +(40) +The initial state of the retarded Green function has been +established in the literature from Gorkov equation18,41,58, +and is written as +gR(0)(E) = +� dξk +π +iτ3(E+ξkτ3+∆0τ1) +(E+i0+)2−ξ2 +k−∆2 +0 += Eτ3+i∆0τ2 +SR(E) +, +(41) +with SR(E) = +� +(E + i0+)2 − ∆2 +0. +By defining E1 = E + Ω and E2 = E + 2Ω, from +Eq. (38), the equation of the first order of retarded Green +function is written as +(E1τ3+i∆0τ2)δgR(1)(E)−δgR(1)(E)(Eτ3+i∆0τ2) += eA0 · vF ΠR(0) +3 +, +(42) +from which one finds the first-order solution (refer to Ap- +pendix B): +δgR(1)(E) = (eA0 · vF )τ3 − gR(0)(E1)τ3gR(0)(E) +SR(E1) + SR(E) +. (43) +Here, ΠR(i) +3 += gR(i)(E1)τ3 − τ3gR(i)(E). Similarly, the +equation of the second order of retarded Green function +from Eq. (39) reads +(E2τ3+i∆0τ2)δgR(2)(E)−δgR(2)(E)(Eτ3+i∆0τ2) += eA0·vF ΠR(1) +3 ++iδ|∆|(2)[gR(0)(E2)τ2−τ2gR(0)(E)],(44) +and gives the +second-order solution (refer +to +Ap- +pendix B): +δgR(2)(E) = iδ|∆|(2) τ2−gR(0)(E2)τ3gR(0)(E) +SR(E2) + SR(E) ++ eA0·vF +E2 +2 −E2 +× +� +SR(E2)gR(0)(E2)ΠR(1) +3 +−ΠR(1) +3 +SR(E)gR(0)(E) +� +. +(45) + +7 +Considering the response expansions, the normaliza- +tion condition [Eq. (35)] for the retarded Green function +is written as +[gR(0)(E)]2 = 1, +(46) +gR(0)(E1)δgR(1)(E)+δgR(1)(E)gR(0)(E) = 0, (47) +gR(0)(E2)δgR(2)(E)+δgR(2)(E)gR(0)(E) ++δgR(1)(E1)δgR(1)(E) = 0. +(48) +The initial-state gR(0)(E) in Eq. (41) naturally satisfies +Eq. (46). Facilitated with Eq. (46), correspondingly sub- +stituting the solutions in Eqs. (43) and (45), one can eas- +ily demonstrate the normalization conditions in Eqs. (47) +and (48). Therefore, as the self-consistent crosscheck, the +obtained solutions of the retarded Green function satisfy +the normalization condition. +Further substituting Eq. (41) into Eqs. (42) and (44), +the specific solution of the τ2 component of δgR(2)(E) is +given by (refer to Appendix B) +δgR(2) +2 +(E)=iδ|∆|(2)�4∆2 +0 − (2Ω)2 +Γ(2)(E) ++ +1 +SR(E) +� ++ i(eA0·vF )2∆0 +2Ω2 +� +1 +SR(E2) + +1 +SR(E) − +2 +SR(E1) +� +,(49) +with Γ(2)(E) = 2SR(E2)SR(E)[SR(E2)+SR(E)]. +It is noted that the involved external optical frequency +in the τ2 component of the second order of the retarded +Green function in Eq. (49) is a continuous variable. To +further consider the gap dynamics at nonzero temper- +ature and eliminate the confusion of the auxiliary Mat- +subara frequency, we next derive the Keldysh Green func- +tion. +B. +Solution of Keldysh Green function +In this part, we derive the Keldysh Green function. We +start with the normalization condition [Eq. (35)] for the +Keldysh Green function: +gR◦gK + gK◦gA = 0. +(50) +It is established that the Keldysh Green function can be +written as a function of the retarded and advanced ones +through a general relation49,58: +gK = gR ◦ h − h ◦ gA, +(51) +where h(t, t′) denotes the distribution function. Substi- +tuting Eq. (51) to Eq. (50), one finds that the normaliza- +tion condition for the Keldysh Green function is immedi- +ately satisfied. Consequently, with the obtained retarded +and hence advanced Green function in Sec. IV A, to solve +the Keldysh Green function, one only needs to solve the +distribution function. +In the previous work to derive the Ginzburg-Landau +equation from Eilenberger equation within the Keldysh +formalism48, the distribution function h(t, t′) is directly +taken as the equilibrium one +� dE +2π h(E)e−iE(t−t′) with the +Fourier component written as +h(E) = tanh +�βE +2 +� +. +(52) +This treatment usually concerns the case near equilib- +rium or in strongly-interacting systems as applied in +the transport theory of normal and superconducting +metals49, and has also been widely used in previous stud- +ies through the Eilenberger equation58,59 and diffusive +Usadel one60,61. In the present work, at clean limit, with +a weak external excitation, we demonstrate Eq. (52) by +seriously taking account of the distribution function. +Specifically, from the expansion of Green function ma- +trices in Eq. (37), the Keldysh Green function reads +gK(t, t′) = +� dE +2π eiE(t′−t)[gK(0)(E)+δgK(1)(E)e−iΩt ++ δgK(2)(E)e−2iΩt], +(53) +and with the general relation in Eq. (51), one has +gK(0)(E)=gR(0)(E)h(0)(E) − h(0)(E)gA(0)(E), +(54) +δgK(1)(E)=gR(0)(E1)δh(1)(E)+δgR(1)(E)h(0)(E) +−h(0)(E1)δgA(1)(E)−δh(1)(E)gA(0)(E),(55) +δgK(2)(E)=δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E) ++gR(0)(E2)δh(2)(E)+δgR(2)(E)h(0)(E) +−h(0)(E2)δgA(2)(E)−δh(2)(E)gA(0)(E).(56) +Here, δh(1)(E) and δh(2)(E) stand for the first- and +second-order responses on the initial-state h(0)(E), re- +spectively. +According to Eq. (33), the Keldysh Green function sat- +isfies the same equation as the retarded/advanced one. +Consequently, by correspondingly replacing gR(0) and +δgR(1) by gK(0) and δgK(1) in Eq. (42) and then substi- +tuting Eq. (55), one finds the equation of the first-order +distribution function (refer to Appendix C): +(E1τ3+i∆0τ2)[gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)] +− [gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)](Eτ3+i∆0τ2) += eA0·vF [h(0)(E1)−h(0)(E)][gR(0)(E1)τ3−τ3gA(0)(E)]. +(57) +From above equation, the solution of the first-order dis- +tribution function reads (refer to Appendix C) +δh(1)(E) = eA0·vF +h(0)(E1)−h(0)(E) +E2 +1 −E2 +[(E1τ3+i∆0τ2)τ3 ++ τ3(Eτ3+i∆0τ2)] += eA0·vF +h(0)(E + Ω)−h(0)(E) +Ω +. +(58) +Similarly, +by correspondingly replacing gR(0) and +δgR(i=1,2) by gK(0) and δgK(i=1,2) in Eq. (44), with + +8 +Eqs. (55) and (56) as well as the help of Eq. (58), the +equation of the second-order distribution function reads +(refer to Appendix C) +(E2τ3+i∆0τ2)[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)] +− [gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)](Eτ3+i∆0τ2) +=eA0·vF [gR(0)(E2)τ3−τ3gA(0)(E)][δh(1)(E1)−δh(1)(E)] ++ iδ|∆|(2)[h(0)(E2)−h(0)(E)][gR(0)(E2)τ2−τ2gA(0)(E)], +(59) +from which the solution of the second-order distribution +function is obtained as (refer to Appendix C) +δh(2)(E) = (eA0·vF )δh(1)(E1)−δh(1)(E) +2Ω +− δ|∆|(2) ∆0 +E+Ω +h(0)(E2)−h(0)(E) +2Ω +. (60) +Then, both the first- and second-order distribution func- +tions are diagonal as they should be. +The initial-state distribution function can be obtained +from the Hamiltonian in Eq. (2) in self-consistent con- +sideration of the Higgs mode and vector potential, and is +written as +h(0)(Ek) = tanh +�β +2 +�� +ξ2 +k+(∆0+δ|∆|)2−eA·vF +�� +≈ tanh +�β +2 +� +Ek + ∆0δ|∆| +Ek +− eA·vF +�� +, (61) +where Ek = +� +ξ2 +k + ∆2 +0 denotes the Bogoliubov quasi- +particle energy. +Following the standard treatment of +energy E = Ek as the previous work in superconduct- +ing state62, with the weak excitation, at low frequency +(Ω < E = Ek), with Eqs. (58) and (60)-(61), the total +distribution function in relative-frequency space reads +h(E) = h(0)(E) + e−iΩtδh(1)(E) + e−2iΩtδh(2)(E) += +� +1+(eA·vF)∂E+ (eA·vF )2∂2 +E +2 +� +h(0)(E) +− δ|∆|∆0 +E ∂Eh(0)(E) ≈ tanh +�βE +2 +� +. +(62) +Then, the drive effect of vector potential and Higgs-mode +part in the initial-state distribution are exactly canceled +by the first- and second-order distribution functions, +leading to the widely applied distribution function +[Eq. (52)] in the literature. +C. +Higgs-mode generation +In this part, with the obtained distribution function in +Eq. (52) and the second-order retarded Green function in +Eq. (49), from the gap equation [Eq. (36)], the second- +order optical response of the Higgs mode at clean limit +is determined by +δ|∆|(2) = −iU +� dE +2π ⟨[h(E)δgR(2) +2 +(E) − h(E2)δgA(2) +2 +(E)]⟩F = −iU +� dE +2π 2h(E)⟨δgR(2) +2 +(E)⟩F += U +� +dE2h(E) +� +δ|∆|(2)�4∆2 +0 − (2Ω)2 +Γ(2)(E) ++ +1 +SR(E) +� ++ (eA0vF )2∆0 +6Ω2 +� +1 +SR(E2) + +1 +SR(E) − +2 +SR(E1) +�� +. (63) +Consequently, one arrives at the equation of motion of +the Higgs mode at clean limit: +[4∆2 +0 − (2Ω)2]δ|∆|(2) = −(eA0vF )22∆0 +3 +λE +βE +, +(64) +similar to the one [Eq. (28)] obtained from the Ginzburg- +Landau Lagrangian. Here, through the standard contour +integral, the amplitude-correlation coefficient reads +βE = − +� dE +2π +h(E) +SR(E2)SR(E)[SR(E2)+SR(E)] += +� +n>0 +1/(4βiΩ) +pn−iΩ +� +1 +� +(pn−2iΩ)2+∆2 +0 +− +1 +� +p2n+∆2 +0 +� +, +(65) +and the essential response coefficient is given by +λE =− +� dE +2π +h(E) +2Ω2 +� +1 +SR(E2) + +1 +SR(E) − +2 +SR(E1) +� += +1 +2βΩ2 +� +n>0 +� +2 +� +(pn−iΩ)2+∆2 +0 +− +1 +� +(pn−2iΩ)2+∆2 +0 +− +1 +� +p2n+∆2 +0 +� +. +(66) +It is noted that both amplitude-correlation coefficient +βE [Eq. (65)] and response one λE [Eq. (66)] derived +here are exactly same as the ones obtained in the previ- +ous work41 by Silaev through Eilenberger equation within +Matsubara formalism. However, in Eqs. (65) and (66), +the fermionic Matsubara frequencies ipn arise from the +singularities in the distribution function h(E) during the +standard contour integral in the complex plane, whereas + +9 +the involved external optical frequency Ω within the +Keldysh formalism is always a continuous variable, in +contrast to the treatment of taking optical frequency as +bosonic Matsubara frequency in Ref. 41. As mentioned +in the introduction, the treatment of taking optical fre- +quency as bosonic Matsubara frequency leads to the van- +ishing response coefficient λE (i.e., zero Higgs-mode gen- +eration) at all Ω ̸= 0, strongly against the Ginzburg- +Landau Lagrangian, whereas the prefactor 1/Ω2 in λE +causes an undefined singularity at zero frequency, and +hence, an unphysical discontinuity between cases at Ω = +0 and Ω → 0. Actually, even for Ω ̸= 0, from Eqs. (65)- +(66), near Tc with a weak gap, one has +βE ≈ +1 +4βΩ2 +� +n>0 +� +2 +pn−iΩ − +1 +pn−2iΩ − 1 +pn +� +, +(67) +and +λE ≈ +1 +2βΩ2 +� +n>0 +� +2 +pn−iΩ − +1 +pn−2iΩ− 1 +pn +� +. +(68) +In this circumstance, as βE = λE/2, the Higgs-mode gen- +eration δ|∆|(2), proportional to λE/βE from the equation +of motion in Eq. (64), becomes undefined at Matsubara +frequency iΩm which leads to λE = βE = 0. This di- +rectly poses a sharp challenge to the study in Ref. 41. +The derivation in the present study, which is performed +in the Keldysh formalism and totally irrelevant to Mat- +subara space, naturally and analytically proves the con- +tinuous variable of the optical frequency in this situa- +tion. +With the continuous optical frequency, near Tc, +from Eqs. (67)-(68), one finds a finite Higgs-mode gener- +ation at all Ω: +δ|∆|(2) = −(eA0vF )2 +3 +4∆0 +[4∆2 +0 − (2Ω)2], +(69) +which exactly recovers the one [Eq. (28)] derived from the +Ginzburg-Landau Lagrangian. +As for the regime with +temperature far below Tc, with the continuous optical +frequency, at low frequency (Ω < ∆0), one finds the co- +efficient βE = 1 +β +� +n>0(p2 +n + ∆2 +0)−3/2 and in particular, a +finite response coefficient: +λE ≈ 1 +2β +� +n>0 +∂2 +pn +� +1 +� +p2n+∆2 +0 +� +, +(70) +implying a finite Higgs-mode generation at clean case. +It is also noted that although the Eilenberger equation +with the continuous optical frequency can recover the +finite Higgs-mode generation at clean limit revealed +by Ginzburg-Landau Lagrangian and gauge-invariant +kinetic equation19,46, this approach fails to derive the +Higgs-mode damping by impurity scattering due to the +generically incomplete scattering integral41. As proved +in Ref. 63, because of the quasiclassical approximation on +τ3-Green function, the scattering integral in Eilenberger +equation only involves the anisotropic part of the Green +function that is related to the transport property, but +generically drops out the isotropic one which determines +the Higgs-mode lifetime. +In this circumstance, the +path-integral approach64 and gauge-invariant kinetic +equation45 provide efficient and separate approaches to +derive the induced damping of the Higgs mode by impu- +rities, which agrees with the analysis through Heisenberg +equation of motion as mentioned in the introduction +and provides a possible origin for the experimentally ob- +served broadening of the Higgs-mode resonance signal as +well as the fast Higgs-mode damping after excitation45,64. +V. +SUMMARY +In summary, we have resolved the current controversy +in the literature that why the previous derivations at +clean limit through the path-integral approach34–36 and +Eilenberger equation41 within the Matsubara formalism +failed to reach the Higgs-mode generation revealed by +Ginzburg-Landau Lagrangian46 and gauge-invariant ki- +netic equation19,46. +The crucial treatment leading to +this controversy lies at an artificial scheme within the +Matsubara formalism that whether the involved external +optical frequency Ω in the gap dynamics is taken as con- +tinuous variable or bosonic Matsubara frequency iΩm. +To resolve this confusion, we derive the effective action +of superconducting gap near Tc in the presence of the +vector potential through the path-integral approach, and +show that only by taking Ω as continuous variable within +Matsubara formalism, one can achieve the fundamental +Ginzburg-Landau superconducting Lagrangian in agree- +ment with Landau phase-transition theory and symme- +try analysis. In addition to this physical justification, we +also perform the formulation of the gap dynamics within +a separate and rigorous framework—Keldysh formalism, +which is totally irrelevant to Matsubara space. By ap- +plying the Eilenberger equation in Keldysh space to cal- +culate the second-order response of the Higgs mode, it is +analytically proved that the involved optical frequency is +always a continuous variable, leading to finite response +coefficient at clean limit. +Consequently, the present study confirms the uni- +fied conclusion, i.e., a finite Higgs-mode generation at +clean limit in the second-order response of supercon- +ductors from three different microscopic approaches (in- +cluding the gauge-invariant kinetic equation, Eilenberger +equation and path-integral approach) as well as from +Ginzburg-Landau Lagrangian, and can therefore help un- +derstanding the experimental findings of the observed +Higgs-mode excitation10–16. +Acknowledgments +The authors acknowledge financial support from the +National Natural Science Foundation of China under + +10 +Grants No. 11334014 and No. 61411136001. +Appendix A: Derivation of correlation coefficients +In this part, we present the specific expressions of the related correlation coefficients in the superconducting La- +grangian in Eq. (14). Firstly, as the Green function G0(p) = +p0+ξkτ3 +p2 +0−ξ2 +k +only consists of the τ0 and τ3 components, +from Eqs. (15)-(18), one immediately finds χ1 = χ13 = χ111 = χ100 = χ1113 = 0. Moreover, within the Matsubara +formalism [p = (ipn, k)], one has +χ113 = +� +p +Tr[G0(p+2q)τ1G0(p+q)τ1G0(p)τ3] = +� +p +2ξk[(ipn+2Ω)(ipn+Ω)−ξ2 +k−ipnΩ] +[(ipn+2Ω)2−ξ2 +k][(ipn+Ω)2−ξ2 +k][(ipn)2−ξ2 +k] = 0, (A1) +χp = 1 +2 +� +p +Tr[G0(p+q)τ1G0(p)τ1]+ 1 +U = +� +p +(ipn+Ω)2+(ipn)2−(ipn+Ω−ipn)2−2ξ2 +k +2[(ipn+Ω)2−ξ2 +k][(ipn)2−ξ2 +k] ++ 1 +U +≈ −Ω2 +2 +� +p +1 +[(ipn+Ω)2−ξ2 +k][(ipn)2−ξ2 +k] + +� +k +f(ξk) − f(−ξk) +2ξk ++ 1 +U += −Ω2 +2 +� +p +1 +[(ipn+Ω)2−ξ2 +k][(ipn)2−ξ2 +k] +D +� ωD +−ωD +dξk +tanh(βcξk/2)−tanh(βξk/2) +2ξk +, +(A2) +χ1111 = +� +p +Tr[G0(p+q)τ1G0(p)τ1G0(p+q)τ1G(p)τ1] = +� +p +2 +(ipn+Ω−ξk)2(ipn+ξk)2 , +χ1010 = +� +p +Tr[G0(p+q)τ1G0(p)τ0G0(p+q)τ1G(p)τ0] = +� +p +2 +[(ipn+Ω)2−ξ2 +k][(ipn)2−ξ2 +k], +(A3) +χ1100+χ0110 = +� +p +� +2 +(ipn+Ω−ξk)2[(ipn)2−ξ2 +k] + +2 +[(ipn+Ω)2−ξ2 +k](ipn−ξk)2 +� +. +(A4) +Here, we have used the gap equation +1 +U = D +� ωD +−ωD dξk +tanh(βcξk/2) +2ξk +at the critical temperature in the BCS theory18. +It is noted that χ113 vanishes as the consequence of the particle-hole symmetry, which eliminates the terms with the +odd order of ξk in the summation of k. +Then, further using the facts: +� +p +1 +[(ipn+Ω)2−ξ2 +k][(ipn)2−ξ2 +k] = D +β +� +ipn>0,η=± +� +dξk +1 +[(ipn+ηΩ)2−ξ2 +k][(ipn)2−ξ2 +k] += 2πiD +βΩ +� +ipn>0,η=± +� +1 +(2ipn+2Ω)(2ipn+Ω) − +1 +2ipn(2ipn+Ω) +� += πiD +βΩ2 +� +ipn>0,η=± +� +4 +2ipn+Ω − 1 +ipn +− +1 +ipn+Ω +� +, (A5) +� +p +2 +(ipn+Ω−ξk)2(ipn+ξk)2 = D +β +� +ipn>0,η=± +� +dξk +2 +(ipn+ηΩ−ξk)2(ipn+ξk)2 = − +� +ipn>0,η=± +8πiD/β +(2ipn+ηΩ)3 , (A6) +� +p +� +2 +(ipn+Ω−ξk)2[(ipn)2−ξ2 +k] + +2 +[(ipn+Ω)2−ξ2 +k](ipn−ξk)2 +� += D +β +� +ipn>0,η=± +� +dξk +� +2 +(ipn+ηΩ−ξk)2[(ipn)2−ξ2 +k] + +2 +[(ipn+ηΩ)2−ξ2 +k](ipn−ξk)2 +� += 2πiD +βΩ2 +� +ipn>0,η=± +� +4 +2ipn+ηΩ − 1 +ipn +− +1 +ipn+ηΩ +� +, +(A7) +the Landau parameters βp [Eq. (21)], λp [Eq. (22)], αp [Eq. (23)] and γp [Eq. (24)] are derived. + +11 +Appendix B: Derivation of retarded Green function from Eilenberger equation +In this part, we present the derivation of the retarded Green function from the Eilenberger equation. With the +initial state of the retarded Green function in Eq. (41), the equation of the first order of retarded Green function in +Eq. (42) is re-written as +SR(E1)gR(0)(E1)δgR(1)(E)−δgR(1)(E)SR(E)gR(0)(E) = eA0 · vF ΠR(0) +3 +. +(B1) +From above equation, considering the normalization condition of gR(0)(E) in Eq. (46), one has +[SR(E1)]2δgR(1)(E)−SR(E1)SR(E)gR(0)(E1)δgR(1)(E)gR(0)(E) = SR(E1)eA0 · vF [τ3−gR(0)(E1)τ3gR(0)(E)], (B2) +SR(E)SR(E1)gR(0)(E1)δgR(1)(E)gR(0)(E)−[SR(E)]2δgR(1)(E) = SR(E)eA0 · vF [gR(0)(E1)τ3gR(0)(E)−τ3]. (B3) +Then, the solution of δgR(1)(E) in Eq. (43) can be easily obtained by adding Eqs. (B2) and (B3). Moreover, substi- +tuting Eq. (41) into Eq. (43), the specific expression of δgR(1)(E) is given by +δgR(1)(E) = (eA0 · vF )τ3 − [SR(E1)SR(E)]−1[E1Eτ3 + i∆0(E0 + E1)τ2 + ∆2 +0τ3] +SR(E1) + SR(E) +. +(B4) +Similarly, the equation of the second order of retarded Green function in Eq. (44) is re-written as +SR(E2)gR(0)(E2)δgR(2)(E)−δgR(2)(E)SR(E)gR(0)(E) = eA0·vF ΠR(1) +3 ++iδ|∆|(2)[gR(0)(E2)τ2−τ2gR(0)(E)], +(B5) +and using the normalization condition of gR(0)(E) in Eq. (46), one easily gets the solution of δgR(1)(E) in Eq. (45). +Substituting Eqs. (41) and (B4) into Eq. (45), the specific expression of the τ2 component of δgR(2)(E) reads +δgR(2) +2 +(E) = i(eA0·vF )2∆0 +�(EE2+E1E+ E1E2+∆2 +0)[SR(E1)+SR(E2)+SR(E)] +SR(E1)SR(E2)SR(E) +− 1 +� +1 +[SR(E)+SR(E2)] +× +1 +[SR(E1)+SR(E2)][SR(E1)+SR(E)] +iδ|∆|(2) 2∆2 +0+2EE2+2SR(E)SR(E2) +Γ(2)(E) +. +(B6) +Further considering +[SR(E1)+SR(E2)+SR(E)][SR(E)−SR(E1)][SR(E1)−SR(E2)][SR(E)−SR(E2)] +SR(E1)SR(E2)SR(E) += E2 +1 −E2 +2 +SR(E) + E2 +2 −E2 +SR(E1) + E2−E2 +1 +SR(E2) , +(B7) +one has +δgR(2) +2 +(E) = +i(eA0·vF )2∆0 +(E2 +1 − E2)(E2 +2 − E2)(E2 +1 − E2 +2) +� +(EE2+E1E+ E1E2+∆2 +0) +�E2 +1 −E2 +2 +SR(E) + E2 +2 −E2 +SR(E1) + E2−E2 +1 +SR(E2) +� +− [SR(E)−SR(E2)][SR(E1)−SR(E2)][SR(E)−SR(E1)] +� ++ iδ|∆|(2) 2∆2 +0+2EE2+2SR(E)SR(E2) +Γ(2)(E) += +i(eA0·vF )2∆0 +(E2 +1 −E2)(E2 +2 −E2)(E2 +1 −E2 +2) +� +(E2 +1 −E2 +2)(E+E2)(E1+E)−[SR(E)]2 +SR(E) ++ (E+E1)(E1+E2)−[SR(E1)]2 +SR(E1) +× (E2 +2 −E2)+(E2−E2 +1)(E+E2)(E2+E1)−[SR(E2)]2 +SR(E2) ++SR(E) +� +[SR(E1)]2−[SR(E2)]2� ++ +� +[SR(E2)]2 +− [SR(E)]2� +SR(E1)+SR(E2) +� +[SR(E)]2−[SR(E1)]2�� ++iδ|∆|(2) 4∆2 +0−(E−E2)2+[SR(E)+SR(E2)]2 +Γ(2)(E) += i(eA0·vF )2∆0 +2Ω2 +� +1 +SR(E2) + +1 +SR(E) − +2 +SR(E1) +� ++ iδ|∆|(2)�4∆2 +0 − (2Ω)2 +Γ(2)(E) ++ +1 +SR(E) +� +. +(B8) +Then, Eq. (49) is derived. + +12 +Appendix C: Derivation of distribution function from Eilenberger equation +In this part, we present the derivation of the distribution function from the Eilenberger equation. As mentioned +in Sec. IV B, according to Eq. (33), the Keldysh Green function satisfies the same equation as the retarded/advanced +one. Therefore, the equation of the first order of the Keldysh Green function reads +(E1τ3+i∆0τ2)δgK(1)(E)−δgK(1)(E)(Eτ3+i∆0τ2) = eA0 · vF ΠK(0) +3 +. +(C1) +Substituting gK(0) [Eq. (54)] and δgK(1)(E) [Eq. (55)], the above equation becomes +(E1τ3+i∆0τ2)[gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)]−[gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)](Eτ3+i∆0τ2)=h(0)(E) +× [δgR(1)(E)(Eτ3+i∆0τ2)−(E1τ3+i∆0τ2)δgR(1)(E)]−h(0)(E1)[δgA(1)(E)(Eτ3+i∆0τ2)−(E1τ3+i∆0τ2)δgA(1)(E)] +− eA0·vF [h(0)(E)τ3gR(0)(E)−gR(0)(E1)h(0)(E1)τ3−h(0)(E)τ3gA(0)(E)+gA(0)(E1)h(0)(E1)τ3]. +(C2) +Then, facilitated with the equations of the first order of the retarded [Eq. (42)] and advanced Green functions, one +arrives at the equation of the first order of the distribution function in Eq. (57). By first multiplying Eq. (57) by +(E1τ3 + i∆0τ2) from the left side and (Eτ3 + i∆0τ2) from the right side respectively and adding the obtained two +equations afterwards, one has +[(E2 +1 +∆2 +0)−(E2+∆2 +0)][gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)] = eA0·vF [h(0)(E1)−h(0)(E)] +× +� +gR(0)(E1)[(E1τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)]−[(E1τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)]gA(0)(E) +� +, +(C3) +from which the solution of the first order of the distribution function reads +δh(1)(E) = eA0·vF [h(0)(E1)−h(0)(E)][(E1τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)] +(E2 +1 +∆2 +0)−(E2+∆2 +0) +, +(C4) +and then, Eq. (58) is derived. +Similarly, the equation of the second order of the Keldysh Green function reads +(E2τ3+i∆0τ2)δgK(2)(E)−δgK(2)(E)(Eτ3+i∆0τ2) = eA0·vF ΠK(1) +3 ++iδ|∆|(2)[gK(0)(E2)τ2−τ2gK(0)(E)]. +(C5) +Substituting gK(0) [Eq. (54)] and δgK(1)(E) [Eq. (55)] as well as δgK(2)(E) [Eq. (56)], the above equation becomes +(E2τ3+i∆0τ2)[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)]−[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)](Eτ3+i∆0τ2) += [δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E)](Eτ3+i∆0τ2)−(E2τ3+i∆0τ2)[δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E)] ++ h(0)(E)[δgR(2)(E)(Eτ3+i∆0τ2)−(E2τ3+i∆0τ2)δgR(2)(E)]−[δgA(2)(E)(Eτ3+i∆0τ2)−(E2τ3+i∆0τ2)δgA(2)(E)] +×h(0)(E2)+eA0·vF [δgR(1)(E1)h(0)(E1)τ3−τ3δgR(1)(E)h(0)(E)+gR(0)(E2)δh(1)(E1)τ3−τ3gR(0)(E1)δh(1)(E)] +− eA0·vF [h(0)(E2)δgA(1)(E1)τ3−τ3h(0)(E1)δgA(1)(E)+δh(1)(E1)gA(0)(E1)τ3−τ3δh(1)(E)gA(0)(E)] +− iδ|∆|(2)[τ2gR(0)(E)h(0)(E)−gR(0)(E2)h(0)(E2)τ2−τ2gA(0)(E)h(0)(E)+gA(0)(E2)h(0)(E2)τ2]. +(C6) +Facilitated with the equations of the second order of the retarded [Eq. (44)] and advanced Green functions, one has +(E2τ3+i∆0τ2)[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)]−[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)](Eτ3+i∆0τ2) += [δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E)](Eτ3+i∆0τ2)−(E2τ3+i∆0τ2)[δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E)] ++ eA0·vF [h(0)(E1) − h(0)(E)]δgR(1)(E1)τ3+eA0·vF [gR(0)(E2)δh(1)(E1)τ3−τ3gR(0)(E1)δh(1)(E)] +− eA0·vF [h(0)(E2)−h(0)(E1)]τ3δgA(1)(E)−eA0·vF [δh(1)(E1)gA(0)(E1)τ3−τ3δh(1)(E)gA(0)(E)] ++ iδ|∆|(2)[h(0)(E2)−h(0)(E)][gR(0)(E2)τ2−τ2gA(0)(E)]. +(C7) +Further using Eq. (58) to replace eA0·vF [h(0)(E1) − h(0)(E)] with Ωδh1(E), the above equation is simplified as +(E2τ3+i∆0τ2)[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)]−[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)](Eτ3+i∆0τ2) += δh(1)(E)[δgR(1)(E1)(E1τ3+i∆0τ2)−(E2τ3+i∆0τ2)δgR(1)(E1)−eA0·vF τ3gR(0)(E1)] +− δh(1)(E1)[δgA(1)(E1)(E1τ3+i∆0τ2)−(E2τ3+i∆0τ2)δgA(1)(E1)+eA0·vF gA(0)(E1)τ3] ++ eA0·vF [gR(0)(E2)δh(1)(E1)τ3+τ3δh(1)(E)gA(0)(E)]+iδ|∆|(2)[h(0)(E2)−h(0)(E)][gR(0)(E2)τ2−τ2gA(0)(E)]. (C8) + +13 +Consequently, substituting equations of the first order of the retarded [Eq. (42)] and advanced Green functions to +above equation, one arrives at the equation of the second order of the distribution function in Eq. (59). By first +multiplying Eq. (57) by (E2τ3 + i∆0τ2) from the left side and (Eτ3 + i∆0τ2) from the right side respectively and +adding the obtained two equations afterwards, one has +[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)]= eA0·vF [δh(1)(E1)−δh(1)(E)] +E2 +2 −E2 +� +gR(0)(E2)[(E2τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)] +− [(E2τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)]gA(0)(E) +� ++iδ|∆|(2) h(0)(E2)−h(0)(E) +E2 +2 −E2 +� +gR(0)(E2)[(E2τ3+i∆0τ2)τ2+τ2(Eτ3 ++ i∆0τ2)]−[(E2τ3+i∆0τ2)τ2+τ2(Eτ3+i∆0τ2)]gA(0)(E) +� +, +(C9) +from which the solution of the first order of the distribution function reads +δh(2)(E) = eA0·vF +[δh(1)(E1)−δh(1)(E)][(E2τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)] +E2 +2 −E2 ++iδ|∆|(2) [h(0)(E2)−h(0)(E)][(E2τ3+i∆0τ2)τ2+τ2(Eτ3+i∆0τ2)] +E2 +2 −E2 += eA0·vF +δh(1)(E1)−δh(1)(E) +E2 − E ++iδ|∆|(2) [h(0)(E2)−h(0)(E)](2i∆0−2iΩτ1) +E2 +2 −E2 +≈ eA0·vF +δh(1)(E1)−δh(1)(E) +E2 − E ++iδ|∆|(2)2i∆0 +h(0)(E2)−h(0)(E) +E2 +2 −E2 +, +(C10) +and then, Eq. (60) is derived. +∗ Electronic address: yfgq@mail.ustc.edu.cn. +† Electronic address: mwwu@ustc.edu.cn. +1 P. B. 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B 106, 144509 (2022). + diff --git a/w9E3T4oBgHgl3EQf_QsV/content/tmp_files/load_file.txt b/w9E3T4oBgHgl3EQf_QsV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f744e0a7100bc394da6d6a539bcde425cddaebc4 --- /dev/null +++ b/w9E3T4oBgHgl3EQf_QsV/content/tmp_files/load_file.txt @@ -0,0 +1,947 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf,len=946 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='04832v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='supr-con] 12 Jan 2023 Optical response of Higgs mode in superconductors at clean limit: formulation through Eilenberger equation and Ginzburg-Landau Lagrangian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Yang∗ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Wu† Hefei National Laboratory for Physical Sciences at Microscale,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' and CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Anhui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' China (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 2023) Both macroscopic Ginzburg-Landau Lagrangian and microscopic gauge-invariant kinetic equation suggest a finite Higgs-mode generation in the second-order optical response of superconductors at clean limit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' whereas the previous derivations through the path-integral approach and Eilenberger equation within the Matsubara formalism failed to give such generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The crucial treatment leading to this controversy lies at an artificial scheme that whether the external optical frequency is taken as continuous variable or bosonic Matsubara frequency to handle the gap dynamics within the Matsubara formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' To resolve this issue, we derive the effective action of the superconducting gap near Tc in the presence of the vector potential through the path-integral approach, to fill the long missing blank of the microscopic derivation of the Ginzburg-Landau Lagrangian in superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' It is shown that only by taking optical frequency as continuous variable within the Matsubara formalism, can one achieve the fundamental Ginzburg-Landau Lagrangian, and in particular, the finite Ginzburg-Landau kinetic term leads to a finite Higgs-mode generation at clean limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' To further eliminate the confusion of the Matsubara frequency through a separate framework, we apply the Eilenberger equation within the Keldysh formalism, which is totally irrelevant to the Matsubara space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' By calculating the gap dynamics in the second-order response, it is analytically proved that the involved optical frequency is a continuous variable rather than bosonic Matsubara frequency, causing a finite Higgs-mode generation at clean limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' PACS numbers: 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='Gh, 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='Gz, 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='N- I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' INTRODUCTION In the past few decades, the Higgs mode in the field of superconductivity, which describes the amplitude fluc- tuation δ|∆| of the superconducting order parameter ∆, has attracted much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' This collective excitation corresponds to the radial excitation in the Mexican-hat potential of free energy1, and hence, exhibits a gapful energy spectrum at a long wavelength1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Owing to the advanced ultrafast terahertz technique in nonlinear op- tics, the Higgs mode has been experimentally observed and identified as the origin of the excited superfluid- density oscillation δρs in the second-order harmonic generation10–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The nonlinear optics in superconduc- tors has since stimulated a lot of experimental interest and inspired a great deal of theoretical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Whereas the existing and growing experimental ob- servations exhibit a very convincing evidence, the the- oretical descriptions concerning the Higgs-mode excita- tion in the literature are filled with controversies, which are detrimental to the understanding of the related ex- perimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The central issue lies at a ques- tion that with the conventional light-matter interactions Hd = ˆp · eAτ0/m (current-vertex-related drive effect) and Hp = e2A2τ3/(2m) (density-vertex-related pump ef- fect) by the vector potential A17–19, whether the Higgs mode can be optically excited at clean limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Here, τi=0,1,2,3 denote the Pauli matrices in Nambu space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Concerning this issue, although the early stage of works through the Bloch10–14,20–26 or Liouville27–31 equa- tion within the Anderson pseudospin picture32 revealed an excited fluctuation of the order parameter by pump effect Hp, a later symmetry analysis33 implies that the pumped order-parameter fluctuation by density-vertex- related Hp is a phase fluctuation rather than the claimed amplitude one19, as the revealed correlation between am- plitude (phase) mode and Hp in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 33 is zero (nonzero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The path-integral approach is naturally capable of dis- tinguishing the excitations of phase and Higgs modes by deriving the corresponding effective actions, and includes both pump and drive effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Using this approach, Cea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' derived a vanishing Higgs-mode generation in second- order response at clean limit34–36, in inconsistency with the previous experimental understanding10–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' To ex- plain the experimental findings, Cea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' pointed out34–36 that the density-vertex-related pump effect Hp can excite a finite fluctuation δn of charge density n, so they speculated that the experimentally observed superfluid-density oscillation δρs is attributed to charge- density fluctuation δn rather than the Higgs mode δ|∆|, since the superfluid density ρs ∝ n|∆|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In the several polarization-resolved measurements afterwards13–16, an isotropic second-order harmonic sig- nal was timely reported and provides a firm evidence to rule out the possible charge-density fluctuation, as the theoretically predicted response of the Higgs mode (charge-density fluctuation) is isotropic (anisotropic)34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Since then, it is believed that the Higgs-mode generation is zero at clean limit and one has to reply on impurity 2 scattering to mediate the Higgs-mode generation31,37–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In this situation, to take account of the microscopic scat- tering, Silaev applied the Eilenberger equation42 within the Matsubara formalism41,43, which solely includes the current-vertex-related drive effect Hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' He also derived a vanishing Higgs-mode generation at clean limit, but with impurities, a finite one to dominate over the charge- density fluctuation is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Particularly, Silaev showed that the impurity scattering only mediate the Higgs-mode excitation and is incapable of causing the damping of this collective excitation41, so the increase of the impurity density can enhance the optical signal of Higgs mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Meanwhile, using a gauge-invariant kinetic equation approach19,44,45 with complete electromagnetic effect and microscopic scattering, Yang and Wu derived totally op- posite results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' They obtain a finite Higgs-mode gener- ation contributed by the drive effect at clean limit19, and show that the charge-density fluctuation in fact van- ishes in the second-order response, as a consequence of the charge conservation and forbidden second-order har- monic current in systems with the spacial inversion sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The revealed Higgs-mode generation at clean limit can capture the experimental observation well19, and in particular, a finite damping/lifetime of the Higgs- mode excitation by impurities is also derived45, provid- ing a possible origin for the experimentally observed broadening of the Higgs-mode resonance signal as well as the fast Higgs-mode damping after optical excita- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' This damping agrees with the analysis of Heisen- berg equation of motion since the Higgs-mode excitation and electron-impurity interaction are non-commutative in Nambu space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' This finite Higgs-mode generation has therefore been in sharp contrast to the aforementioned vanishing one from Eilenberger equation and path-integral approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Actu- ally, at clean limit, the finite Higgs-mode generation in second-order response of superconductors is a direct con- sequence of the Ginzburg-Landau Lagrangian46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' This is because that from the time-dependent Ginzburg-Landau superconducting Lagrangian at clean limit, which was proposed by Pekker and Varma through the symmetry analysis and Lorentz invariance from the Landau phase- transition theory6, one can directly reveal the equation of motion of the Higgs mode by considering the amplitude fluctuation of the Landau order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, a fi- nite Higgs-mode generation in the second-order response is immediately obtained46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' This directly leads to a par- ticularly bizarre question that why both path-integral approach47 and Eilenberger equation48 can recover the Ginzburg-Landau equation but reach a zero Higgs-mode generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' To resolve this controversy, by re-examining the pre- vious derivations within the path-integral approach in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 34–36 and Eilenberger equation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 41, it is pointed out by Yang and Wu46 that both previous deriva- tions contain flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Specifically, the previous works34–36 within the path-integral approach only kept the per- turbation expansion of the action up to second order, whereas the essential coupling of the Higgs mode to the second order of the drive effect Hd emerges in the third- order perturbation expansion of the action46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' So this cou- pling and hence finite second-order harmonic generation of Higgs mode by the drive effect are excessively over- looked in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 34–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Within the Matsubara formalism, picking up this coupling in the path-integral approach, one can find the exactly same amplitude-response coeffi- cient as the one derived from Eilenberger equation41: λE = T 2Ω2 � pn � 2 � (pn−iΩ)2+∆2 0 − 1 � (pn−2iΩ)2+∆2 0 − 1 � p2n+∆2 0 � , (1) where pn = (2n+1)πT represents the fermionic Matsub- ara frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Nevertheless, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 41, the involved optical frequency Ω is taken as bosonic Matsubara frequency iΩm, lead- ing to a vanishing response coefficient λE in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (1) strongly against the finite one from gauge-invariant ki- netic equation19,46 and Ginzburg-Landau Lagrangian46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Moreover, because of this treatment, the prefactor 1/Ω2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (1) causes an undefined singularity at zero fre- quency, and an unphysical discontinuity between cases at Ω = 0 and Ω → 0 emerges46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In contrast, the previ- ous work in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 46 takes Ω as continuous variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' A finite Higgs-mode generation at clean limit in agreement with the Ginzburg-Landau Lagrangian is then derived, and the obtained λE from both Eilenberger equation and path-integral approach becomes exactly same as the one from gauge-invariant kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Actually, the Matsubara formalism is developed as an auxiliary-function technique in the finite-temperature Green function approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In this framework, whether taking the external optical frequency Ω as continuous variable or bosonic Matsubara frequency iΩm can not be self-justified by method itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The treatment of Ω as iΩm in the calculations of conductivity and dielectric function in normal metals50 can be cross-justified by many other methods irrelevant to Matsubara space, such as equa- tion of motion, zero-temperature Green function, Boltz- mann equation as well as Keldysh Green function ap- proaches, whereas such justification in the gap dynamics of superconductors has not been performed in the liter- ature so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Physically, any treatment leading to result strongly against the Ginzburg-Landau superconducting Lagrangian can not be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Nevertheless, this justi- fication in superconductors has been challenged, arguing that the Ginzburg-Landau superconducting Lagrangian is a phenomenological model near Tc and is unimportant in microscopic studies, as the microscopic derivation of this Lagrangian is absent in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Theoretically, this Lagrangian is a fundamental model by symmetry analysis and Lorentz invariance as well as Landau phase- transition theory, whereas to fill the long missing blank of the microscopic derivation, in the present work, we de- 3 rive the Lagrangian in superconductors through the basic path-integral approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In addition to this physical jus- tification, a natural and rigorous framework developed in the literature to eliminate the confusion of the Mat- subara frequency in superconductors is to perform the formulation within the Keldysh formalism49, which is a well-established systematic approach for studying non- equilibrium properties and is totally irrelevant to Mat- subara space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' We therefore apply the Eilenberger equa- tion within the Keldysh formalism to calculate the gap dynamics at clean limit to cross-justify the treatment of the external optical frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Specifically, through the path-integral approach within the Matsubara formalism, we derive the effective action of superconducting gap near Tc in the presence of the vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' We show that to recover the Ginzburg- Landau kinetic term, one needs to keep the perturba- tion expansion of the action up to the fourth order and formulate the fourth-order correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Par- ticularly, during our calculation of the correlation coeffi- cient, it is shown that only by taking optical frequency as continuous variable, one can recover the finite coef- ficient in the Ginzburg-Landau kinetic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, if one considers the gap fluctuation to obtain the equation of motion of the Higgs mode from the Ginzburg-Landau Lagrangian, it is clearly seen that the finite coefficient in the Ginzburg-Landau kinetic term directly leads to the finite response coefficient in the equation of motion of the Higgs mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Furthermore, with the vector potential alone, we apply the Eilenberger equation within the Keldysh formalism to derive the equation of motion of the Higgs mode at clean limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' It is established in the literature that the Keldysh Green function can be written as a function of the re- tarded and advanced ones through a general relation via the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' We prove that this relation- ship makes the Keldysh Green function directly satisfying the normalization condition of the Eilenberger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, we solve the retarded Green function and distri- bution function to obtain the Keldysh Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' With the derived Keldysh Green function, we obtain the equation of motion of the Higgs mode, which is exactly same as the one derived through Eilenberger equation within the Matsubara formalism41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In contrast, as our derivation is performed in the Keldysh formalism and is irrelevant to the Matsubara space, it is clearly shown that the involved optical frequency in the response coefficient in the equation of motion of the Higgs mode is a con- tinuous variable rather than the Matsubara frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Particularly, with the continuous optical frequency, the response coefficient in the equation of motion of the Higgs mode does not vanish, leading to a finite Higgs mode gen- eration at clean limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' HAMILTONIAN We first present the general Bogoliubov-de Gennes Hamiltonian of the conventional s-wave superconductors in the presence of the electromagnetic potential17–19,51: H = � dx ψ†(x) � ξˆp−eA + eφ ∆(x) ∆∗(x) −ξˆp+eA − eφ � ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (2) Here, ψ(x) = [ψ↑(x), ψ† ↓(x)]T is the field operator in the Nambu space and x = (t, x) represents the space-time vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' φ and A denote the scalar and vector potentials, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' the momentum operator ˆp = −iℏ∇ and ξˆp = ˆp2/(2m) − µ with m being the effective mass and µ denoting the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' It is noted that the scalar potential can be generally written as φ = ¯φ0 + Eφ · x + δφ, where ¯φ0 denotes the effect of the electric voltage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Eφ · x concerns the drive effect by electric field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' δφ is the induced scalar potential related to the long- range Coulomb interaction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=', Hartree field caused by charge density fluctuation)17,19,51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In consideration of the phase δθ(x) and amplitude δ|∆(x)| fluctuations around the equilibrium gap ∆0, the superconducting order parameter reads: ∆(x) = |∆(x)|eiδθ(x) = [∆0 + δ|∆(x)|]eiδθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (3) It is established that the phase mode δθ in Hamil- tonian above can be effectively removed by a unitary transformation51–53 ψ(x)→eiτ3δθ(x)/2ψ(x), (4) and then, one has H = � dx ψ†(x)(H0 + HLM)ψ(x), (5) where the free BCS Hamiltonian H0 is written as H0 = ξˆpτ3 + |∆(x)|τ1, (6) and the light-matter interaction reads HLM = ps · ˆp m + p2 s 2mτ3 + µeffτ3 (7) with the gauge-invariant superconducting momentum ps = ∇δθ/2 − eA and effective field µeff = eφ + ∂tδθ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' It has been revealed in the literature19,51 that thanks to the Coulomb screening, in the linear regime, at long- wavelength limit, the induced scalar potential eδφ can- cels the original longitudinal part e¯φ0 + ∂tδθ/2 in the effective field µeff = e¯φ0 + eEφ · x + eδφ + ∂tδθ/2, leaving only the drive effect of scalar-potential-induced electric field Eφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In general, the inclusion of this retained effect from scalar potential is essential to capture the optical- electric-field effect in a gauge-invariant manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Whereas considering the fact that the vector potential character- izes the Meissner effect/Ginzburg-Landau kinetic term in 4 addition to the optical-electric-field effect19,44,46,64, in the present work, we only focus on the electromagnetic effect from vector potential, similar to the previous works by Cea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='34–36 and Silaev41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The light-matter interaction then becomes HLM = ps · ˆp m + p2 s 2mτ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (8) Moreover, it has been established in the literature1,8,9,17,19,51,54–56 that the linear response of the phase mode, as a scalar quantity, responds to the longitudinal electromagnetic field and hence experiences the Coulomb screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The resonance pole of this response is then effectively lifted from the original gapless spectrum up to the high-energy plasma frequency ωp as a consequence of the Anderson-Higgs mechanism57, and hence, no effective linear response of the phase mode occurs at frequency Ω ≪ ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Because of this effect, the linear response of the phase mode cancels the unphysical longitudinal vector potential in ps, and the superconducting momentum that appears in the previous theoretical descriptions in the literature only involves the physical transverse vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The light-matter interaction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (8) then consists of the drive effect Hd and pump one Hp III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' DERIVATION OF GINZBURG-LANDAU LAGRANGIAN In this section, we present the derivations of the time- dependent Ginzburg-Landau Lagrangian and its non- equilibrium variation (equation of motion of the Higgs mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Specifically, the action of superconductors in the presence of vector potential after the Hubbard- Stratonovich transformation is written as17,47,55,56 S[ψ, ψ∗] = � dx � � s=↑,↓ ψ∗ s(x)(i∂t−ξˆp−eA)ψs(x) +ψ∗ ↑(x)ψ∗ ↓(x)∆(x)+ψ↓(x)ψ↑(x)∆∗(x)− |∆(x)|2 U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (9) Considering the spatial dependence of the gap, one has |∆(x)| = � q |∆q|eiq·x in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, applying the unitary transformation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (4), the above action in momentum space becomes S[ψ, ψ∗] = � dt � � kq � ψ∗ k+ q 2 ↑(i∂t−ξk+ q 2 +ps)ψk+ q 2 ↑ + ψ∗ −k+ q 2 ↓(i∂t−ξ−k+ q 2 +ps)ψ−k+ q 2 ↓ + (ψ−k+ q 2 ↓ψk+ q 2 ↑ + ψ∗ k+ q 2 ↑ψ∗ −k+ q 2 ↓)|∆q| � − � q |∆q|2 U � , (10) and in Nambu space, one finds the action related to gap: S[ψ, ψ∗] = � dt � q � � k ψ† kq(G−1 0 −Σ)ψkq− |∆q|2 U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (11) Here, the field operator ψ† kq = (ψ∗ k+ q 2 ↑, ψ−k+ q 2 ↓);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' G−1 0 = i∂t − ξkτ3 which gives the Green function G0(p) = (p0−ξkτ3)−1 in frequency space with the four-vector mo- mentum p = (p0, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' the self-energy reads Σ = |∆q|τ1 + k · (q/2 + ps) m + (q/2 + ps)2 2m τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (12) Considering the small gap near critical temperature Tc as well as the weak strength and spatial variation of the vector potential, the self-energy can be treated as small quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, after the standard integration over the Fermi field, one has S = S0− ∞ � n=1 1 n ¯Tr[(G0Σ)n]− � dt � q |∆q|2 U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (13) To derive the Lagrangian related to the superconduc- tivity, one needs to formulate the expansions of the action with respect to the fourth order of the self-energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=', keep the expansions up to n = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, with expan- sions up to n = 4, by only keeping the terms related to the gap, one can obtain the effective action Ss = � dqL with the frequency-momentum four vector q = (Ω, q) and the Lagrangian of the superconductivity: L = −χ1|∆q|− �1 2χ11 + 1 U � |∆q|2−χ13 (q/2+ps)2|∆q| 2m −χ111 |∆q|3 3 −χ100 k2 F (q/2+ps)2|∆q| 3m2 − χ1111|∆q|4 4 −χ113 (q/2+ps)2|∆q|2 2m −χ1113 (q/2+ps)2|∆q|3 2m −(χ1100 + χ0110 + χ1010)k2 F (q/2+ps)2|∆q|2 6m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (14) Here, we have neglected the terms proportional to odd orders of k · (q/2 + ps) as these anisotropic terms vanish after the summation of the momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The correla- tion coefficients are determined by χi = � p Tr[G0(p)τj], (15) χij = � p Tr[G0(p+q)τiG0(p)τj], (16) χijk = � p Tr[G0(p+2q)τiG0(p+q)τjG0(p)τk], (17) χijkl = � p Tr[G0(p+q)τiG0(p)τjG0(p+q)τkG(p)τl].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (18) Since the Green function only consists of the τ0 and τ3 components, one immediately finds χ1 = χ13 = χ111 = χ100 = χ1113 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Moreover, the coefficient χ113 van- ishes due to the particle-hole symmetry, and the forth- order correlation coefficient χ1010 = χ0110 + χ1100 (refer to Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, one finds an embryonic form of the Ginzburg-Landau Lagrangian of superconductors: L = −χp|∆q|2 − βp|∆q|4 2 − λp(q/2+ps)2|∆q|2 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (19) 5 with the parameters χp = χ11/2+1/U and βp = χ1111/2 as well as λp = k2 F χ1010/(3m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Within the Matsubara formalism [p = (ipn, k)], the retained correlation coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (19) are given by (refer to Appendix A) χp = αp − Ω2γp/2, (20) βp = − � ipn>0,η=± 2πiD β 2 (2ipn+ηΩ)3 , (21) λp = � ipn>0,η=± k2 F 3m 2πiD βΩ2 � 4 2ipn+ηΩ − 1 ipn − 1 ipn+ηΩ � , (22) with parameters: αp = D � ωD −ωD dξk tanh( βcξk 2 )−tanh( βξk 2 ) 2ξk =D ln T Tc ,(23) γp = � ipn>0,η=± πiD βΩ2 � 4 2ipn+ηΩ − 1 ipn − 1 ipn+ηΩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (24) Here, D and ωD denote the density of states and Debye frequency, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' f(x) stands for the Fermi dis- tribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' β = kBT and βc = kBTc with kB being the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' At low frequency (Ω < kBTc), one finds the specific parameters: γp = βp ≈ � n>0 πD βp3n = 7Dζ(3) 8(πT )2 , (25) λp ≈ � n>0 k2 F 3m 2πD 4β 4 p3n = k2 F 3m 7Dζ(3) (2πT )2 , (26) with ζ(x) being the Riemann zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, through the transformation into time-space coordinate, the time-dependent Ginzburg-Landau Lagrangian of su- perconductors is obtained: L = γp|i∂t∆|2 2 − � αp|∆|2+ βp|∆|4 2 + λp|(∇−2ieA)∆|2 4m � , (27) which is exactly same as the one obtained by Pekker and Varma through symmetry analysis and Lorentz invari- ance from the general Ginzburg-Landau free energy6 as it should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Moreover, the parameters αp [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (23)] and βp [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (25)] as well as λp [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (26)] here are exactly same as the obtained Landau parameters in the previous work by Gorkov18 in which the Ginzburg-Landau equation is derived from Gorkov equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' It is noted that in the derivation of the gap dynam- ics within the Matsubara formalism in the present work, we take the optical frequency Ω as continuous variable rather than bosonic Matsubara frequency iΩm, and only with continuous Ω in this circumstance, can one re- cover the Ginzburg-Landau Lagrangian as demonstrated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Actually, as seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (22), if Ω is taken as iΩm = 2miπT , different irrational response coefficients are obtained at cases with odd and even m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' For odd m, a divergent pole (2n + 1 = m and η = −1) which is unable to circumvent emerges in the formulation of the first term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' As for even m, through the frequency displacement in the Matsubara frequency summation, the response coefficient λp directly vanishes for nonzero m, but due to the prefactor 1/Ω2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (22), there exists an undefined singularity at zero frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, an unphysical discontinuity between Ω = 0 and Ω → 0 emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The difference between cases with odd and even m in bosonic Matsubara frequency iΩm = 2miπT is totally unreasonable within the Mat- subara formalism, and neither of them can recover the finite Landau parameter at low frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Clearly, any treatment that leads to consequence strongly against the fundamental model of symmetry analysis and Lorentz in- variance as well as Landau phase-transition theory can not be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' All of the irrationalities here simply sug- gest that the treatment of taking Ω as iΩm can not be correct in the derivation of gap dynamics in supercon- ductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Furthermore, from the Lagrangian above, with the equilibrium gap ∆0 = � −αp/βp, by considering the gap fluctuation δ|∆| through |∆| = ∆0 +δ|∆|, its equation of motion at long-wavelength limit is directly obtained: � (2∆0)2 − ∂2 t � δ|∆| = −λp γp e2A2 m 2∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (28) Then, one immediately finds the Higgs-mode energy spectrum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=', resonance pole) ωH = 2∆0 on the left- hand side of the equation above, and in particular, a finite second-order response of Higgs mode at clean limit on the right-hand side of the equation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' It is pointed out that although the derivation of the equation of mo- tion of the Higgs mode here is based on the small gap and only holds near Tc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' the serious derivation in regime ex- tending to T = 0 through three different microscopic ap- proaches including the gauge-invariant kinetic equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Eilenberger equation as well as path-integral approach also obtains the similar equation of motion as a conse- quence of the renormalization group (scaling) theory or basic local Abelian U(1) model (complex scalar field cou- pled to an electromagnetic potential) in the field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' DERIVATION OF HIGGS-MODE RESPONSE THROUGH EILENBERGER EQUATION To eliminate the confusion of the Matsubara frequency from a separate framework in addition to the physical justification above, in this section, we perform the for- mulation of the gap dynamics within the Keldysh for- malism, which is totally irrelevant to Matsubara space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Specifically, we apply the Eilenberger equation within the Keldysh formalism to derive the second-order optical response of the Higgs mode at clean limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The Eilen- berger equation42,58 is derived from the basic Gorkov 6 equation of τ3-Green function through the quasiclassical approximation49: gR/K/A R,kF (t, t′) = i π � dξk � drτ3GR/K/A(x, x′)e−ik·(x−x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (29) Here, R = (x + x′)/2 represents the center-of-mass spa- tial coordinate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' the retarded (R), advanced (A) and Keldysh (K) Green functions are defined by49,58 GR(x, x′) = −i⟨{ψ(x), ψ†(x′)}⟩θ(t − t′), (30) GA(x, x′) = i⟨{ψ(x), ψ†(x′)}⟩θ(t′ − t), (31) GK(x, x′) = −i⟨[ψ(x), ψ†(x′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (32) The Eilenberger equation within the Keldysh formal- ism at clean limit reads42,58: i{τ3∂t, ˆg}t−[(∆0+δ|∆|)τ1τ3, ˆg]t+[eA·vFτ3, ˆg]t =0, (33) where the green function matrices ˆg is defined as ˆg = � gR gK 0 gA � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (34) Here, the operators [X, ˆg]t = X(t1)ˆg(t1, t2) − ˆg(t1, t2)X(t2) and {X, ˆg}t = X(t1)ˆg(t1, t2) + ˆg(t1, t2)X(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Moreover, to guarantee the unique solution, the Eilen- berger equation must be supplemented by the normaliza- tion condition42,58: ˆg ◦ ˆg = 1, (35) where the operator ◦ is defined by relation A ◦ B = � dtA(t1, t)B(t, t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The corresponding gap equation is written as ∆0 + δ|∆| = −iUTr[⟨gK R,kF(t, t)⟩F τ2/2], (36) with ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='⟩F denoting the angular average over the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Considering an external optical field with A(t) = A0e−iΩt, by self-consistently solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (33)-(36), one can formulate the Higgs-mode generation at clean limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Specifically, in this circumstance, one can expand the quasiclassical Green function matrices as ˆg = ˆg(0) + δˆg(1) + δˆg(2), (37) with the m-th order response δˆg(m) on the initial state ˆg(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Correspondingly, the Higgs-mode generation δ|∆| = δ|∆|(1)e−iΩt + δ|∆|(2)e−2iΩt from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (36) with δ|∆|(1) and δ|∆|(2) being the excitations in first- and second- order optical responses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Particularly, the first-order response of Keldysh Green function must be anisotropic in momentum space, leading to a vanishing δ|∆|(1) after the angular average over the Fermi surface in the gap equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' We then directly take δ|∆|(1) = 0 for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Consequently, the Eilenberger equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (33) be- comes a chain of equations: {τ3∂t, δˆg(1)}t+[∆0τ2, δˆg(1)]t−i[eA·vF τ3, ˆg(0)]t =0, (38) {τ3∂t, δˆg(2)}t + [∆0τ2, δˆg(2)]t − i[eA · vF τ3, δˆg(1)]t + [δ|∆|(2)e−2iΩtτ2, ˆg(0)]t =0, (39) and one can solve δˆg(1) and δˆg(2) in sequence with the given initial state ˆg(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then, with the obtained solution of the Keldysh Green function, one can further derive the response of the Higgs mode from the gap equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Solution of retarded Green function In this part, we first solve the retarded Green functions from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' From the expansion of Green function ma- trices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (37), the retarded Green function is written as gR(t, t′) = gR(0)(t, t′) + δgR(1)(t, t′) + δgR(2)(t, t′) = � dE 2π eiE(t′−t)[gR(0)(E)+δgR(1)(E)e−iΩt + δgR(2)(E)e−2iΩt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (40) The initial state of the retarded Green function has been established in the literature from Gorkov equation18,41,58, and is written as gR(0)(E) = � dξk π iτ3(E+ξkτ3+∆0τ1) (E+i0+)2−ξ2 k−∆2 0 = Eτ3+i∆0τ2 SR(E) , (41) with SR(E) = � (E + i0+)2 − ∆2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' By defining E1 = E + Ω and E2 = E + 2Ω, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (38), the equation of the first order of retarded Green function is written as (E1τ3+i∆0τ2)δgR(1)(E)−δgR(1)(E)(Eτ3+i∆0τ2) = eA0 · vF ΠR(0) 3 , (42) from which one finds the first-order solution (refer to Ap- pendix B): δgR(1)(E) = (eA0 · vF )τ3 − gR(0)(E1)τ3gR(0)(E) SR(E1) + SR(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (43) Here, ΠR(i) 3 = gR(i)(E1)τ3 − τ3gR(i)(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Similarly, the equation of the second order of retarded Green function from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (39) reads (E2τ3+i∆0τ2)δgR(2)(E)−δgR(2)(E)(Eτ3+i∆0τ2) = eA0·vF ΠR(1) 3 +iδ|∆|(2)[gR(0)(E2)τ2−τ2gR(0)(E)],(44) and gives the second-order solution (refer to Ap- pendix B): δgR(2)(E) = iδ|∆|(2) τ2−gR(0)(E2)τ3gR(0)(E) SR(E2) + SR(E) + eA0·vF E2 2 −E2 × � SR(E2)gR(0)(E2)ΠR(1) 3 −ΠR(1) 3 SR(E)gR(0)(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (45) 7 Considering the response expansions, the normaliza- tion condition [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (35)] for the retarded Green function is written as [gR(0)(E)]2 = 1, (46) gR(0)(E1)δgR(1)(E)+δgR(1)(E)gR(0)(E) = 0, (47) gR(0)(E2)δgR(2)(E)+δgR(2)(E)gR(0)(E) +δgR(1)(E1)δgR(1)(E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (48) The initial-state gR(0)(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (41) naturally satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Facilitated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (46), correspondingly sub- stituting the solutions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (43) and (45), one can eas- ily demonstrate the normalization conditions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (47) and (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Therefore, as the self-consistent crosscheck, the obtained solutions of the retarded Green function satisfy the normalization condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Further substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (41) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (42) and (44), the specific solution of the τ2 component of δgR(2)(E) is given by (refer to Appendix B) δgR(2) 2 (E)=iδ|∆|(2)�4∆2 0 − (2Ω)2 Γ(2)(E) + 1 SR(E) � + i(eA0·vF )2∆0 2Ω2 � 1 SR(E2) + 1 SR(E) − 2 SR(E1) � ,(49) with Γ(2)(E) = 2SR(E2)SR(E)[SR(E2)+SR(E)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' It is noted that the involved external optical frequency in the τ2 component of the second order of the retarded Green function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (49) is a continuous variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' To further consider the gap dynamics at nonzero temper- ature and eliminate the confusion of the auxiliary Mat- subara frequency, we next derive the Keldysh Green func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Solution of Keldysh Green function In this part, we derive the Keldysh Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' We start with the normalization condition [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (35)] for the Keldysh Green function: gR◦gK + gK◦gA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (50) It is established that the Keldysh Green function can be written as a function of the retarded and advanced ones through a general relation49,58: gK = gR ◦ h − h ◦ gA, (51) where h(t, t′) denotes the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Substi- tuting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (51) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (50), one finds that the normaliza- tion condition for the Keldysh Green function is immedi- ately satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Consequently, with the obtained retarded and hence advanced Green function in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' IV A, to solve the Keldysh Green function, one only needs to solve the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In the previous work to derive the Ginzburg-Landau equation from Eilenberger equation within the Keldysh formalism48, the distribution function h(t, t′) is directly taken as the equilibrium one � dE 2π h(E)e−iE(t−t′) with the Fourier component written as h(E) = tanh �βE 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (52) This treatment usually concerns the case near equilib- rium or in strongly-interacting systems as applied in the transport theory of normal and superconducting metals49, and has also been widely used in previous stud- ies through the Eilenberger equation58,59 and diffusive Usadel one60,61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In the present work, at clean limit, with a weak external excitation, we demonstrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (52) by seriously taking account of the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Specifically, from the expansion of Green function ma- trices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (37), the Keldysh Green function reads gK(t, t′) = � dE 2π eiE(t′−t)[gK(0)(E)+δgK(1)(E)e−iΩt + δgK(2)(E)e−2iΩt], (53) and with the general relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (51), one has gK(0)(E)=gR(0)(E)h(0)(E) − h(0)(E)gA(0)(E), (54) δgK(1)(E)=gR(0)(E1)δh(1)(E)+δgR(1)(E)h(0)(E) −h(0)(E1)δgA(1)(E)−δh(1)(E)gA(0)(E),(55) δgK(2)(E)=δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E) +gR(0)(E2)δh(2)(E)+δgR(2)(E)h(0)(E) −h(0)(E2)δgA(2)(E)−δh(2)(E)gA(0)(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (56) Here, δh(1)(E) and δh(2)(E) stand for the first- and second-order responses on the initial-state h(0)(E), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (33), the Keldysh Green function sat- isfies the same equation as the retarded/advanced one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Consequently, by correspondingly replacing gR(0) and δgR(1) by gK(0) and δgK(1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (42) and then substi- tuting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (55), one finds the equation of the first-order distribution function (refer to Appendix C): (E1τ3+i∆0τ2)[gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)] − [gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)](Eτ3+i∆0τ2) = eA0·vF [h(0)(E1)−h(0)(E)][gR(0)(E1)τ3−τ3gA(0)(E)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (57) From above equation, the solution of the first-order dis- tribution function reads (refer to Appendix C) δh(1)(E) = eA0·vF h(0)(E1)−h(0)(E) E2 1 −E2 [(E1τ3+i∆0τ2)τ3 + τ3(Eτ3+i∆0τ2)] = eA0·vF h(0)(E + Ω)−h(0)(E) Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (58) Similarly, by correspondingly replacing gR(0) and δgR(i=1,2) by gK(0) and δgK(i=1,2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (44), with 8 Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (55) and (56) as well as the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (58), the equation of the second-order distribution function reads (refer to Appendix C) (E2τ3+i∆0τ2)[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)] − [gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)](Eτ3+i∆0τ2) =eA0·vF [gR(0)(E2)τ3−τ3gA(0)(E)][δh(1)(E1)−δh(1)(E)] + iδ|∆|(2)[h(0)(E2)−h(0)(E)][gR(0)(E2)τ2−τ2gA(0)(E)], (59) from which the solution of the second-order distribution function is obtained as (refer to Appendix C) δh(2)(E) = (eA0·vF )δh(1)(E1)−δh(1)(E) 2Ω − δ|∆|(2) ∆0 E+Ω h(0)(E2)−h(0)(E) 2Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (60) Then, both the first- and second-order distribution func- tions are diagonal as they should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The initial-state distribution function can be obtained from the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (2) in self-consistent con- sideration of the Higgs mode and vector potential, and is written as h(0)(Ek) = tanh �β 2 �� ξ2 k+(∆0+δ|∆|)2−eA·vF �� ≈ tanh �β 2 � Ek + ∆0δ|∆| Ek − eA·vF �� , (61) where Ek = � ξ2 k + ∆2 0 denotes the Bogoliubov quasi- particle energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Following the standard treatment of energy E = Ek as the previous work in superconduct- ing state62, with the weak excitation, at low frequency (Ω < E = Ek), with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (58) and (60)-(61), the total distribution function in relative-frequency space reads h(E) = h(0)(E) + e−iΩtδh(1)(E) + e−2iΩtδh(2)(E) = � 1+(eA·vF)∂E+ (eA·vF )2∂2 E 2 � h(0)(E) − δ|∆|∆0 E ∂Eh(0)(E) ≈ tanh �βE 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (62) Then, the drive effect of vector potential and Higgs-mode part in the initial-state distribution are exactly canceled by the first- and second-order distribution functions, leading to the widely applied distribution function [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (52)] in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Higgs-mode generation In this part, with the obtained distribution function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (52) and the second-order retarded Green function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (49), from the gap equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (36)], the second- order optical response of the Higgs mode at clean limit is determined by δ|∆|(2) = −iU � dE 2π ⟨[h(E)δgR(2) 2 (E) − h(E2)δgA(2) 2 (E)]⟩F = −iU � dE 2π 2h(E)⟨δgR(2) 2 (E)⟩F = U � dE2h(E) � δ|∆|(2)�4∆2 0 − (2Ω)2 Γ(2)(E) + 1 SR(E) � + (eA0vF )2∆0 6Ω2 � 1 SR(E2) + 1 SR(E) − 2 SR(E1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (63) Consequently, one arrives at the equation of motion of the Higgs mode at clean limit: [4∆2 0 − (2Ω)2]δ|∆|(2) = −(eA0vF )22∆0 3 λE βE , (64) similar to the one [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (28)] obtained from the Ginzburg- Landau Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Here, through the standard contour integral, the amplitude-correlation coefficient reads βE = − � dE 2π h(E) SR(E2)SR(E)[SR(E2)+SR(E)] = � n>0 1/(4βiΩ) pn−iΩ � 1 � (pn−2iΩ)2+∆2 0 − 1 � p2n+∆2 0 � , (65) and the essential response coefficient is given by λE =− � dE 2π h(E) 2Ω2 � 1 SR(E2) + 1 SR(E) − 2 SR(E1) � = 1 2βΩ2 � n>0 � 2 � (pn−iΩ)2+∆2 0 − 1 � (pn−2iΩ)2+∆2 0 − 1 � p2n+∆2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (66) It is noted that both amplitude-correlation coefficient βE [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (65)] and response one λE [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (66)] derived here are exactly same as the ones obtained in the previ- ous work41 by Silaev through Eilenberger equation within Matsubara formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' However, in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (65) and (66), the fermionic Matsubara frequencies ipn arise from the singularities in the distribution function h(E) during the standard contour integral in the complex plane, whereas 9 the involved external optical frequency Ω within the Keldysh formalism is always a continuous variable, in contrast to the treatment of taking optical frequency as bosonic Matsubara frequency in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' As mentioned in the introduction, the treatment of taking optical fre- quency as bosonic Matsubara frequency leads to the van- ishing response coefficient λE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=', zero Higgs-mode gen- eration) at all Ω ̸= 0, strongly against the Ginzburg- Landau Lagrangian, whereas the prefactor 1/Ω2 in λE causes an undefined singularity at zero frequency, and hence, an unphysical discontinuity between cases at Ω = 0 and Ω → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Actually, even for Ω ̸= 0, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (65)- (66), near Tc with a weak gap, one has βE ≈ 1 4βΩ2 � n>0 � 2 pn−iΩ − 1 pn−2iΩ − 1 pn � , (67) and λE ≈ 1 2βΩ2 � n>0 � 2 pn−iΩ − 1 pn−2iΩ− 1 pn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (68) In this circumstance, as βE = λE/2, the Higgs-mode gen- eration δ|∆|(2), proportional to λE/βE from the equation of motion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (64), becomes undefined at Matsubara frequency iΩm which leads to λE = βE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' This di- rectly poses a sharp challenge to the study in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The derivation in the present study, which is performed in the Keldysh formalism and totally irrelevant to Mat- subara space, naturally and analytically proves the con- tinuous variable of the optical frequency in this situa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' With the continuous optical frequency, near Tc, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (67)-(68), one finds a finite Higgs-mode gener- ation at all Ω: δ|∆|(2) = −(eA0vF )2 3 4∆0 [4∆2 0 − (2Ω)2], (69) which exactly recovers the one [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (28)] derived from the Ginzburg-Landau Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' As for the regime with temperature far below Tc, with the continuous optical frequency, at low frequency (Ω < ∆0), one finds the co- efficient βE = 1 β � n>0(p2 n + ∆2 0)−3/2 and in particular, a finite response coefficient: λE ≈ 1 2β � n>0 ∂2 pn � 1 � p2n+∆2 0 � , (70) implying a finite Higgs-mode generation at clean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' It is also noted that although the Eilenberger equation with the continuous optical frequency can recover the finite Higgs-mode generation at clean limit revealed by Ginzburg-Landau Lagrangian and gauge-invariant kinetic equation19,46, this approach fails to derive the Higgs-mode damping by impurity scattering due to the generically incomplete scattering integral41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' As proved in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 63, because of the quasiclassical approximation on τ3-Green function, the scattering integral in Eilenberger equation only involves the anisotropic part of the Green function that is related to the transport property, but generically drops out the isotropic one which determines the Higgs-mode lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In this circumstance, the path-integral approach64 and gauge-invariant kinetic equation45 provide efficient and separate approaches to derive the induced damping of the Higgs mode by impu- rities, which agrees with the analysis through Heisenberg equation of motion as mentioned in the introduction and provides a possible origin for the experimentally ob- served broadening of the Higgs-mode resonance signal as well as the fast Higgs-mode damping after excitation45,64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' SUMMARY In summary, we have resolved the current controversy in the literature that why the previous derivations at clean limit through the path-integral approach34–36 and Eilenberger equation41 within the Matsubara formalism failed to reach the Higgs-mode generation revealed by Ginzburg-Landau Lagrangian46 and gauge-invariant ki- netic equation19,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' The crucial treatment leading to this controversy lies at an artificial scheme within the Matsubara formalism that whether the involved external optical frequency Ω in the gap dynamics is taken as con- tinuous variable or bosonic Matsubara frequency iΩm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' To resolve this confusion, we derive the effective action of superconducting gap near Tc in the presence of the vector potential through the path-integral approach, and show that only by taking Ω as continuous variable within Matsubara formalism, one can achieve the fundamental Ginzburg-Landau superconducting Lagrangian in agree- ment with Landau phase-transition theory and symme- try analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' In addition to this physical justification, we also perform the formulation of the gap dynamics within a separate and rigorous framework—Keldysh formalism, which is totally irrelevant to Matsubara space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' By ap- plying the Eilenberger equation in Keldysh space to cal- culate the second-order response of the Higgs mode, it is analytically proved that the involved optical frequency is always a continuous variable, leading to finite response coefficient at clean limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Consequently, the present study confirms the uni- fied conclusion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=', a finite Higgs-mode generation at clean limit in the second-order response of supercon- ductors from three different microscopic approaches (in- cluding the gauge-invariant kinetic equation, Eilenberger equation and path-integral approach) as well as from Ginzburg-Landau Lagrangian, and can therefore help un- derstanding the experimental findings of the observed Higgs-mode excitation10–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Acknowledgments The authors acknowledge financial support from the National Natural Science Foundation of China under 10 Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 11334014 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 61411136001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Appendix A: Derivation of correlation coefficients In this part, we present the specific expressions of the related correlation coefficients in the superconducting La- grangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Firstly, as the Green function G0(p) = p0+ξkτ3 p2 0−ξ2 k only consists of the τ0 and τ3 components, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (15)-(18), one immediately finds χ1 = χ13 = χ111 = χ100 = χ1113 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' within the Matsubara formalism [p = (ipn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' k)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' one has χ113 = � p Tr[G0(p+2q)τ1G0(p+q)τ1G0(p)τ3] = � p 2ξk[(ipn+2Ω)(ipn+Ω)−ξ2 k−ipnΩ] [(ipn+2Ω)2−ξ2 k][(ipn+Ω)2−ξ2 k][(ipn)2−ξ2 k] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (A1) χp = 1 2 � p Tr[G0(p+q)τ1G0(p)τ1]+ 1 U = � p (ipn+Ω)2+(ipn)2−(ipn+Ω−ipn)2−2ξ2 k 2[(ipn+Ω)2−ξ2 k][(ipn)2−ξ2 k] + 1 U ≈ −Ω2 2 � p 1 [(ipn+Ω)2−ξ2 k][(ipn)2−ξ2 k] + � k f(ξk) − f(−ξk) 2ξk + 1 U = −Ω2 2 � p 1 [(ipn+Ω)2−ξ2 k][(ipn)2−ξ2 k] +D � ωD −ωD dξk tanh(βcξk/2)−tanh(βξk/2) 2ξk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (A2) χ1111 = � p Tr[G0(p+q)τ1G0(p)τ1G0(p+q)τ1G(p)τ1] = � p 2 (ipn+Ω−ξk)2(ipn+ξk)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' χ1010 = � p Tr[G0(p+q)τ1G0(p)τ0G0(p+q)τ1G(p)τ0] = � p 2 [(ipn+Ω)2−ξ2 k][(ipn)2−ξ2 k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (A3) χ1100+χ0110 = � p � 2 (ipn+Ω−ξk)2[(ipn)2−ξ2 k] + 2 [(ipn+Ω)2−ξ2 k](ipn−ξk)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (A4) Here, we have used the gap equation 1 U = D � ωD −ωD dξk tanh(βcξk/2) 2ξk at the critical temperature in the BCS theory18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' It is noted that χ113 vanishes as the consequence of the particle-hole symmetry, which eliminates the terms with the odd order of ξk in the summation of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' further using the facts: � p 1 [(ipn+Ω)2−ξ2 k][(ipn)2−ξ2 k] = D β � ipn>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='η=± � dξk 1 [(ipn+ηΩ)2−ξ2 k][(ipn)2−ξ2 k] = 2πiD βΩ � ipn>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='η=± � 1 (2ipn+2Ω)(2ipn+Ω) − 1 2ipn(2ipn+Ω) � = πiD βΩ2 � ipn>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='η=± � 4 2ipn+Ω − 1 ipn − 1 ipn+Ω � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (A5) � p 2 (ipn+Ω−ξk)2(ipn+ξk)2 = D β � ipn>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='η=± � dξk 2 (ipn+ηΩ−ξk)2(ipn+ξk)2 = − � ipn>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='η=± 8πiD/β (2ipn+ηΩ)3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (A6) � p � 2 (ipn+Ω−ξk)2[(ipn)2−ξ2 k] + 2 [(ipn+Ω)2−ξ2 k](ipn−ξk)2 � = D β � ipn>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='η=± � dξk � 2 (ipn+ηΩ−ξk)2[(ipn)2−ξ2 k] + 2 [(ipn+ηΩ)2−ξ2 k](ipn−ξk)2 � = 2πiD βΩ2 � ipn>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='η=± � 4 2ipn+ηΩ − 1 ipn − 1 ipn+ηΩ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (A7) the Landau parameters βp [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (21)], λp [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (22)], αp [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (23)] and γp [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (24)] are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 11 Appendix B: Derivation of retarded Green function from Eilenberger equation In this part, we present the derivation of the retarded Green function from the Eilenberger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' With the initial state of the retarded Green function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (41), the equation of the first order of retarded Green function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (42) is re-written as SR(E1)gR(0)(E1)δgR(1)(E)−δgR(1)(E)SR(E)gR(0)(E) = eA0 · vF ΠR(0) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (B1) From above equation, considering the normalization condition of gR(0)(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (46), one has [SR(E1)]2δgR(1)(E)−SR(E1)SR(E)gR(0)(E1)δgR(1)(E)gR(0)(E) = SR(E1)eA0 · vF [τ3−gR(0)(E1)τ3gR(0)(E)], (B2) SR(E)SR(E1)gR(0)(E1)δgR(1)(E)gR(0)(E)−[SR(E)]2δgR(1)(E) = SR(E)eA0 · vF [gR(0)(E1)τ3gR(0)(E)−τ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (B3) Then, the solution of δgR(1)(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (43) can be easily obtained by adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (B2) and (B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Moreover, substi- tuting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (41) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (43), the specific expression of δgR(1)(E) is given by δgR(1)(E) = (eA0 · vF )τ3 − [SR(E1)SR(E)]−1[E1Eτ3 + i∆0(E0 + E1)τ2 + ∆2 0τ3] SR(E1) + SR(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (B4) Similarly, the equation of the second order of retarded Green function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (44) is re-written as SR(E2)gR(0)(E2)δgR(2)(E)−δgR(2)(E)SR(E)gR(0)(E) = eA0·vF ΠR(1) 3 +iδ|∆|(2)[gR(0)(E2)τ2−τ2gR(0)(E)], (B5) and using the normalization condition of gR(0)(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (46), one easily gets the solution of δgR(1)(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (41) and (B4) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (45), the specific expression of the τ2 component of δgR(2)(E) reads δgR(2) 2 (E) = i(eA0·vF )2∆0 �(EE2+E1E+ E1E2+∆2 0)[SR(E1)+SR(E2)+SR(E)] SR(E1)SR(E2)SR(E) − 1 � 1 [SR(E)+SR(E2)] × 1 [SR(E1)+SR(E2)][SR(E1)+SR(E)] +iδ|∆|(2) 2∆2 0+2EE2+2SR(E)SR(E2) Γ(2)(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (B6) Further considering [SR(E1)+SR(E2)+SR(E)][SR(E)−SR(E1)][SR(E1)−SR(E2)][SR(E)−SR(E2)] SR(E1)SR(E2)SR(E) = E2 1 −E2 2 SR(E) + E2 2 −E2 SR(E1) + E2−E2 1 SR(E2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='(B7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='one has ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='δgR(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='(E) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='i(eA0·vF )2∆0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='(E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 − E2)(E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2 − E2)(E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 − E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='(EE2+E1E+ E1E2+∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='�E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 −E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E) + E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2 −E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E1) + E2−E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='− [SR(E)−SR(E2)][SR(E1)−SR(E2)][SR(E)−SR(E1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='+ iδ|∆|(2) 2∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='0+2EE2+2SR(E)SR(E2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='Γ(2)(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='i(eA0·vF )2∆0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='(E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 −E2)(E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2 −E2)(E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 −E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='(E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 −E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2)(E+E2)(E1+E)−[SR(E)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='+ (E+E1)(E1+E2)−[SR(E1)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='× (E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2 −E2)+(E2−E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1)(E+E2)(E2+E1)−[SR(E2)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='+SR(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='[SR(E1)]2−[SR(E2)]2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='[SR(E2)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='− [SR(E)]2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E1)+SR(E2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='[SR(E)]2−[SR(E1)]2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='+iδ|∆|(2) 4∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='0−(E−E2)2+[SR(E)+SR(E2)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='Γ(2)(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='= i(eA0·vF )2∆0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2Ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E2) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='+ iδ|∆|(2)�4∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='0 − (2Ω)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='Γ(2)(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='SR(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (B8) Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (49) is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 12 Appendix C: Derivation of distribution function from Eilenberger equation In this part, we present the derivation of the distribution function from the Eilenberger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' IV B, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (33), the Keldysh Green function satisfies the same equation as the retarded/advanced one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Therefore, the equation of the first order of the Keldysh Green function reads (E1τ3+i∆0τ2)δgK(1)(E)−δgK(1)(E)(Eτ3+i∆0τ2) = eA0 · vF ΠK(0) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C1) Substituting gK(0) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (54)] and δgK(1)(E) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (55)], the above equation becomes (E1τ3+i∆0τ2)[gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)]−[gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)](Eτ3+i∆0τ2)=h(0)(E) × [δgR(1)(E)(Eτ3+i∆0τ2)−(E1τ3+i∆0τ2)δgR(1)(E)]−h(0)(E1)[δgA(1)(E)(Eτ3+i∆0τ2)−(E1τ3+i∆0τ2)δgA(1)(E)] − eA0·vF [h(0)(E)τ3gR(0)(E)−gR(0)(E1)h(0)(E1)τ3−h(0)(E)τ3gA(0)(E)+gA(0)(E1)h(0)(E1)τ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C2) Then, facilitated with the equations of the first order of the retarded [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (42)] and advanced Green functions, one arrives at the equation of the first order of the distribution function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' By first multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (57) by (E1τ3 + i∆0τ2) from the left side and (Eτ3 + i∆0τ2) from the right side respectively and adding the obtained two equations afterwards,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' one has [(E2 1 +∆2 0)−(E2+∆2 0)][gR(0)(E1)δh(1)(E)−δh(1)(E)gA(0)(E)] = eA0·vF [h(0)(E1)−h(0)(E)] × � gR(0)(E1)[(E1τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)]−[(E1τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)]gA(0)(E) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C3) from which the solution of the first order of the distribution function reads δh(1)(E) = eA0·vF [h(0)(E1)−h(0)(E)][(E1τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)] (E2 1 +∆2 0)−(E2+∆2 0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C4) and then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (58) is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Similarly, the equation of the second order of the Keldysh Green function reads (E2τ3+i∆0τ2)δgK(2)(E)−δgK(2)(E)(Eτ3+i∆0τ2) = eA0·vF ΠK(1) 3 +iδ|∆|(2)[gK(0)(E2)τ2−τ2gK(0)(E)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C5) Substituting gK(0) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (54)] and δgK(1)(E) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (55)] as well as δgK(2)(E) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (56)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' the above equation becomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='(E2τ3+i∆0τ2)[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)]−[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)](Eτ3+i∆0τ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='= [δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E)](Eτ3+i∆0τ2)−(E2τ3+i∆0τ2)[δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='+ h(0)(E)[δgR(2)(E)(Eτ3+i∆0τ2)−(E2τ3+i∆0τ2)δgR(2)(E)]−[δgA(2)(E)(Eτ3+i∆0τ2)−(E2τ3+i∆0τ2)δgA(2)(E)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='×h(0)(E2)+eA0·vF [δgR(1)(E1)h(0)(E1)τ3−τ3δgR(1)(E)h(0)(E)+gR(0)(E2)δh(1)(E1)τ3−τ3gR(0)(E1)δh(1)(E)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='− eA0·vF [h(0)(E2)δgA(1)(E1)τ3−τ3h(0)(E1)δgA(1)(E)+δh(1)(E1)gA(0)(E1)τ3−τ3δh(1)(E)gA(0)(E)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='− iδ|∆|(2)[τ2gR(0)(E)h(0)(E)−gR(0)(E2)h(0)(E2)τ2−τ2gA(0)(E)h(0)(E)+gA(0)(E2)h(0)(E2)τ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C6) Facilitated with the equations of the second order of the retarded [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (44)] and advanced Green functions, one has (E2τ3+i∆0τ2)[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)]−[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)](Eτ3+i∆0τ2) = [δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E)](Eτ3+i∆0τ2)−(E2τ3+i∆0τ2)[δgR(1)(E1)δh(1)(E)−δh(1)(E1)δgA(1)(E)] + eA0·vF [h(0)(E1) − h(0)(E)]δgR(1)(E1)τ3+eA0·vF [gR(0)(E2)δh(1)(E1)τ3−τ3gR(0)(E1)δh(1)(E)] − eA0·vF [h(0)(E2)−h(0)(E1)]τ3δgA(1)(E)−eA0·vF [δh(1)(E1)gA(0)(E1)τ3−τ3δh(1)(E)gA(0)(E)] + iδ|∆|(2)[h(0)(E2)−h(0)(E)][gR(0)(E2)τ2−τ2gA(0)(E)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C7) Further using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (58) to replace eA0·vF [h(0)(E1) − h(0)(E)] with Ωδh1(E), the above equation is simplified as (E2τ3+i∆0τ2)[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)]−[gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)](Eτ3+i∆0τ2) = δh(1)(E)[δgR(1)(E1)(E1τ3+i∆0τ2)−(E2τ3+i∆0τ2)δgR(1)(E1)−eA0·vF τ3gR(0)(E1)] − δh(1)(E1)[δgA(1)(E1)(E1τ3+i∆0τ2)−(E2τ3+i∆0τ2)δgA(1)(E1)+eA0·vF gA(0)(E1)τ3] + eA0·vF [gR(0)(E2)δh(1)(E1)τ3+τ3δh(1)(E)gA(0)(E)]+iδ|∆|(2)[h(0)(E2)−h(0)(E)][gR(0)(E2)τ2−τ2gA(0)(E)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C8) 13 Consequently, substituting equations of the first order of the retarded [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (42)] and advanced Green functions to above equation, one arrives at the equation of the second order of the distribution function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' By first multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (57) by (E2τ3 + i∆0τ2) from the left side and (Eτ3 + i∆0τ2) from the right side respectively and adding the obtained two equations afterwards,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' one has [gR(0)(E2)δh(2)(E)−δh(2)(E)gA(0)(E)]= eA0·vF [δh(1)(E1)−δh(1)(E)] E2 2 −E2 � gR(0)(E2)[(E2τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)] − [(E2τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)]gA(0)(E) � +iδ|∆|(2) h(0)(E2)−h(0)(E) E2 2 −E2 � gR(0)(E2)[(E2τ3+i∆0τ2)τ2+τ2(Eτ3 + i∆0τ2)]−[(E2τ3+i∆0τ2)τ2+τ2(Eτ3+i∆0τ2)]gA(0)(E) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C9) from which the solution of the first order of the distribution function reads δh(2)(E) = eA0·vF [δh(1)(E1)−δh(1)(E)][(E2τ3+i∆0τ2)τ3+τ3(Eτ3+i∆0τ2)] E2 2 −E2 +iδ|∆|(2) [h(0)(E2)−h(0)(E)][(E2τ3+i∆0τ2)τ2+τ2(Eτ3+i∆0τ2)] E2 2 −E2 = eA0·vF δh(1)(E1)−δh(1)(E) E2 − E +iδ|∆|(2) [h(0)(E2)−h(0)(E)](2i∆0−2iΩτ1) E2 2 −E2 ≈ eA0·vF δh(1)(E1)−δh(1)(E) E2 − E +iδ|∆|(2)2i∆0 h(0)(E2)−h(0)(E) E2 2 −E2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (C10) and then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' (60) is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' ∗ Electronic address: yfgq@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' † Electronic address: mwwu@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Littlewood and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Varma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 47, 811 (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' B 26, 4883 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 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(1974) [Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' JETP 38, 1018 (1974)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 3 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Yuzbashyan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Dzero, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Lett 96, 230404 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 4 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Gurarie, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 103, 075301 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Moor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Volkov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Volkov, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Efetov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' B 90, 024511 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Pekker and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Varma, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 6, 269 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 7 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Tsuji, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Murakami, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Aoki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' B 94, 224519 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 8 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Yanagisawa, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 23, 459 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 9 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Fogler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Basov, and Andrew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Millis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Research 2, 023413 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 10 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Matsunaga and R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Hamada, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Makise, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Uzawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Terai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Wang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Shimano, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Sugioka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Makise, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Uzawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Terai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Aoki, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Shimano, Science 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Hamada, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Matsunaga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Schneeloch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Zhong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Gu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Aoki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Gallais, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Shimano, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 120, 117001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Bawatna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Christiani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Logvenov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Gallais, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Raffy, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Gallais, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Shimano, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' B 102, 054510 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 17 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 7, 11921 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 31 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Seibold, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Udina, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Castellani, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Benfatto, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Barone, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Castellani, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Benfatto, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' B 97, 094516 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 37 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} 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Eilenberger, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 214, 195 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 43 To justify the result from the Eilenberger equation, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 41 also performed the diagrammatic formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Neverthe- less, as referred to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' IV A in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 41, a special proce- dure that follows the exact step to derive the Eilenberger equation by first taking the commutation between the self-energy and τ3-Green function and applying the quasi- classical approximation afterwards, was applied to handle the calculation within the diagrammatic formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' Con- sequently, the applied diagrammatic formalism approach with this procedure becomes exactly same as the Eilen- berger equation and is not a separate nontrivial techinique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9E3T4oBgHgl3EQf_QsV/content/2301.04832v1.pdf'} +page_content=' 44 F.' metadata={'source': 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